Further aspects of Ik-convergence in topological spaces

[EN] In this paper, we obtain some results on the relationships between different ideal convergence modes namely, I K, I K∗ , I, K, I ∪ K and (I ∪K) ∗ . We introduce a topological space namely I K-sequential space and show that the class of I K-sequential spaces contain the sequential spaces. Further I K-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space X, we characterize the set of I K-cluster points of the sequence as closed subsets of X.The first author would like to thank the University Grants Comission (UGC) for awarding the junior research fellowship vide UGC Ref. No.: 1115/(CSIR-UGC NET DEC. 2017), India.Sharmah, A.; Hazarika, D. (2021). Further aspects of Ik-convergence in topological spaces. Applied General Topology. 22(2):355-366. https://doi.org/10.4995/agt.2021.14868OJS355366222A. K. Banerjee and M. Paul, A note on IK and IK∗-convergence in topological spaces, arXiv:1807.11772v1 [math.GN], 2018.B. K. Lahiri and P. Das, I and I⋆-convergence of nets, Real Anal. Exchange 33 (2007), 431-442. https://doi.org/10.14321/realanalexch.33.2.0431B. K. Lahiri and P. Das, I and I⋆-convergence in topological spaces, Math. Bohemica 130 (2005), 153-160. https://doi.org/10.21136/MB.2005.134133H. Fast, Sur la convergence statistique, Colloq. Math, 2(1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244K. P. Hart, J. Nagata and J. E. Vaughan. Encyclopedia of General Topology, Elsevier Science Publications, Amsterdam-Netherlands, 2004.M. Macaj and M. Sleziak, IK-convergence, Real Anal. Exchange 36 (2011), 177-194. https://doi.org/10.14321/realanalexch.36.1.0177P. Das, M. Sleziak and V. Tomac, IK-Cauchy functions, Topology Appl. 173 (2014), 9-27. https://doi.org/10.1016/j.topol.2014.05.008P. Das, S. Dasgupta, S. Glab and M. Bienias, Certain aspects of ideal convergence in topological spaces, Topology Appl. 275 (2020), 107005. https://doi.org/10.1016/j.topol.2019.107005P. Das, S. Sengupta, J and Supina, IK-convergence of sequence of functions, Math. Slovaca 69, no. 5(2019), 1137-1148. https://doi.org/10.1515/ms-2017-0296P. Halmos and S. Givant, Introduction to Boolean Algebras, Undergraduate Texts in Mathematics, Springer, New York, 2009. https://doi.org/10.1007/978-0-387-68436-9P. Kostyrko, T. Salat and W. Wilczynski, I-convergence, Real Anal. Exchange 26 (2001), 669-685. https://doi.org/10.2307/44154069S. K. Pal, I-sequential topological spaces, Appl. Math. E-Notes 14 (2014), 236-241.X. Zhou, L. Liu and L. Shou, On topological space defined by I-convergence, Bull. Iranian Math. Soc. 46 (2020), 675-692. https://doi.org/10.1007/s41980-019-00284-

The Residual Method for Regularizing Ill-Posed Problems

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals. We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur

Modal Logics that Bound the Circumference of Transitive Frames

For each natural number $n$ we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than $n$ and no strictly ascending chains. The case $n=0$ is the G\"odel-L\"ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When $n=1$, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the $n$-th member of the sequence of extensions of S4 is the logic of hereditarily $n+1$-irresolvable spaces when the modality $\Diamond$ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of $\Diamond$ as the derived set (of limit points) operation. The variety of modal algebras validating the $n$-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by $n$. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters

We use Coulomb branch indices of Argyres-Douglas theories on S1×L(k,1) to quantize moduli spaces M_H of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of M_H, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in M_H under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform STkS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising

The topology of asymptotically locally flat gravitational instantons

In this letter we demonstrate that the intersection form of the Hausel--Hunsicker--Mazzeo compactification of a four dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kahler forms of the hyper-Kahler gravitational instanton metric is exact. This leads to the topological classification of these spaces. The proof exploits the relationship between L^2 cohomology and U(1) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU(2) anti-instanton moduli space over the compactification leading to the identification of its intersection form. This observation on the intersection form might be a useful tool in the full geometric classification of various asymptotically locally flat gravitational instantons.Comment: 9 pages, LaTeX, no figures; Some typos corrected, slightly differs from the published versio

On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action $(G,X)$ we construct a zero-dimensional compactification $\mu X$ of $X$ with the properties: (a) there exists an extension of the action on $\mu X$, (b) if $\mu L\subseteq \mu X\setminus X$ is the set of the limit points of the orbits of the initial action in $\mu X$, then the restricted action $(G,\mu X\setminus \mu L)$ remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on $\mu X\setminus \mu L$ by that of $\mu X$, and (c) $\mu X$ is the maximal among the zero-dimensional compactifications of $X$ with these properties. Proper actions are usually embedded in the end point compactification $\epsilon X$ of $X$, in order to obtain topological invariants concerning the cardinality of the space of the ends of $X$, provided that $X$ has an additional "nice" property of rather local character ("property Z", i.e., every compact subset of $X$ is contained in a compact and connected one). If the considered space has this property, our new compactification coincides with the end point one. On the other hand, we give an example of a space not having the "property Z" for which our compactification is different from the end point compactification. As an application, we show that the invariant concerning the cardinality of the ends of $X$ holds also for a class of actions strictly containing the properly discontinuous ones and for spaces not necessarily having "property Z".Comment: 18 page