Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 364 papers for “total boundedness”

Lebesgue Number and Total Boundedness

Ajit Kumar Gupta, Saikat Mukherjee · arXiv (Cornell University) · 2022

A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric spaces which is a generalization of connectedness and Menger convexity is introduced. It is observed that Atsujiness and compactness are equivalent for a metric space with this introduced property as well as for a chainable metric space.

Cited by 0Open access

Total boundedness in metrizable spaces

Gerald, Beer, Camillo, Costantini, Sandro, Levi · Institutional Research Information System University of Turin (University of Turin) · 2011

Cited by 0Open access

Lebesgue number and total boundedness

Ajit Kumar Gupta · Proceedings of the International Geometry Center · 2024

A generalization of the Lebesgue number lemma is obtained. For a metric space X in the class of strongly metrizable spaces, sufficient conditions for each open cover of X with a Lebesgue number has a finite subcover are obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space X has a Lebesgue number, then X is totally bounded. A property for metric spaces which is a generalization of connectedness and Menger convexity is introduced. It is observed that Atsujiness and compactness are equivalent for a metric space with this introduced property as well as for a chainable metric space.

Cited by 1Open access

Total boundedness in probabilistic normed spaces

Lafuerza Guillén, Bernardo, Saadati, Reza, Zhang, Gaoxun · Repositorio de Patentes de la Universidad de Almería (Universidad de Almería) · 2014

In this paper, we study total boundedness in probabilistic normed space and we give criterion for total boundedness and D-boundedness in these spaces. Also we show that in general a totally bounded set is not D-bounded.

Cited by 0Open access

Boundedness and Unboundedness in Total Variation Regularization

Kristian Bredies, José A. Iglesias, Gwenaël Mercier · Applied Mathematics & Optimization · 2023

Abstract We consider whether minimizers for total variation regularization of linear inverse problems belong to $$L^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math> even if the measured data does not. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for sufficiently small noise under a source condition and adequate a priori parameter choices. To show that such a result cannot be expected for every fidelity term and dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving the equivalence of weighted one-dimensional denoising with a generalized taut string problem. Finally, we discuss the possibility of extending such results to related higher-order regularization functionals, obtaining a positive answer for the infimal convolution of first and second order total variation.

Cited by 3Open access

Completeness and total boundedness of the Hausdorff metric

Jeff Henrikson · 1999

Abstract. This paper defines and discusses the Hausdorff metric on the space of nonempty, closed, and bounded subsets of a given metric space. We consider two important topological properties, completeness and total boundedness. We prove that each of these properties is posessed by a Hausdorff metric space if the property is possesed by the underlying metric space. Finally, we explore applications of the Hausdorff metric, including fractal geometry. 1. Introduction. The Hausdorff

Cited by 0

Boundedness of Some Commutators in Total Fofana Spaces

Pokou Nagacy · European Journal of Mathematical Analysis · 2024

In this paper, we find necessary and sufficient conditions for the boundedness of the commutator of the Hardy-Littlewood maximal operator in total Fofana spaces. We also give in these spaces the boundedness of some sublinear operators and their commutators.

Cited by 0Open access

The anti‐Specker property, positivity, and total boundedness

Douglas Bridges, Hannes Diener · Mathematical logic quarterly · 2010

Abstract Working within Bishop‐style constructive mathematics, we examine some of the consequences of the anti‐Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Cited by 9

CONTROL FUNCTIONS AND TOTAL BOUNDEDNESS IN THE SPACE L0

Caponetti, Diana, Lewicki, Grzegorz, Trombetta, Giulio · Jagiellonian University Repository (Jagiellonian University) · 2014

Abstract. In this paper we introduce a parameter ω ∞ in the space of functions L0. The parameter measures the lack of equimeasurability using a sequence of functions which controls the oscillations of every function of a given subset of L0. We estimate the Hausdorff measure of noncompactness in terms of ω ∞ and the parameter σ (see [3]) and characterize the totally bounded subsets of L0. A criterion of compactness given in [5] for subsets of the space BC(Ω, R) is extended to the case of the space

Cited by 0Open access

Total Poisson Boundedness and Total Oscillability of Solutions of Systems of Differential Equations

K. S. Lapin · Владикавказский математический журнал · 2022

В работах автора было начато изучение особого вида ограниченности решений систем дифференциальных уравнений, а именно, их ограниченности по Пуассону. Понятие ограниченности по Пуассону решения обобщает классическое понятие ограниченности решения и состоит в том, что в фазовом пространстве найдутся такой шар и на временн$\mbox{о}^{\!\!'}$й полуоси такая счетная система непересекающихся интервалов, последовательность правых концов которых стремится к плюс бесконечности, что решение при всех значениях времени из этих интервалов содержится в данном шаре. Далее в работах автора на основе методов функций Ляпунова, вектор-функций Ляпунова и высших производных функций Ляпунова были получены достаточные условия различных видов ограниченности по Пуассону всех решений. В частности, были получены достаточные условия тотальной ограниченности (ограниченности при малых возмущениях) по Пуассону, частичной тотальной ограниченности по Пуассону, а также частичной тотальной ограниченности по Пуассону решений с частично контролируемыми начальными условиями. В настоящей работе автором была получена асимптотическая или, как еще говорят, финальная характеризация понятия ограниченности по Пуассону решения, которая позволила установить связь между понятием ограниченного по Пуассону решения и понятием осциллирующего решения. Далее в работе введены понятия тотальной осциллируемости решений, частичной тотальной осциллируемости решений и частичной тотальной осциллируемости решений с частично контролируемыми начальными условиями. На основе указанной выше финальной характеризации понятия ограниченности по Пуассону решения, а также на основе метода вектор-функций Ляпунова с системами сравнений в работе получены достаточные условия тотальной осциллируемости, частичной тотальной осциллируемости, а также частичной тотальной осциллируемости решений с частично контролируемыми начальными условиями. Как следствия получены достаточные условия указанных выше видов тотальной осциллируемости решений в терминах функций Ляпунова.

Cited by 0Open access

Poisson Total Boundedness and Total Oscillability of Solutions to Systems of Differential Equations

K. S. Lapin · Siberian Mathematical Journal · 2023

This is a continuation of the author’s study of some special form of boundedness of solutions to systems of differential equations, namely, their Poisson boundedness. The latter concept generalizes the classical boundedness of a solution and means that there are a ball in the phase space and countably many disjoint intervals on the time half-axis such that the sequence of right endpoints of the intervals tends to the plus infinity and the values of a solution on these intervals lie in the ball. Sufficient conditions for various types of Poisson boundedness of solutions were obtained by using Lyapunov functions, Lyapunov vector-functions, and higher-order derivatives of Lyapunov functions. In particular, there were established some sufficient conditions for the Poisson total boundedness (the Poisson boundedness under small perturbations), the Poisson partial total boundedness, and also the Poisson partial total boundedness of solutions with partially controlled initial conditions. In the present article, we obtain an asymptotic or, in other words, a final characterization of the Poisson boundedness of a solution, which made it possible to establish a connection between the latter concept and an oscillating solution. Further, we introduce the total oscillability of solutions, partial total oscillability of solutions, and partial total oscillability of solutions with partially controlled initial conditions. Using the final characterization of the Poisson boundedness of a solution, and Lyapunov vector-functions with comparison systems, we obtain some sufficient conditions for total oscillability, partial total oscillability, and partial total oscillability of solutions with partially controlled initial conditions. As a consequence, we obtain sufficient conditions for the above types of total oscillability in terms of Lyapunov functions.

Cited by 0

Higher Lyapunov Functions Derivatives and Total Poisson Boundedness of Solutions

K. S. Lapin · Russian Mathematics · 2019

Using the method of higher Lyapunov functions derivatives we obtain sufficient conditions for total Poisson boundedness of solutions, partial total Poisson boundedness of solutions, and partial total Poisson boundedness of solutions with partially controlled initial conditions.

Cited by 2