Convergence In Topological Spaces Research Articles

Statistically multiplicative convergence on locally solid Riesz algebras

In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to algebra multiplication and solid topology. We study this concept and the notion of $st_m$-bounded sequences, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.

On n-Convergence of Filters

In this work, we have present a new form of sets named - open sets. We have study and described some of the properties and relationships between these sets. We have obtained some results and relationships between by the theories inferring through the style set (n-open). Furthermore, we presented an analysis of the classes of convergence in topological spaces. We called it a name n convergence of filters beneficiaries of the idea of -open sets and its possibilities of - cluster point of filters has been studied.

On B * c-Convergence of Filters

In this paper we introduce and study types of convergence in topological spaces namely, convergence of filters by using the concept of open sets. Also, some properties of cluster points of filters has been studied.

Certain aspects of ideal convergence in topological spaces

In this note we investigate certain aspects of ideal convergence of functions, in a very general context. We consider the situation, given an arbitrary infinite set S, free ideals I, K(⊂I) and a family F⊂[S]≥ω, when an I convergent function f from S to a topological space X will have a F-subfunction which is K convergent to the same limit. We present certain observations on such topological spaces. Further we also introduce the notion of I-cluster points of functions and make some observations.

On the structure of ultrafilters and properties related to convergence in topological spaces

We consider properties of broadly understood measurable spaces that provide the preservation of maximality when ultrafilters are restricted to filters of the corresponding subspace. We study conditions that guarantee the convergence of images of ultrafilters consisting of open sets under continuous mappings.

Some further results on ideal convergence in topological spaces

In this paper we make some further investigations on ideal convergence and in particular we concentrate on I-limit points and I-cluster points. We try to establish the characterization of the set of I-limit points (which has not been done in any structure so far) and show that this set can be characterized as an Fσ-set for a large class of ideals, namely analytic P-ideals and then make certain interesting observations on I-cluster points.

Neighborhoods and convergence with respect to a closure operator

Abstract We study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.

Compactness and convergence with respect to a neighborhood operator

We introduce a concept of neighborhood operator on a category. Such an operator is obtained by assigning to every atom of the subobject lattice of a given object a centered stack of subobjects of the object subject to two axioms. We study separation, compactness and convergence defined in a natural way by the help of a neighborhood operator. We show that they behave analogously to the separation, compactness and convergence in topological spaces. We also investigate relationships between the separation and compactness as defined on one hand and those with respect to the closure operator induced by the neighborhood operator considered on the other hand.

Is nature quantum non-local, complex holistic, or what?, I: Theory & analysis

Is Quantum Mechanics a general theory that applies to everything from subatomic particles to galaxies; is Nature governed by entanglements of linear superposition in Hilbert space or is it an expression of the nonlinear holism of emergence, self-organization, and complexity? This paper explores these issues in the context of extensions of the recently formulated approach to Chaos-Nonlinearity-compleXity [A. Sengupta, Chaos, nonlinearity, complexity: A unified perspective, in: A. Sengupta (Ed.), Chaos, Nonlinearity, and Complexity: The Dynamical Paradigm of Nature, in: StudFuzz, vol. 206, Springer-Verlag, Berlin, 2006, pp. 270–352] and proposes that Nature manifests itself only through a bi-directional, contextually objective adaptation of the Second Law of Thermodynamics. The mathematics developed is that of convergence in topological spaces, multifunctions, exclusion topology, and discontinuities, jumps, and multiplicities induced by difference equations rather than the smoothness and continuity of the Newtonian world of differential equations. In the second Part of this paper [A. Sengupta, Is Nature Quantum Non-local, Complex Holistic, or What? II—Applications, Nonlinear Analysis: RWA (2009), in press ( doi:10.1016/j.nonrwa.2008.09.002)] to be referenced “II-”, we examine the bi-directionality of a self-organized, emergent, engine-pump system with specific reference of the role of gravity as the compressive agent responsible for generation and sustenance of the delicate homeostatic balance in life, structures, and patterns appearing in this Participatory Universe. Holism of “systems and processes that interact with themselves and produce themselves from themselves” calls for radically different techniques and philosophy from that adapted in the mainstream reductionist Newtonian paradigm that we are accustomed to.

$I$ and $I^*$-convergence in topological spaces

$I$ and $I^*$-convergence in topological spaces

TOWARD A THEORY OF CHAOS

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

A topological category suited for approximation theory?

This paper presents a framework in which basic concepts of approximation theory arise as canonically as convergence in topological spaces.

Random Elements and Their Convergence in Topological Spaces

Previous article Next article Random Elements and Their Convergence in Topological SpacesV. V. BuldyginV. V. Buldyginhttps://doi.org/10.1137/1126007PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Kiyosi Itô and , Makiko Nisio, On the convergence of sums of independent Banach space valued random variables, Osaka J. Math., 5 (1968), 35–48 38:3897 0177.45102 Google Scholar[2] V. V. Buldygin, Random series in a Banach space, Teor. Verojatnost. i Primenen., 18 (1973), 491–504 48:7321 0303.60004 Google Scholar[3] J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Studia Math., 52 (1974), 159–186 50:8626 0265.60005 CrossrefGoogle Scholar[4] S. D. Chatterji, Martingales of Banach-valued random variables, Bull. Amer. Math. Soc., 66 (1960), 395–398 22:10008 0102.13601 CrossrefGoogle Scholar[5] W. A. Woyczynski, Strong laws of large numbers in certain linear spaces, Ann. Inst. Fourier (Grenoble), 24 (1974), vii, 205–223 53:9318 0275.60039 CrossrefGoogle Scholar[6] V. V. Buldygin, Convergence of random elements in topological spaces, Naukova Dumka, Kiev, 1980, (In Russian.) 0512.60009 Google Scholar[7] H. Schaefer, Topological vector spaces, The Macmillan Co., New York, 1966ix+294 33:1689 0141.30503 Google Scholar[8] J. Hoffmann-Jorgensen, The theory of analytic spaces, Matematisk Institut, Aarhus Universitet, Aarhus, 1970vi+314, Denmark 53:13500 Google Scholar[9] S. D. Chatterji, Vector-valued martingales and their applicationsProbability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975), Springer, Berlin, 1976, 33–51. Lecture Notes in Math., Vol. 526 58:24529 0336.60049 CrossrefGoogle Scholar[10] A. Korzeniovski, Martingales in Banach spaces for which the convergence with probability one, in probability and in law coincide, Colloq. Math., 39 (1978), 153–159 80a:60064 CrossrefGoogle Scholar[11] Jacques Neveu, Mathematical foundations of the calculus of probability, Translated by Amiel Feinstein, Holden-Day Inc., San Francisco, Calif., 1965xiii+223 33:6660 0137.11301 Google Scholar[12] R. Ya. Maidanyuk, Masters Thesis, Absolute continuity of measures corresponding to random processes, dissertation resumé, In-t Matem. AN USSR, Kiev, 1971, (In Russian.) Google Scholar[13] Yu. G. Kondrat'ev, Positive definite functions on certain sequence spaces, Ukrain. Mat. Ž., 28 (1976), 27–35, 141, (In Russian.) 53:6665 0347.42022 Google Scholar[14] Albert Badrikian and , Simone Chevet, Mesures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes, Springer-Verlag, Berlin, 1974x+383, Lecture Notes in Mathematics 379 54:8772 0288.60009 CrossrefGoogle Scholar[15] V. V. Buldygin, The convergence of random sequences with values in a Banach space, Dokl. Akad. Nauk SSSR, 213 (1973), 1012–1014, (In Russian.) 49:1546 0303.60003 Google Scholar[16] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[17] V. V. Buldygin, Convergence of random elements in Banach spaces in Limit Theorems for Random Processes, Kiev, 1977, (In Russian.) 0394.60006 Google Scholar[18] V. V. Buldygin and , Yu. V. Kozachenko, On a question of the applicability of the Fourier method for solving problems with random boundary conditionsRandom processes in problems of mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1979, 4–35, 150, (In Russian.) 83i:35080 0438.60016 Google Scholar[19] T. Inglot and , A. Weron, On Gaussian random elements in some non-Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 22 (1974), 1039–1043 52:6812 0319.60003 Google Scholar[20] V. V. Buldygin and , V. S. Donchenko, An oscillation theorem for Gaussian sequences in a Banach spaceProbability distributions in infinite-dimensional spaces (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1978, 38–44, 175, (In Russian.) 81g:60010 0409.60011 Google Scholar[21] E. I. Ostrovskii, Covariance operators and some estimates for Gaussian vectors in a Banach space, Dokl. Akad. Nauk SSSR, 236 (1977), 541–543, (In Russian.) 57:7752 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails ReferencesGaussian Measures in Hilbert Space | 27 December 2019 Cross Ref Volume 26, Issue 1| 1981Theory of Probability & Its Applications1-217 History Submitted:17 January 1978Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126007Article page range:pp. 85-96ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics