COEFFICIENT INVERSE PROBLEMS FOR THE PARABOLIC EQUATION WITH GENERAL WEAK DEGENERATION

It is investigated the inverse problems for the degenerate parabolic equation. The mi- nor coeffcient of this equation is a linear polynomial with respect to space variable with two unknown time-dependent functions. The degeneration of the equation is caused by the monotone increasing function at the time derivative. It is established conditions of existence and uniqueness of the classical solutions to the named problems in the case of weak degeneration.

PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

ON APPROXIMATION OF ALMOST-PERIODIC SOLUTIONS FOR A NON-LINEAR COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS BY QUASI-PERIODIC SOLUTIONS FOR SOME LINEAR SYSTEM

It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.

ON SEPARATE ORDER CONTINUITY OF ORTHOGONALLY ADDITIVE OPERATORS

Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).

THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS

In this paper we investigate the differential-operator equation $$ \partial u (t, x) / \partial t + \varphi (i \partial / \partial x) u (t, x) = 0, \quad (t, x) \in (0, + \infty) \times \mathbb {R} \equiv \Omega, $$ where the function $ \varphi \in C ^ {\infty} (\mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i \partial / \partial x $, in $ L_2 (\mathbb {R}) $ it is established that the operator $ \varphi (i \partial / \partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ \partial u / \partial t + \sqrt {I- \Delta} u = 0 $, $ \Delta = D_x ^ 2 $, with the fractionation differentiation operator $ \sqrt { I- \Delta} = \varphi (i \partial / \partial x) $, where $ \varphi (\sigma) = (1+ \sigma ^ 2) ^ {1/2} $, $ \sigma \in \mathbb {R} $ is attributed to the considered equation. Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, \cdot) $ for $ t \to + \infty $ (solution stabilization) in spaces of type $ S '$.

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

SYMOIN STOILOV (1887-1961): DETAILS OF SCIENTIfiC CAREER

The present article covers topics of life, scientific, pedagogical and social activities of the famous Romanian mathematician Simoin Stoilov (1887-1961), professor of Chernivtsi and Bucharest universities. Stoilov was working at Chernivtsi University during 1923-1939 (at this interwar period Chernivtsi region was a part of royal Romania. The article is aimed on the occasion of honoring professors’ memory and his managerial abilities in the selection of scientific and pedagogical staff to ensure the educational process and research in Chernivtsi University in the interwar period. In addition, it is noted that Simoin Stoilov has made a significant contribution to the development of mathematical science, in particular he is the founder of the Romanian school of complex analysis and the theory of topological analysis of analytic functions; the main directions of his research are: partial differential equation; set theory; general theory of real functions and topology; topological theory of analytic functions; issues of philosophy and foundation of mathematics, scientific research methods, Lenin’s theory of cognition. The article focuses on the active socio-political and state activities of Simoin Stoilov in terms of restoring scientific and cultural ties after the Second World War.

WIMAN’S TYPE INEQUALITY FOR SOME DOUBLE POWER SERIES

By $\mathcal{A}^2$ denote the class of analytic functions of the formBy $\mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $\mathbb{T}=\{z=(z_1,z_2)\in\mathbb C^2\colon|z_1|<1,\ |z_2|<+\infty\}=\mathbb{D}\times\mathbb{C}$ and$\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $f\in\mathcal{A}^2$. Let a function $h\colon \mathbb R^2_+\to \mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)\geq 10$ for all $r\in T:=(0,1)\times (0,+\infty)$and $\iint_{\Delta_\varepsilon}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+\infty$ for some $\varepsilon\in(0,1)$, where $\Delta_{\varepsilon}=\{(t_1, t_2)\in T\colon t_1>\varepsilon,\ t_2> \varepsilon\}$.We say that $E\subset T$ is a set of asymptotically finite $h$-measure on\ ${T}$if $\nu_{h}(E){:=}\iint\limits_{E\cap\Delta_{\varepsilon}}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+\infty$ for some $\varepsilon>0$. For $r=(r_1,r_2)\in T$ and a function $f\in\mathcal{A}^2$ denote\begin{gather*}M_f(r)=\max \{|f(z)|\colon |z_1|\leq r_1,|z_2|\leq r_2\},\\mu_f(r)=\max\{|a_{nm}|r_1^{n} r_2^{m}\colon(n,m)\in{\mathbb{Z}}_+^2\}.\end{gather*}We prove the following theorem:{\sl Let $f\in\mathcal{A}^2$. For every $\delta>0$ there exists a set $E=E(\delta,f)$ of asymptotically finite $h$-measure on\ ${T}$ such that for all $r\in (T\cap\Delta_{\varepsilon})\backslash E$ we have \begin{equation*} M_f(r)\leq\frac{h^{3/2}(r)\mu_f(r)}{(1-r_1)^{1+\delta}}\ln^{1+\delta} \Bigl(\frac{h(r)\mu_f(r)}{1-r_1}\Bigl)\cdot\ln^{1/2+\delta}\frac{er_2}{\varepsilon}. \end{equation*}}

A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

HYBRID MODEL OF SELF-ORGANIZING MAP AND ADAPTIVE NEURO FUZZY INFERENCE SYSTEM IN STOCK INDEXES FORECASTING

The paper investigates methods of artificial intelligence in the prognostication and analysis of financial data time series. It is uncovered that scholars and practitioners face some difficulties in modelling complex system such as the stock market because it is nonlinear, chaotic, multi- dimensional, and spatial in nature, making forecasting a complex process. Models estimating nonstationary financial time series may include noise and errors. The relationship between the input and output parameters of the models is essentially non-linear, where stock prices include higher-level variables, which complicates stock market modeling and forecasting. It is also revealed that financial time series are multidimensional and they are influenced by many factors, such as economics, politics, environment and so on. Analysis and evaluation of multi- dimensional systems and their forecasting should be carried out by machine learning models. The problem of forecasting the stock market and obtaining quality forecasts is an urgent task, and the methods and models of machine learning should be the main mathematical tools in solving the above problems. First, we proposed to use self-organizing map, which is used to visualize multidimensional data by configuring neurons to quantize or cluster the input space in the topological structure. These characteristics of this algorithm make it attractive in solving many problems, including clustering, especially for forecasting stock prices. In addition, the methods discussed, encourage us to apply this cluster approach to present a different data structure for forecasting. Thus, models of adaptive neuro-fuzzy inference system combine the characteristics of both neural networks and fuzzy logic. Given the fact that the rule of hybrid learning and the theory of logic is a clear advantage of adaptive neuro-fuzzy inference system, which has computational advantages over other methods of parameter identification, we propose a new hybrid algorithm for integrating self-organizing map with adaptive fuzzy inference system to forecast stock index prices. This algorithm is well suited for estimating the relationship between historical prices in stock markets. The proposed hybrid method demonstrated reduced errors and higher overall accuracy.