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.
{"title":"COEFFICIENT INVERSE PROBLEMS FOR THE PARABOLIC EQUATION WITH GENERAL WEAK DEGENERATION","authors":"N. Huzyk, O. Brodyak","doi":"10.31861/bmj2021.01.08","DOIUrl":"https://doi.org/10.31861/bmj2021.01.08","url":null,"abstract":"It is investigated the inverse problems for the degenerate parabolic equation. The mi-\u0000nor coeffcient of this equation is a linear polynomial with respect to space variable with\u0000two 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.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116187653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-infty,r_1(s)),,(r_1(s),r_2(s)),ldots,,(r_{n}(s),infty)$ separated by points $r_i(s),(i=1,ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s),(i=1,ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.
{"title":"THE NONLOCAL CONJUGATION PROBLEM FOR A LINEAR SECOND ORDER PARABOLIC EQUATION OF KOLMOGOROV'S TYPE WITH DISCONTINUOUS COEFFICIENTS","authors":"R. Shevchuk, Ivan Savka","doi":"10.31861/bmj2022.02.20","DOIUrl":"https://doi.org/10.31861/bmj2022.02.20","url":null,"abstract":"In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-infty,r_1(s)),,(r_1(s),r_2(s)),ldots,,(r_{n}(s),infty)$ separated by points $r_i(s),(i=1,ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary.\u0000\u0000The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s),(i=1,ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129988618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Pratsiovytyi, S. Ratushniak, Yu. Symonenko, D. Shpytuk
We consider distribution of random variable $xi=tau+eta$, where $tau$ and $eta$ independent random variables, moreover $tau$ has classic Cantor type distribution and $eta$ is a random variable with independent identically distributed digits of the nine-digit representation. With additional conditions for the distributions of the digits $eta$, sufficient conditions for the singularity of the Cantor type of the distribution $xi$ are specified. To substantiate the statements, a topological-metric analysis of the representation of numbers $xin [0;2]$ in the numerical system with base $9$ and a seventeen-symbol alphabet (a set of numbers) is carried out. The geometry (positional and metric) of this representation is described by the properties of the corresponding cylindrical sets.
{"title":"CONVOLUTION OF TWO SINGULAR DISTRIBUTIONS: CLASSIC CANTOR TYPE AND RANDOM VARIABLE WITH INDEPENDENT NINE DIGITS","authors":"M. Pratsiovytyi, S. Ratushniak, Yu. Symonenko, D. Shpytuk","doi":"10.31861/bmj2022.02.16","DOIUrl":"https://doi.org/10.31861/bmj2022.02.16","url":null,"abstract":"We consider distribution of random variable $xi=tau+eta$, where $tau$ and $eta$ independent random variables, moreover $tau$ has classic Cantor type distribution and $eta$ is a random variable with independent identically distributed digits of the nine-digit representation. With additional conditions for the distributions of the digits $eta$, sufficient conditions for the singularity of the Cantor type of the distribution $xi$ are specified. To substantiate the statements, a topological-metric analysis of the representation of numbers $xin [0;2]$ in the numerical system with base $9$ and a seventeen-symbol alphabet (a set of numbers) is carried out. The geometry (positional and metric) of this representation is described by the properties of the corresponding cylindrical sets.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133890645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could infinitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.
{"title":"REPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE","authors":"H. Pasichnyk, S. Ivasyshen","doi":"10.31861/bmj2021.01.16","DOIUrl":"https://doi.org/10.31861/bmj2021.01.16","url":null,"abstract":"The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could infinitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128852334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"SYMOIN STOILOV (1887-1961): DETAILS OF SCIENTIfiC CAREER","authors":"O. Martynyuk, I. Zhytaryuk","doi":"10.31861/bmj2021.01.12","DOIUrl":"https://doi.org/10.31861/bmj2021.01.12","url":null,"abstract":"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.\u0000The 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.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129908305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.
{"title":"FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS","authors":"M. Zarichnyi","doi":"10.31861/bmj2021.01.14","DOIUrl":"https://doi.org/10.31861/bmj2021.01.14","url":null,"abstract":"Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction.\u0000The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets.\u0000Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122367017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"HYBRID MODEL OF SELF-ORGANIZING MAP AND ADAPTIVE NEURO FUZZY INFERENCE SYSTEM IN STOCK INDEXES FORECASTING","authors":"M. Kushnir, K. Tokarieva","doi":"10.31861/bmj2021.02.05","DOIUrl":"https://doi.org/10.31861/bmj2021.02.05","url":null,"abstract":"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.\u0000The 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.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128787107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation $$ frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n), $$ with fixed $c$, $c in mathbb{R}$, $a_k in mathbb{R}$, $1 leq k leq n$, stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains.
本文的目的是研究一类时滞变量线性自治微分方程解的稳定性。稳定性的研究可以简化为特征方程的根定位问题。对于具有多个时滞的线性微分方程，得到了特征方程的所有根均具有负实部(因而零解渐近稳定)的充分必要条件。对于固定$c$, $c in mathbb{R}$, $a_k in mathbb{R}$, $1 leq k leq n$的标量延迟微分方程$$frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n),$$，得到了参数平面上的稳定域。研究了一类多时滞线性自治微分方程的有界性条件，构造了一个稳定定域。利用d划分法、参数原理和数值方法构造了稳定域。
{"title":"CONSTRUCTION OF STABILITY DOMAINS FOR LINEAR DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS","authors":"I. Klevchuk, M. Hrytchuk","doi":"10.31861/bmj2022.01.06","DOIUrl":"https://doi.org/10.31861/bmj2022.01.06","url":null,"abstract":"The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation\u0000$$\u0000frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n),\u0000$$\u0000with fixed $c$, $c in mathbb{R}$, $a_k in mathbb{R}$, $1 leq k leq n$,\u0000stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125175639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concepts of the pseudostarlikeness of order $alphain [0,,1)$ and type $betain (0,,1]$ and the pseudoconvexity of the order $alpha$ and type $beta$ are introduced for Dirichlet series of the form $F(s)=e^{-sh}+sum_{j=1}^{n}a_jexp{-sh_j}+sum_{k=1}^{infty}f_kexp{slambda_k}$, where $h>h_n>dots>h_1ge 1$ and $(lambda_k)$ is an increasing to $+infty$ sequence of positive numbers. Criteria for pseudostarlikeness and pseudoconvexity in terms of coefficients are proved. The obtained results are applied to the study of meromorphic starlikeness and convexity of the Laurent series break $f(s)=1/z^p+sum_{j=1}^{p-1}a_j/z^j+sum_{k=1}^{infty}f_kz^k$. Conditions, under which the differential equation $w''+gamma w'+(delta e^{2sh}+tau)w=0$ has a pseudostarlike or pseudoconvex solution of the order $alpha$ and the type $beta=1$ are investigated.
{"title":"ON PSEUDOSTARLIKE AND PSEUDOCONVEX DIRICHLET SERIES","authors":"M. Sheremeta","doi":"10.31861/bmj2021.01.07","DOIUrl":"https://doi.org/10.31861/bmj2021.01.07","url":null,"abstract":"The concepts of the pseudostarlikeness of order $alphain [0,,1)$ and type $betain (0,,1]$ and the pseudoconvexity of the order $alpha$ and type $beta$ are introduced for Dirichlet series of the form $F(s)=e^{-sh}+sum_{j=1}^{n}a_jexp{-sh_j}+sum_{k=1}^{infty}f_kexp{slambda_k}$,\u0000where $h>h_n>dots>h_1ge 1$ and $(lambda_k)$ is an increasing to $+infty$ sequence of positive numbers. Criteria for pseudostarlikeness and pseudoconvexity in terms of coefficients are proved. The obtained results are applied to the study of meromorphic starlikeness and convexity of the Laurent series break $f(s)=1/z^p+sum_{j=1}^{p-1}a_j/z^j+sum_{k=1}^{infty}f_kz^k$.\u0000Conditions, under which the differential equation $w''+gamma w'+(delta e^{2sh}+tau)w=0$ has a pseudostarlike or pseudoconvex solution of the order $alpha$ and the type $beta=1$ are investigated.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113942864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of the Cauchy problem for uniformly parabolic equations of the second order with limited coefficients is sufficiently fully investigated, for example, in the works of S.D. Eidelman and S.D. Ivasyshen, in contrast to such equations with unlimited coefficients. One of the areas of research of Professor S.D. Ivasyshen and students of his scientific school are finding fundamental solutions and investigating the correctness of the Cauchy problem for classes of degenerate equations, which are generalizations of the classical Kolmogorov equation of diffusion with inertia and contain for the main variables differential expressions, parabolic according to I.G. Petrovskyi and according to S.D. Eidelman (S.D. Ivasyshen, L.M. Androsova, I.P. Medynskyi, O.G. Wozniak, V.S. Dron, V.V. Layuk, G.S. Pasichnyk and others). Parabolic Petrovskii equations with the Bessel operator were also studied (S.D. Ivasyshen, V.P. Lavrenchuk, T.M. Balabushenko, L.M. Melnychuk). The article considers a parabolic equation of the second order with increasing coefficients and Bessel operators. In this equation, the some of coefficients for the lower derivatives of one group of spatial variables $xin mathbb{R}^n $ are components of these variables, therefore, grow to infinity. In addition, the equation contains Bessel operators of different orders in another group of spatial variables $yin mathbb{R}^m_+ $, due to which the coefficients in the first derivatives of these variables are unbounded around the point y=0. The paper defines a modified Fourier-Bessel transform that takes into account different orders of Bessel operators on different variables. With the help of this transformation and the method of characteristics, the solution of the Cauchy problem of the specified equation is found in the form of the Poisson integral, and its kernel, which is the fundamental solution of the Cauchy problem, is written out in an explicit form. Some properties of the found fundamental solution, in particular, estimates of its derivatives, have been established. They will be used to establish the correctness of the Cauchy problem.
{"title":"FUNDAMENTAL SOLUTION OF THE CAUCHY PROBLEM FOR PARABOLIC EQUATION OF THE SECOND ORDER WITH INCREASING COEFFICIENTS AND WITH BESSEL OPERATORS OF DIFFERENT ORDERS","authors":"L. Melnychuk","doi":"10.31861/bmj2022.02.13","DOIUrl":"https://doi.org/10.31861/bmj2022.02.13","url":null,"abstract":"The theory of the Cauchy problem for uniformly parabolic equations of the second order with limited coefficients is sufficiently fully investigated, for example, in the works of S.D. Eidelman and S.D. Ivasyshen, in contrast to such equations with unlimited coefficients. One of the areas of research of Professor S.D. Ivasyshen and students of his scientific school are finding fundamental solutions and investigating the correctness of the Cauchy problem for classes of degenerate equations, which are generalizations of the classical Kolmogorov equation of diffusion with inertia and contain for the main variables differential expressions, parabolic according to I.G. Petrovskyi and according to S.D. Eidelman (S.D. Ivasyshen, L.M. Androsova, I.P. Medynskyi, O.G. Wozniak, V.S. Dron, V.V. Layuk, G.S. Pasichnyk and others). Parabolic Petrovskii equations with the Bessel operator were also studied (S.D. Ivasyshen, V.P. Lavrenchuk, T.M. Balabushenko, L.M. Melnychuk).\u0000\u0000The article considers a parabolic equation of the second order with increasing coefficients and Bessel operators. In this equation, the some of coefficients for the lower derivatives of one group of spatial variables $xin mathbb{R}^n $ are components of these variables, therefore, grow to infinity. In addition, the equation contains Bessel operators of different orders in another group of spatial variables $yin mathbb{R}^m_+ $, due to which the coefficients in the first derivatives of these variables are unbounded around the point y=0.\u0000\u0000The paper defines a modified Fourier-Bessel transform that takes into account different orders of Bessel operators on different variables. With the help of this transformation and the method of characteristics, the solution of the Cauchy problem of the specified equation is found in the form of the Poisson integral, and its kernel, which is the fundamental solution of the Cauchy problem, is written out in an explicit form. Some properties of the found fundamental solution, in particular, estimates of its derivatives, have been established. They will be used to establish the correctness of the Cauchy problem.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131106157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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