Мiри на банахових многовидах з рiвномiрною структурою

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dc.contributor.authorМоравецька, Катерина Вiталiївна
dc.date.accessioned2019-01-25T13:47:37Z
dc.date.available2019-01-25T13:47:37Z
dc.date.issued2018
dc.description.abstractenThe thesis is devoted to differentiable measures on Banach manifolds with uniform structure. A construction of associated surface measures on embedded surfaces with finite codimension is proposed. A criterion for weak differentiability of measures is generalized to the case of differentiation along vector fields on Banach manifolds with uniform structure. The main part of the thesis consists of four sections. Section 1 is devoted to the review of works related to the topic of the dissertation research. A class of Banach manifolds with a special structure, namely Banach manifolds with uniform structure is considered, and some of their properties are indicated. Various existing approaches to the definition of differentiable measures are indicated and an overview of some known results on the theory of measure differentiability is presented. The Fomin and Skorokhod differentiability along directions and along vector fields on linear spaces and nonlinear manifolds is considered, as well as the connection between them is determined. An overview of approaches to the construction of surface measures and surface integration in infinite-dimensional Hilbert spaces and nonlinear manifolds is presented, several variants of the generalization of the Gauss-Ostrogradsky formula are presented. Section 2 presents the construction of surface measures in Banach manifolds with uniform structure. A method for constructing associate measures of the first and second type on an embedded surface of finite codimension is proposed. The concepts of an embedded surface of finite codimension, an associated surface form and strictly transversal to the surface set of vector fields are introduced. Examples are given and some elementary properties are noted. For a Borel measure on a Banach manifold with bounded structure a construction of the associated surface measure on embedded surface is proposed with the usage of strictly transversal to the surface set of mutually commuting vector fields. Under the action of the flow of a given set of vector fields, any subset of the surface can be transformed into a certain area on a manifold for which the initial measure is determined. Passing to the limit yields the value of the desired surface measure (the first type). Sufficient conditions for the existence of a first type surface measure are given. The concepts of a coherent and coherent in the broad sense triple pS, ⃗ Z, μq are introduced (the triple consists of a surface, a set of vector fields, and measure) for which a surface measure is defined. Some properties of the proposed construction are proved. The main result of the section is a consistency theorem, according to which in case of Banach manifold with uniform structure the first type surface measure depends only on associated differential form and does not depend on a specific strictly transversal set of vector fields used during construction. Due to this property, the concept of surface type of the second type, independent of a set of vector fields, is correctly introduced. Section 3 deals with the study of the transitive properties of associated surface measures and the study of the coherence of the proposed construction with classical results. The case of double embedding of surfaces is considered, that is, the situation when the Σ is an embedded into M surface of finite codimension and the surface S is embedded into Σ. In this case, S can also be regarded as a surface (with finite codimension) embedded in M. It is proved that the proposed construction of surface measures is transitive, that is, the direct construction of the surface measure on S when considering its embedding in M leads to the same result as the two-stage construction with the surface measure on Σ. Two examples of the usage of proposed construction of associated surface measures are considered. The surface measure associated with the Lebesgue measure is constructed on a parametrically defined surface in a finite-dimensional space Rn, which, in the case of normalization, coincides with the classical area. For the Riemann volume measure on a Riemann manifold with uniform structure first and second type associated surface measures are obtained for embedded Riemann submanifold. It is shown that the obtained measure coincides with the volume measure on the surface given by the induced tensor. Thus, the adequacy of the proposed approach to the construction of surface measures on finite-dimensional surfaces in infinite-dimensional spaces is substantiated. Section 4 is devoted to the study of the measure differentiability along vector fields. Plenty of results from the classical theory, which is based on measure shifts along constant direction, is generalized to the case of shifts along integral curves of vector fields. In particular, for the Radon measure, the criterion of differentiability through the integration by parts formula is obtained, where the derivative is transferred from function to measure. The main result of the last section is the criterion of weak measure differentiability along bounded vector field, which generalizes the known result of V.I. Bogachev, obtained for differentiability along direction on a linear space. In accordance with this criterion, the weak differentiability of the measure μ is equivalent to the Lipschity of all shift functions t ÞÑ μtpAq.uk
dc.description.abstractruДиссертация посвящена дифференцируемым мерам на банаховых многообразиях с равномерной структурой. Предложено метод построения ассоциированных мер на поверхностях конечной коразмерности, вложенных в банахово многообразие с равномерной структурой. Введено понятие ассоциированной дифференциальной формы поверхности и строго трансверсального к поверхности набора векторных полей. Доказано теорему “про согласованность”, в соответствии с которой поверхностная мера задается однозначно ассоциированной дифференциальной формой поверхности. Показано транзитивность предложенной конструкции. На примере меры Лебега в конечномерном пространстве Rn и меры объема на римановом многообразии с равномерной структурой обосновано ее адекватность. Получено обобщение ряда результатов классической теории дифференцируемых мер на линейных пространствах на случай банаховых многообразий с равномерной структурой. Доказано критерий слабой дифференцируемости меры вдоль ограниченного векторного поля, который обобщает известный результат В.И. Богачева.uk
dc.description.abstractukДисертацiя присвячена диференцiйовним мiрам на банахових многовидах з рiвномiрною структурою. Запропоновано метод побудови асоцiйованих мiр на поверхнях скiнченної корозмiрностi, вкладених у банахiв многовид з рiвномiрною структурою. Введено поняття асоцiйованої диференцiальної форми поверхнi та строго трансверсального до поверхнi набору векторних полiв. Доведено теорему “про узгодженiсть”, згiдно з якою поверхнева мiра задається однозначно асоцiйованою диференцiальною формою поверхнi. Показано транзитивнiсть запропонованої конструкцiї. На прикладi мiри Лебега в скiнченновимiрному просторi Rn та мiри об’єму на рiмановому многовидi з рiвномiрною структурою обґрунтовано її адекватнiсть. Отримано узагальнення низки результатiв з теорiї диференцiйовних мiр на лiнiйних просторах на випадок банахових многовидiв з рiвномiрною структурою. Доведено критерiй слабкої диференцiйовностi мiри уздовж обмеженого векторного поля, що узагальнює вiдомий результат В.I. Богачова.uk
dc.format.page22 с.uk
dc.identifier.citationМоравецька, К. В. Мiри на банахових многовидах з рiвномiрною структурою : автореф. дис. … канд. техн. наук : 01.01.01 – математичний аналiз / Моравецька Катерина Вiталiївна. – Київ, 2018. – 22 с.uk
dc.identifier.urihttps://ela.kpi.ua/handle/123456789/26063
dc.language.isoukuk
dc.publisherКПІ ім. Ігоря Сікорськогоuk
dc.publisher.placeКиївuk
dc.subjectбанахiв многовид з рiвномiрною структуроюuk
dc.subjectборелiвська мiраuk
dc.subjectдиференцiйовнiсть мiр уздовж векторних полiв за Фомiним та за Скороходомuk
dc.subjectповерхнева мiраuk
dc.subjectвкладена поверхняuk
dc.subjectасоцiйована форма поверхнiuk
dc.subjectтрансверсальний набiр векторних полiвuk
dc.subjectBanach manifold with uniform structureuk
dc.subjectBorel measureuk
dc.subjectFomin and Skorokhod measure differentiability along vector fieldsuk
dc.subjectsurface measureuk
dc.subjectembedded surfaceuk
dc.subjectassociated surface formuk
dc.subjecttransversal set of vector fieldsuk
dc.subjectбанахово многообразие с равномерной структуройuk
dc.subjectборелевская мераuk
dc.subjectдифференцируемость мер вдоль векторных полей по Фомину и по Скороходуuk
dc.subjectповерхностная мераuk
dc.subjectвложенная поверхностьuk
dc.subjectассоциированная форма поверхностиuk
dc.subjectтрансверсальный набор векторных полейuk
dc.subject.udc517.98+515.164](043.3)uk
dc.titleМiри на банахових многовидах з рiвномiрною структуроюuk
dc.typeThesisuk

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