Asymmetric variations of multifunctions with application to functional inclusions
Functional Analysis. arXiv [math.FA]. Cornell University, NY, USA, 2019. No. 1901.09722.
Vyacheslav V. Chistyakov.
Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\subset F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$.
Research target: Mathematics
Priority areas: mathematics
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Keywords: multifunctionbounded variationcompact setHausdorff metricPompeiu excessset-valued selectorselection principle
Publication based on the results of: Модулярные пространства, являющиеся модулярно полными (Modular spaces that are modular complete)(2017)
Обобщение теоремы Хелли для функций со значениями в равномерном пространстве
Третьяченко Ю. В. Известия высших учебных заведений. Математика. 2010. № 5. С. 41-54.
In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.
Совместный модуль вариации функций и условно регулярный принцип выбора
С.А.Чистякова, В.В.Чистяков В кн.: Труды Математического центра имени Н.И.Лобачевского. Т. 54: Теория функций, ее приложения и смежные вопросы. Каз.: Издательство Казанского математического общества и Академии наук РТ, 2017. С. 399-402.
Given a closed interval $I=[a,b]$ and a metric space $(M,d)$, we introduce a
nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^I$ (the set of all
functions from $I$ into $M$), called the {\it joint modulus of variation}. We show that
if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^I$ are such that
$\{f_j\}$ is pointwise relatively compact on $I$, $\{g_j\}$ is pointwise convergent on $I$,
and $\limsup_{j\to\infty}\nu_n(f_j,g_j)=o(n)$ as $n\to\infty$, then $\{f_j\}$ admits
a pointwise convergent subsequence whose limit on $I$ is a conditionally regulated function.
The approximate variation to pointwise selection principles
Vyacheslav V. Chistyakov. Functional Analysis. arXiv [math.FA]. Cornell University Library, NY, USA, 2019. No. arXiv: 1910.08490.
Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was introduced by Fra{\v n}kov{\'a} [Math. Bohem. 116 (1991), 20--59] for intervals $T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the general case by Chistyakov and Chistyakova [Studia Math. 238 (2017), 37--57]. We prove directly the following basic pointwise selection principle: If a sequence of functions $\{f_j\}_{j=1}^\infty$ from $M^T$ is such that the closure in $M$ of the set $\{f_j(t):j\in\mathbb{N}\}$ is compact for all $t\in T$ and $\limsup_{j\to\infty}V_\varepsilon(f_j)$ is finite for all $\varepsilon>0$, then it contains a subsequence, which converges pointwise on $T$ to a bounded regulated function $f\in M^T$. We establish several variants of this result for sequences of regulated and nonregulated functions, for functions with values in reflexive separable Banach spaces, for the almost everywhere convergence and weak pointwise convergence of extracted subsequences, and comment on the necessity of assumptions in the selection principles. The sharpness of all assertions is illustrated by examples.
Dynamics of Information Systems: Mathematical Foundations
Iss. 20. NY: Springer, 2012.
This proceedings publication is a compilation of selected contributions from the "Third International Conference on the Dynamics of Information Systems" which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Added: Dec 19, 2012
The joint modulus of variation of metric space valued functions and pointwise selection principles
Vyacheslav V. Chistyakov, Svetlana A. Chistyakova. arxiv.org. math. Cornell University, 2016. No. 1601.07298.
Given a subset T of real numbers and a metric space M, we introduce a nondecreasing sequence {v_n} of pseudometrics on the set M^T of all functions from T into M, called the joint modulus of variation. We prove that if two sequences of functions {f_j} and {g_j} from M^T are such that {f_j} is pointwise precompact, {g_j} is pointwise convergent, and the limit superior of v_n(f_j,g_j) as j → ∞ is o(n) as n → ∞, then {f_j} admits a pointwise convergent subsequence whose limit is a conditionally regulated function. We illustrate the sharpness of this result by examples (in particular, the assumption on the limsup is necessary for uniformly convergent sequences {f_j} and {g_j}, and 'almost necessary' when they converge pointwise) and show that most of the known Helly-type pointwise selection theorems are its particular cases.
Maps of several variables of finite total variation. I. Mixed differences and the total variation
Chistyakov V., Tretyachenko Y. Journal of Mathematical Analysis and Applications. 2010. Vol. 370. No. 2. P. 672-686.
Given two points a=(a1,…,an) and b=(b1,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61–71] (n∈N) for real-valued functions of n variables.
A Generalization of the Helly Theorem for Functions with Values in a Uniform Space
Tretyachenko Y. Russian Mathematics. 2010. Vol. 54. No. 5. P. 35-46.
In this paper we consider sequences of functions that are defined on a subset of the real line and take on values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.
Model for organizing cargo transportation with an initial station of departure and a final station of cargo distribution
Khachatryan N., Akopov A. S. Business Informatics. 2017. No. 1(39). P. 25-35.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the "correct" extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Added: Apr 18, 2017
Nullstellensatz over quasi-fields
Trushin D. Russian Mathematical Surveys. 2010. Vol. 65. No. 1. P. 186-187.
Деловой климат в оптовой торговле во II квартале 2014 года и ожидания на III квартал
Лола И. С., Остапкович Г. В. Современная торговля. 2014. № 10.
Прикладные аспекты статистики и эконометрики: труды 8-ой Всероссийской научной конференции молодых ученых, аспирантов и студентов
Вып. 8. МЭСИ, 2011.
Laminations from the Main Cubioid
Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.
According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.
Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles
Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.
Bounded limit cycles of polynomial foliations of ℂP²
Goncharuk N. B., Kudryashov Y. arxiv.org. math. Cornell University, 2015. No. 1504.03313.
In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.
Метод параметрикса для диффузий и цепей Маркова
Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.
Added: Dec 5, 2012
Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?
Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem
Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1.
We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.
Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау
Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.
Hypercommutative operad as a homotopy quotient of BV
Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749.
We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.
Cross-sections, quotients, and representation rings of semisimple algebraic groups
V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3. P. 827-856.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.
Математическое моделирование социальных процессов
Edited by: А. Михайлов Вып. 14. М.: Социологический факультет МГУ, 2012.