Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542021-11-25T18:46:10.358925ZWerkzeugOn modal logics arising from scattered locally compact Hausdorff spaceshttps://zbmath.org/1472.030152021-11-25T18:46:10.358925Z"Bezhanishvili, Guram"https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Bezhanishvili, Nick"https://zbmath.org/authors/?q=ai:bezhanishvili.nick"Lucero-Bryan, Joel"https://zbmath.org/authors/?q=ai:lucero-bryan.joel-gregory"van Mill, Jan"https://zbmath.org/authors/?q=ai:van-mill.janSummary: For a topological space \(X\), let \(\mathsf{L}(X)\) be the modal logic of \(X\) where \(\square\) is interpreted as interior (and hence \(\diamond\) as closure) in \(X\). It was shown in [the first author and \textit{J. Harding}, Order 29, No. 2, 271--292 (2012; Zbl 1259.03030)] that the modal logics \textsf{S4}, \textsf{S4.1}, \textsf{S4.2}, \textsf{S4.1.2}, \textsf{S4.Grz}, \(\mathsf{S4} . \mathsf{Grz}_n\) (\(n \geq 1\)), and their intersections arise as \(\mathsf{L}(X)\) for some Stone space \(X\). We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [loc. cit., Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either \textsf{S4.Grz} or \(\mathsf{S4} . \mathsf{Grz}_n\) for some \(n \geq 1\). In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.Modal logic axioms valid in quotient spaces of finite CW-complexes and other families of topological spaceshttps://zbmath.org/1472.030182021-11-25T18:46:10.358925Z"Nogin, Maria"https://zbmath.org/authors/?q=ai:nogin.maria-s"Xu, Bing"https://zbmath.org/authors/?q=ai:xu.bing.1Summary: In this paper we consider the topological interpretations of \(\mathcal{L}_\square\), the classical logic extended by a ``box'' operator \(\square\) interpreted as interior. We present extensions of S4 that are sound over some families of topological spaces, including particular point topological spaces, excluded point topological spaces, and quotient spaces of finite CW-complexes.Derived topologies on ordinals and stationary reflectionhttps://zbmath.org/1472.030482021-11-25T18:46:10.358925Z"Bagaria, Joan"https://zbmath.org/authors/?q=ai:bagaria.joanSummary: We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor's derivative operator on sets of ordinals, starting from the usual interval topology. We characterize the non-isolated points in the \( \xi \)th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call \( \xi \)-simultaneous-reflection. We prove some properties of the ideals of non-\( \xi \)-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of \( \Pi ^1_\xi \)-indescribability, for any ordinal \( \xi \), which extends to the transfinite the usual notion of \( \Pi ^1_n\)-indescribability, and prove that in the constructible universe \( L\), a regular cardinal is \( (\xi +1)\)-simultaneously-reflecting if and only if it is \( \Pi ^1_\xi \)-indescribable, a result that generalizes to all ordinals \( \xi \) previous results of \textit{R. B. Jensen} [Ann. Math. Logic 4, 229--308 (1972; Zbl 0257.02035)] in the case \( \xi =2\), and \textit{J. Bagaria} et al. [Isr. J. Math. 208, 1--11 (2015; Zbl 1371.03069)] in the case \( \xi =n\). This yields a complete characterization in \( L\) of the non-discreteness of the \( \xi \)-topologies, both in terms of iterated stationary reflection and in terms of indescribability.Haar null and Haar meager sets: a survey and new resultshttps://zbmath.org/1472.030512021-11-25T18:46:10.358925Z"Elekes, Márton"https://zbmath.org/authors/?q=ai:elekes.marton"Nagy, Donát"https://zbmath.org/authors/?q=ai:nagy.donatThis is a~survey of results about Haar null sets in Polish groups. The notion of a~Haar measure zero set is well defined in locally compact groups although a~Haar measure is not unique. This notion can be generalized for non-locally compact groups although the Haar measure cannot be generalized for groups that are not locally compact. Two such generalizations are considered: the notion of a~Haar null set and the notion of a~generalized Haar null set. Another notion of smallness considered in the paper is the notion of a~Haar meager set. The Haar meager sets were defined in the literature as a~topological counterpart to the Haar null sets. Every Haar meager set is meager but there are Polish groups in which the converse is not true. However, in locally compact Polish groups the Haar meager sets coincide with the meager sets. All these notions define translation invariant \(\sigma\)-ideals of sets. A~number of characterizations of these notions are presented in the survey. The authors present several results by which they discuss the possibility of generalizations of the following theorems for non-locally compact Polish groups: Fubini's theorem, Kuratowski-Ulam theorem, the Steinhaus theorem, the countable chain condition, and a~decomposition into a~Haar null set and a~Haar meager set. They present a~brief outlook of applications of Haar null sets in various fields of mathematics.Absoluteness theorems for arbitrary Polish spaceshttps://zbmath.org/1472.030522021-11-25T18:46:10.358925Z"Mejía, Diego Alejandro"https://zbmath.org/authors/?q=ai:mejia.diego-alejandro"Rivera-Madrid, Ismael E."https://zbmath.org/authors/?q=ai:rivera-madrid.ismael-eSummary: By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield's absoluteness theorem.$G_\delta $-topology and compact cardinalshttps://zbmath.org/1472.030642021-11-25T18:46:10.358925Z"Usuba, Toshimichi"https://zbmath.org/authors/?q=ai:usuba.toshimichiSummary: For a topological space $X$, let $X_\delta $ be the space $X$ with the $G_\delta $-topology of $X$. For an uncountable cardinal $\kappa $, we prove that the following are equivalent: (1) $\kappa $ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_\delta $ is $\le \kappa $. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_\delta $ is $\le \kappa $. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the $G_\delta $-topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the $G_\delta $-topology. For the square of a Lindelöf space, using a weak $G_\delta $-topology, we prove that the following are consistent: (1) The least $\omega_1$-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular $T_1$ Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelöf spaces.pr-convergence of filters and nets in frameshttps://zbmath.org/1472.060122021-11-25T18:46:10.358925Z"Reshma, M. T. Elizabeth"https://zbmath.org/authors/?q=ai:reshma.m-t-elizabeth"Johnson, T. P."https://zbmath.org/authors/?q=ai:johnson.t-pThe notions of convergence and clustering of filters in frames are defined in terms of how a filter relates to all covers of a given frame. To recall, a filters in a frame is said to converge if it meets every cover of the frame. On the other hand, if \(F\) is a filter in a frame \(L\) and \(\mathrm{Sec}(F)\) is the set
\[
\mathrm{Sec}(F)=\{x\in L : x\wedge a\ne 0 \text{ for every } a\in F\}
\]
then \(F\) is said to cluster if \(\mathrm{Sec}(F)\) meets every cover of \(L\).
The paper under review discusses variants of convergence and clustering of filters by replacing covers with covers consisting of prime elements on one hand, and covers consisting of semi-prime elements on the other. They also constrain the notion of compactness by replacing covers with covers restricted as just mentioned. Then, mimicking the proofs in a paper of \textit{S. S. Hong} [Kyungpook Math. J. 35, No. 1, 85--91 (1995; Zbl 0829.54001)] that deals with convergence and clustering, they characterise their variants of convergence and clustering analogously to Hong's characterisations [loc. cit.]. They also study a variant of Čech-completeness defined using one of their variants of clustering. Again this mimics an already existing notion of Čech-completeness in frames [\textit{T. Dube} et al., Quaest. Math. 37, No. 1, 49--65 (2014; Zbl 1397.54007)] which is defined in terms of the standard notion of clustering.
Reviewer's comments: In Corollary 3.1, the authors speak of a ``finite spatial frame''. This is tautological because all finite frames are spatial. In Theorem 5.1, they present a longish proof on frames hypothesised to be Boolean and pr-regular, where the latter means every element is the join of prime elements rather below it. The result is actually a non-result because the only Boolean frames that are pr-regular are the two-element frame and the four-element Boolean algebra. To see this, observe first that every element in a Boolean pr-regular frame is either \(0\), \(1\), or a co-atom. Now let \(a\) and \(b\) be distinct non-zero non-top elements in a Boolean pr-regular frame. Then \(a\) and \(b\) are co-atoms and hence \(a\vee b=1\) and \(a\wedge b=0\), so that \(a=b^*\).Local extension property for finite height spaceshttps://zbmath.org/1472.060172021-11-25T18:46:10.358925Z"Correa, Claudia"https://zbmath.org/authors/?q=ai:correa.claudia"Tausk, Daniel V."https://zbmath.org/authors/?q=ai:tausk.daniel-victorSummary: We introduce a new technique for the study of the local extension property (${\mathop \mathrm{LEP}}$) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has $\mathop \mathrm{LEP}$. This implies, under appropriate additional assumptions on $K$ and Martin's Axiom, that every twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by \textit{W. Marciszewski} and \textit{G. Plebanek} [J. Funct. Anal. 274, No. 5, 1491--1529 (2018; Zbl 1390.46016)].Alexandroff unitization of a truncated vector latticehttps://zbmath.org/1472.060222021-11-25T18:46:10.358925Z"Boulabiar, Karim"https://zbmath.org/authors/?q=ai:boulabiar.karim"Hafsi, Hamza"https://zbmath.org/authors/?q=ai:hafsi.hamza"Mahfoudhi, Mounir"https://zbmath.org/authors/?q=ai:mahfoudhi.mounirSummary: Recently Ball introduced the remarkable concept of truncated vector lattices and studied their representations by continuous functions. Weakening the original Ball definition, we introduce the notion of an unitization of a truncated vector lattice \(T\) and we prove that \(T\) has a smallest unitization \(T^{*}\), called the Alexandroff unitization of \(T\). We show that \(T^{*}\) has universal properties and we prove that \(T\) is dense in \(T^{*}\) if and only if \(T\) is not unital. Otherwise, \(T\) and its polar are cardinal summands for \(T^{*}\). All these facts are also interpreted in category-theoretic terms.Systems of polynomial equations, higher-order tensor decompositions, and multidimensional harmonic retrieval: a unifying framework. Part II: The block term decompositionhttps://zbmath.org/1472.130472021-11-25T18:46:10.358925Z"Vanderstukken, Jeroen"https://zbmath.org/authors/?q=ai:vanderstukken.jeroen"Kürschner, Patrick"https://zbmath.org/authors/?q=ai:kurschner.patrick"Domanov, Ignat"https://zbmath.org/authors/?q=ai:domanov.ignat"De Lathauwer, Lieven"https://zbmath.org/authors/?q=ai:de-lathauwer.lievenRings in which idempotents generate maximal or minimal idealshttps://zbmath.org/1472.160362021-11-25T18:46:10.358925Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Ghirati, Mojtaba"https://zbmath.org/authors/?q=ai:ghirati.mojtaba"Nazari, Sajad"https://zbmath.org/authors/?q=ai:nazari.sajad"Taherifar, Ali"https://zbmath.org/authors/?q=ai:taherifar.aThe authors characterize rings with identity in which every left ideal generated by an idempotent different from \(0\) and \(1\) is either a maximal left ideal or a minimal left ideal. These rings are called IMm-rings. Several special classes of IMm-rings are considered. In particular, if \(R\) is a semiprimitive commutative ring that has infinitely many maximal ideals, \(R\) being an IMm-ring is characterized by means of the Zariski topology of the maximal spectrum \(\text{Max}(R):=\{M\subseteq R: M \text{ is a maximal ideal of } R\}\). Finally the authors study rings with a weaker form of the ``IMm-property''.When is the sum of two closed subgroups closed in a locally compact abelian group?https://zbmath.org/1472.220032021-11-25T18:46:10.358925Z"Herfort, Wolfgang"https://zbmath.org/authors/?q=ai:herfort.wolfgang-n"Hofmann, Karl H."https://zbmath.org/authors/?q=ai:hofmann.karl-heinrich"Russo, Francesco G."https://zbmath.org/authors/?q=ai:russo.francesco-giuseppeThe authors study the classification of the locally compact abelian grous in which the sum \(U+V\) of closed subgroups \(U,V\) is again closed. The results in this paper extend an old classification of Yu.~N.~Mukhin. In particular the case of totally disconnected groups is studied more closely.Expansive actions of automorphisms of locally compact groups \(G\) on \(\text{Sub}_G\)https://zbmath.org/1472.220052021-11-25T18:46:10.358925Z"Prajapati, Manoj B."https://zbmath.org/authors/?q=ai:prajapati.manoj-b"Shah, Riddhi"https://zbmath.org/authors/?q=ai:shah.riddhiIn the theory of dynamical systems, a homeomorphism \(T\colon X\to X\) of a metric space \((X,d)\) is called expansive if there exists \(\epsilon >0\) such that, for all \(x,y\in X\) with \(x\not=y\), the iterates \(T^n(x)\) and \(T^n(y)\) have distance \(d(T^n(x),T^n(y))>\epsilon\) for some integer~\(n\). For topological groups with left-invariant metrics, results concerning expansive automorphisms emerged in the 1980s and 1990s, mostly for compact groups. The structure theory of totally disconnected locally compact groups, initiated in 1994 by \textit{G. Willis} [Math. Ann. 300, No. 2, 341--363 (1994; Zbl 0811.22004)], enabled progress concerning expansive automorphisms of the latter groups (and the special case of contractive automorphisms) in the last decade, and most recently (via work of \textit{R. Shah} [New York J. Math. 26, 285--302 (2020; Zbl 1435.22005)]) also for general locally compact groups. If \(G\) is a metrizable locally compact group, then the set \(\text{Sub}_G\) of all closed subgroups of~\(G\) is a compact, metrizable topological space in a natural way, using the Chabauty topology introduced in the 1950s. In the last decade, \(\text{Sub}_G\) and the Chabauty topology thereon attracted considerable interest (see [\textit{P.-E. Caprace} (ed.) and \textit{N. Monod} (ed.), New directions in locally compact groups. Cambridge: Cambridge University Press (2018; Zbl 1390.22004)]). Every automorphism \(T\) of~\(G\) induces a homeo\-morphism \(\text{Sub}_G\to \text{Sub}_G\), \(H \mapsto T(H)\) and a corresponding homeomorphism of the compact subset \(\text{Sub}^a_G\) of abelian closed subgroups of \(G\).
Among other topics, the article investigates the question for which \(G\) an automorphism \(T\) can be found such that the corresponding homeomorphism of \(\text{Sub}_G\) (resp., \(\text{Sub}^a_G\)) is expansive. The authors show that whenever a locally compact group \(G\) admits an automorphism \(T\) which acts expansively on \(\text{Sub}_G\), then \(G\) is totally disconnected, \(T\) is expansive and the contraction group \(C(T):=\{g\in G\colon \lim_{n\to\infty} T^n(g)=e\}\) is closed in \(G\) (see Theorem 4.1). Moreover, \(C(T)\cong {\mathbb Q}_{p_1}\times\cdots\times {\mathbb Q_{p_k}}\) for certain primes \(p_1<\cdots <p_k\) (cf.\ Theorem 4.2). If \(T\) acts expansively on \(\text{Sub}_G\), then it acts expansively on \(\text{Sub}^a_G\). Already expansivity of the \(T\)-action on \(\text{Sub}^a_G\) imposes restrictions on~\(G\). For example, a connected Lie group \(G\) must be trivial if it admits an automorphism acting expansively on \(\text{Sub}^a_G\) (Theorem 3.1).
Reviewer's comment: In Lemmas 2.4 and 2.5, \(X\) should be assumed compact.Distance between sets -- a surveyhttps://zbmath.org/1472.280062021-11-25T18:46:10.358925Z"Conci, Aura"https://zbmath.org/authors/?q=ai:conci.aura"Kubrusly, Carlos"https://zbmath.org/authors/?q=ai:kubrusly.carlos-sSummary: The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which is followed by a review on applications used in many areas of knowledge, ranging from theoretical to practical applications.Growth theorems for metric spaces with applications to PDEhttps://zbmath.org/1472.350712021-11-25T18:46:10.358925Z"Safonov, M. V."https://zbmath.org/authors/?q=ai:safonov.mikhail-vIn this article, the author focuses mainly on the extent of the results and techniques of him, obtained with N. V. Krylov, for the second order elliptic and parabolic equations in the form of non-divergence, which date back to 1978--1980 to a more general abstract setting in terms of metric spaces. The author deals only with the elliptic versions of these results, the extension of the results to the parabolic (time-dependent) case requires more assumptions, which are not considered here. Some elements of the techniques are similar to those previously introduced by De Giorgi, Nash and Moser for the elliptic and parabolic equations in the form of divergence. The main tools are special Landis-type growth theorems.Fine metrizable convex relaxations of parabolic optimal control problemshttps://zbmath.org/1472.354152021-11-25T18:46:10.358925Z"Roubíček, Tomáš"https://zbmath.org/authors/?q=ai:roubicek.tomasThe paper deals with fine metrizable convex relaxations of parabolic optimal control problems. In particular, a compromising convex compactification is devised. The basic idea consists in combining classical techniques for Young measures with Choquet theory. Therefore, the proposed approach works under classical \(\sigma\)-additive measures and standard sequences. At the same time, it allows for dealing with a wider class of nonlinearities than only affine. The controls \(u\) are valued in the set \(S_p\) of the form \[ S_p = \{ u \in L^p(\Omega;\mathbb{R}^m): u(x) \in B \; \text{for a.a.} \; x \in \Omega\}, \] with \(\Omega \subset \mathbb{R}^d\), \(d \in \mathbb{N}\), \(1 \leq p < +\infty\) and \(B \subset \mathbb{R}^m\) bounded and closed. In addition, some generalization to unbounded domain \(B\) by considering a general \(S_p\) bounded in \(L^p(\Omega;\mathbb{R}^m)\), with \(1 \leq p < \infty\) fixed but not necessarily bounded in \(L^\infty(\Omega;\mathbb{R}^m)\), is also discussed. Finally, an application to optimal control of a system of semilinear parabolic differential equations is presented for the reader convenience, together with other relaxation strategies as well as more general nonlinearities, showing that the finds reported in the paper are useful for practical applications.Erratum to: ``Nonseparable closed vector subspaces of separable topological vector spaces''https://zbmath.org/1472.460012021-11-25T18:46:10.358925Z"Kąkol, Jerzy"https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, Arkady G."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Morris, Sidney A."https://zbmath.org/authors/?q=ai:morris.sidney-aErratum to the authors' paper [ibid. 182, No. 1, 39--47 (2017; Zbl 1361.46001)].Fixed point theorems for inward mappings in \(\mathbb{R}\)-treeshttps://zbmath.org/1472.470462021-11-25T18:46:10.358925Z"Markin, Jack"https://zbmath.org/authors/?q=ai:markin.jack-t"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseerSummary: In an \(\mathbb R\)-tree setting, we develop fixed point theorems for multivalued mappings that are stricly contractiove, nonexpansive or upper continuous and satisfy an inward condition. As applications, we obtain common fixed point theorems for a point-valued and a multivalued mapping that commute.Existence and approximations for order-preserving nonexpansive semigroups over \(\mathrm{CAT}(\kappa)\) spaceshttps://zbmath.org/1472.470492021-11-25T18:46:10.358925Z"Chaipunya, Parin"https://zbmath.org/authors/?q=ai:chaipunya.parinSummary: In this paper, we discuss the fixed point property for an infinite family of order-preserving mappings which satisfy the Lipschitz condition on comparable pairs. The underlying framework of our main results is a metric space of any global upper curvature bound \(\kappa\in\mathbb{R}\), i.e., a \(\mathrm{CAT}(\kappa)\) space. In particular, we prove the existence of a fixed point for a nonexpansive semigroup on comparable pairs. Then, we propose and analyze two algorithms to approximate such a fixed point.
For the entire collection see [Zbl 1470.47001].Fixed point approximation of nonexpansive mappings on a nonlinear domainhttps://zbmath.org/1472.470742021-11-25T18:46:10.358925Z"Khan, Safeer Hussain"https://zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: We use a three-step iterative process to prove some strong and \(\Delta\)-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.Convergence theorems on total asymptotically demicontractive and hemicontractive mappings in CAT(0) spaceshttps://zbmath.org/1472.470782021-11-25T18:46:10.358925Z"Liu, Xin-dong"https://zbmath.org/authors/?q=ai:liu.xindong"Chang, Shih-sen"https://zbmath.org/authors/?q=ai:chang.shih-senSummary: The purpose of this paper is to introduce the concepts of \textit{total asymptotically demicontractive mappings} and \textit{total asymptotically hemicontractive mappings}. Under suitable conditions some strong convergence theorems for these two kinds of mappings to converge to their fixed points in \textit{CAT(0) space} are proved. The results presented in the paper extend and improve some recent results announced in the current literature.Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaceshttps://zbmath.org/1472.470842021-11-25T18:46:10.358925Z"Tang, Jinfang"https://zbmath.org/authors/?q=ai:tang.jinfangSummary: The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.\(\Delta\)-convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in \(W\)-hyperbolic spaceshttps://zbmath.org/1472.470892021-11-25T18:46:10.358925Z"Yi, Li"https://zbmath.org/authors/?q=ai:yi.li"Bo, Liu Hong"https://zbmath.org/authors/?q=ai:bo.liu-hongSummary: Throughout this paper, we introduce a class of asymptotically nonexpansive nonself-mapping and modify the classical Kuhfittig iteration algorithm in the general setup of hyperbolic space. Also, convergence results are obtained under a limit condition. The results presented in the paper extend various results in the existing literature.Approximating fixed points of Suzuki \((\alpha,\beta)\)-nonexpansive mappings in ordered hyperbolic metric spaceshttps://zbmath.org/1472.470932021-11-25T18:46:10.358925Z"Martínez-Moreno, Juan"https://zbmath.org/authors/?q=ai:martinez-moreno.juan"Calderón, Kenyi"https://zbmath.org/authors/?q=ai:calderon.kenyi"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Rojas, Edixon"https://zbmath.org/authors/?q=ai:rojas.edixon-mSummary: In this chapter, we define the class of monotone \((\alpha,\beta)\)-nonexpansive mappings and prove that they have an approximate fixed point sequence in partially ordered hyperbolic metric spaces. We prove the \(\Delta\) and strong convergence of the CR-iteration scheme.
For the entire collection see [Zbl 1470.47001].Approximating fixed points for generalized nonexpansive mapping in CAT(0) spaceshttps://zbmath.org/1472.470942021-11-25T18:46:10.358925Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Dalal, Sumitra"https://zbmath.org/authors/?q=ai:dalal.sumitra"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: \textit{W. Takahashi} and \textit{G.-E. Kim} [Math. Japon. 48, No. 1, 1--9 (1998; Zbl 0913.47056)] used the Ishikawa iteration process to prove some convergence theorems for nonexpansive mappings in Banach spaces. The aim of this paper is to prove similar results in CAT(0) spaces for generalized nonexpansive mappings, which, in turn, generalize the corresponding results of Takahashi and Kim [loc.\,cit.], \textit{T. Laokul} and \textit{B. Panyanak} [Int. J. Math. Anal., Ruse 3, No. 25--28, 1305--1315 (2009; Zbl 1196.54077)], \textit{A. Razani} and \textit{H. Salahifard} [Bull. Iran. Math. Soc. 37, No. 1, 235--246 (2011; Zbl 1302.47096)], and some others.Some convergence theorems for a hybrid pair of generalized nonexpansive mappings in CAT(0) spaceshttps://zbmath.org/1472.470952021-11-25T18:46:10.358925Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: In [Fixed Point Theory Appl. 2010, Article ID 618767, 9 p. (2010; Zbl 1208.47070)], \textit{K. Sokhuma} and \textit{A. Kaewkhao} introduced a modified Ishikawa iteration scheme for a pair of hybrid mappings in Banach spaces and utilized the same to prove psome convergence theorems. In this paper, we study the convergence of a modified Ishikawa iteration process involving a hybrid pair of generalised nonexpansive mappings in CAT(0) spaces. In process, the result of Sokhuma and Kaewkhao [loc. cit.], \textit{K. Sokhuma} et al. [Int. J. Math. Anal., Ruse 6, No. 17--20, 923--932 (2012; Zbl 1296.47091)] and \textit{I. Uddin} et al. [Bull. Malays. Math. Sci. Soc. (2) 38, No. 2, 695--705 (2015; Zbl 1312.54035)] are generalized and improved.Ekeland type variational principle for set-valued maps in quasi-metric spaces with applicationshttps://zbmath.org/1472.490352021-11-25T18:46:10.358925Z"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Sharma, Pradeep Kumar"https://zbmath.org/authors/?q=ai:sharma.pradeep-kumarSummary: In this paper, we derive a fixed point theorem, minimal element theorems and Ekeland type variational principle for set-valued maps with generalized variable set relations in quasi-metric spaces. These generalized variable set relations are the generalizations of set relations with constant ordering cone, and form the modern approach to compare sets in set-valued optimization with respect to variable domination structures under some appropriate assumptions. At the end, we give application of these variational principles to the capability theory of well-beings via variational rationality.Regularity of implicit solution mapping to parametric generalized equationhttps://zbmath.org/1472.490372021-11-25T18:46:10.358925Z"Ouyang, Wei"https://zbmath.org/authors/?q=ai:ouyang.wei"Zhang, Binbin"https://zbmath.org/authors/?q=ai:zhang.binbinSummary: This paper concerns the study of both local and global metric regularity/Lipschitz-like properties concerning the behavior of the implicit solution mapping associated to parametric generalized equation in metric space. We extend some implicit multifunction results to the addition of two multifunctions both depending on parameters. Through the approach of inverse mapping iteration, several results are established regarding the relations between the (partial) metric regularity/Lipschitz-like moduli of multifunctions used as the defining form of the generalized equation and the corresponding implicit solution mapping, the proof of which is completely self-contained. Finally, a local Lyusternik-Graves Theorem is obtained as an application.A novel construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametrichttps://zbmath.org/1472.510072021-11-25T18:46:10.358925Z"Wan, Zhengchao"https://zbmath.org/authors/?q=ai:wan.zhengchaoThis paper shows that the collection of all compact ultrametric spaces \(\mathcal U\) equipped with the Gromov-Hausdorff metric is universal. This means that any Polish ultrametric space is isometrically embeddable into \(\mathcal U\). This space is in addition ultra-homogeneous which means that for any finite ultrametric space \(B\), a subset \(A\subset B\) and an isometric embedding \(\varphi:A\to\mathcal U\) there exists an isometric extension \(\psi:B\to \mathcal U\) of \(\varphi\).
If \(R\subset \mathbb R_{\ge 0}\) is a countable set containing \(0\) an ultrametric space \(X\) is called \(R\)-ultrametric if \(d\) takes only values in \(R\). The author shows the collection of all compact \(R\)-ultrametric spaces equipped with the Gromov-Hausdorff metric is a Polish ultrametric space universal for Polish \(R\)-ultrametric spaces and homogeneous for all finite \(R\)-ultrametric spaces.\(\mathbb{B}\)-spaces are KKM spaceshttps://zbmath.org/1472.520032021-11-25T18:46:10.358925Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: A subset \(B\) of \(\mathbb{R}_+^n\) is \(\mathbb{B}\)-convex if for all \(x_1,x_2\in B\) and all \(t\in[0,1]\) one has \(tx_1\vee x_2\in B\). These sets were first investigated in [\textit{W. Briec} and \textit{C. Horvath}, Optimization 53, No. 2, 103--127 (2004; Zbl 1144.90506)]. In this paper, we show that any finite dimensional \(\mathbb{B}\)-space is a KKM space, that is, a space satisfying the abstract form of the celebrated Knaster-Kuratowski-Mazurkiewicz theorem appeared in 1929 and its open-valued version. Therefore, a \(\mathbb{B}\)-space satisfies a large number of the KKM theoretic results appeared in the literature.Fuzzy conditional topological entropyhttps://zbmath.org/1472.540012021-11-25T18:46:10.358925Z"Afsan, B. M. Uzzal"https://zbmath.org/authors/?q=ai:afsan.b-m-uzzalSummary: In [\textit{I. Tok}, ``On the fuzzy topological entropy function'', J. File Syst. 28, 74--80 (2005)] the author defined fuzzy topological entropy for fuzzy compact topological spaces that was further studied by \textit{B. M. U. Afsan} and \textit{C. K. Basu} [Appl. Math. Lett. 24, No. 12, 2030--2033 (2011; Zbl 1269.54003)] for arbitrary topological spaces. In the present paper, we have initiated a new notion of topological entropy in fuzzy settings, namely fuzzy conditional entropy based on the notion of fuzzy compactness \textit{C. L. Chang} [J. Math. Anal. Appl. 24, 182--190 (1968; Zbl 0167.51001)] that actually, has fuzzified the notion of the topological conditional entropy of \textit{M. Misiurewicz} [Stud. Math. 55, 175--200 (1976; Zbl 0355.54035)]. We have studied many basic properties of this notion and we have achieved a formula of the fuzzy conditional topological entropy for fuzzy \(C\)-compact topological spaces in terms of its fuzzy closed covers. Further, We have established a relationship between the fuzzy conditional topological entropy and the fuzzy topological entropies due to Tok [loc. cit.] and Afsan and Basu [loc. cit.].Infra soft compact spaces and application to fixed point theoremhttps://zbmath.org/1472.540022021-11-25T18:46:10.358925Z"Al-shami, Tareq M."https://zbmath.org/authors/?q=ai:al-shami.tareq-mohammedSummary: Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the ``transmission'' of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.Star and strong star-type versions of Rothberger and Menger principles for hit-and-miss topologyhttps://zbmath.org/1472.540032021-11-25T18:46:10.358925Z"Cruz-Castillo, Ricardo"https://zbmath.org/authors/?q=ai:cruz-castillo.ricardo"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandro"Tenorio, Jesús F."https://zbmath.org/authors/?q=ai:tenorio.jesus-fLet \(X\) be a space and denote by \(\mathcal{D}\) the family of dense subsets in the hyperspace of nonempty closed subsets with the hit-and-miss topology. The reviewer in [\textit{Z. Li}, Topology Appl. 212, 90--104 (2016; Zbl 1355.54014)] defined the concepts of \(\pi_{F}\)-network and \(\pi_{V}\)-network, obtaining many interesting results. In this paper, the authors introduce some selection principles, which are motivated by star and strong star Rothberger and star and strong star Menger principles to characterize the star and strong star Rothberger-type properties in spaces using the family \(\mathbb{C}_{\Delta}(\Lambda)\) of \(C_{\Delta}(\Lambda)\)-covers of a space \(X\). The main results are the following:
\textbf{Theorem 1.} Let \((X,\tau)\) be a topological space. Denote by \(\mathcal{D}\) the family of all the dense subsets in the hyperspace \((\Lambda,\tau_{\Delta}^{+})\). The following conditions are equivalent:
(1) \((\Lambda,\tau_{\Delta}^{+})\) satisfies \(\textbf{S}_{1}(\mathcal{D}, \mathcal{D})\);
(2) \((X,\tau)\) satisfies \(\textbf{S}_{1}(\mathbb{C}_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda))\).
\textbf{Theorem 2.} Let \((X,\tau)\) be a topological space. Denote by \(\mathcal{D}\) the family of all the dense subsets in the hyperspace \((\Lambda,\tau_{\Delta}^{+})\). The following conditions are equivalent:
(1) \((\Lambda,\tau_{\Delta}^{+})\) satisfies \(\textbf{DS}_{1}(\mathcal{D}, \mathcal{D})\);
(2) \((X,\tau)\) satisfies \(\textbf{CS}_{1}(\mathbb{C}_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda))\).
\textbf{Theorem 3.} Let \((X,\tau)\) be a topological space. Denote by \(\mathcal{D}\) the family of all the dense subsets in the hyperspace \((\Lambda,\tau_{\Delta}^{+})\). The following conditions are equivalent:
(1) \((\Lambda,\tau_{\Delta}^{+})\) satisfies \(\textbf{SDS}_{1}(\mathcal{D}, \mathcal{D})\);
(2) \((X,\tau)\) satisfies \(\textbf{SCS}_{1}(\mathbb{C}_{\Delta}(\Lambda), \mathbb{C}_{\Delta}(\Lambda))\).Correction to: ``Distinguished \(C_p(X)\) spaces''https://zbmath.org/1472.540042021-11-25T18:46:10.358925Z"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"Kąkol, J."https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, A."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Saxon, S. A."https://zbmath.org/authors/?q=ai:saxon.stephen-aCorrection to the authors' paper [ibid. 115, No. 1, Paper No. 27, 18 p. (2021; Zbl 1460.54011)].There are \(2^{\mathfrak{c}}\) quasicontinuous non Borel functions on uncountable Polish spacehttps://zbmath.org/1472.540052021-11-25T18:46:10.358925Z"Holá, Ľubica"https://zbmath.org/authors/?q=ai:hola.lubicaA function \(f : X \to Y\) between spaces \(X\) and \(Y\) is said to be quasicontinuous if \(f^{-1}(V) \subseteq \mathrm{cl}_X(\mathrm{int}_X(f^{-1}(V)))\) for every open \(V \subseteq Y\) where \(\mathrm{cl}_X\) and \(\mathrm{int}_X\) represent the closure and interior operators in \(X\), respectively.
\textit{S. Marcus} proved [Colloq. Math. 8, 47--53 (1961; Zbl 0099.04501)], that there is a quasicontinuous function from the interval \([0,1]\) to \(\mathbb R\) which is not Lebesgue measurable. In [\textit{Ľ. Holá}, Am. Math. Mon. 128, No. 5, 457--460 (2021; Zbl 1471.28001)], it was shown that there are \(2^{\mathfrak c}\) such functions. In this paper, using embeddings of the irrationals or the interval \([0,1]\), it is shown that, if \(X\) is either an uncountable Polish space or a locally pathwise connected perfectly normal space with at least one non-isolated point, then there are \(2^{\mathfrak c}\) quasicontinuous non-Borel functions from \(X\) to \([0,1]\).
In the remainder of the paper, properties of cliquish functions and set-valued maps are investigated.
For a space \(X\) and a metric space \((Y,d)\), a function \(f: X \to Y\) is cliquish if, for any \(x \in X\), every \(\varepsilon > 0\), and every neighborhood \(U\) of \(x\), there is a nonempty open set \(G \subseteq U\) so that \(d(f(x_1),f(x_2)) < \varepsilon\) for every \(x_1,x_2 \in G\). Though quasicontinuous functions into a metric space are cliquish, not all cliquish functions are quasicontinuous. In this paper it is established that, if \(X\) is a space in which every open set is an \(F_\sigma\), then \(X\) is a Baire space if and only if every Borel measurable function of the first class from \(X\) to \(\mathbb R\) is cliquish.
The final results of this paper establish new characterizations of minimal usco set-valued mappings in metric spaces by studying quasicontinuous Borel measurable functions.The space consisting of uniformly continuous functions on a metric measure space with the \(L^p\) normhttps://zbmath.org/1472.540062021-11-25T18:46:10.358925Z"Koshino, Katsuhisa"https://zbmath.org/authors/?q=ai:koshino.katsuhisaLet \(\mathbf{s} = (-1,1)^\mathbb{N}\) be a countable infinite product of lines endowed with the product topology and let \(c_0\) be the subspace of \(\mathbf{s}\) consisting of those sequences converging to zero. Kadec, Bessaga, Pelczynski and other well-known mathematicians studied homeomorphisms between various infinite dimensional Banach and Fréchet spaces motivated by several questions posed by Fréchet and Banach. A classical celebrated result due to \textit{R. D. Anderson} [Bull. Am. Math. Soc. 72, 515--519 (1966; Zbl 0137.09703)] and \textit{M. I. Kadets} [Funkts. Anal. Prilozh. 1, No. 1, 61--70 (1967; Zbl 0166.10603)] states that every separable infinite dimensional Banach space or Fréchet space is homeomorphic to \(\mathbf{s}\).
In contrast with this result, \textit{R. Cauty} showed in [Fundam. Math. 139, No. 1, 23--36 (1991; Zbl 0793.46008)] that the subspace of \(L^p[0,1]\) consisting of those elements having a representative which is continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\). In this paper the author generalizes the aforementioned result of Cauty showing that if \(X\) is a metric measure space satisfying some natural conditions then the subspace \(C_u(X)\) of \(L^p(X)\) consisting of those elements having a representative which is uniformly continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\).Intermediate rings of complex-valued continuous functionshttps://zbmath.org/1472.540072021-11-25T18:46:10.358925Z"Acharyya, Amrita"https://zbmath.org/authors/?q=ai:acharyya.amrita"Acharyya, Sudip Kumar"https://zbmath.org/authors/?q=ai:acharyya.sudip-kumar"Bag, Sagarmoy"https://zbmath.org/authors/?q=ai:bag.sagarmoy"Sack, Joshua"https://zbmath.org/authors/?q=ai:sack.joshuaThe ring of complex-valued continuous functions on \(X\), where \(X\) is a completely regular Hausdorff topological space, is denoted by \(C(X, \mathbb{C})\). \(C^*(X, \mathbb{C})\) is its subring of bounded functions, \(\Sigma (X, C)\) is the collection of rings lying between \(C^*(X, \mathbb{C})\) and \(C(X, \mathbb{C})\). This paper shows extensive complex analogues of parallel results on absolutely convex, prime, maximal, \(z\)-, \(z^0\)-ideals of the intermediate rings of real-valued continuous functions on \(X\). Using a complex analogue of the structure space on the set of all maximal ideals on a commutative ring with unity (where a family of sets of maximal ideals forms a base for closed sets of the hull-kernel topology), the authors show that the complex analogue of the structure space of an intermediate ring \(P(X, \mathbb{C}) \in \Sigma(X, C)\) is the \textit{Stone-Čech} compactification \(\beta X\) of \(X\).
Extending the notion of real-valued \(C\)-type intermediate rings to rings of complex-valued continuous functions, \(P(X, \mathbb{C})\) is a \(C\)-type ring if it is isomorphic to a ring \(C(Y, \mathbb{C})\) for some Tychonoff space \(Y\). The ring \(C^*(X, \mathbb{C}) + I\), where \(I\) is a \(z\)-ideal in \(C(X, \mathbb{C})\), is a \(C\)-type intermediate ring of \(C(X, \mathbb{C})\). Those are the only \(C\)-type rings between \(C^*(X, \mathbb{C})\) and \(C(X, \mathbb{C})\) if and only if \(X\) is pseudocompact. The paper shows that for any maximal ideal \(M\) in \(C(X)\) and its complex analogue \(M_c\), the residue class field \(C(X, \mathbb{C} )/M_c\) is an algebraically closed field, as well as the algebraic closure of \(C(X)/M\). Some special cases are examined.
Ideals \(C_\mathcal{P}(X)\), where \(\mathcal{P}\) is an ideal of closed sets in \(X\), were introduced in [\textit{S. K. Acharyya} and \textit{S. K. Ghosh}, Topol. Proc. 35, 127--148 (2010; Zbl 1180.54040)] and investigated in [ibid. 40, 297--301 (2012; Zbl 1266.54057); \textit{S. Bag} et al., Appl. Gen. Topol. 20, No. 1, 109--117 (2019; Zbl 1429.54024)]. Here, a necessary and sufficient condition is found for the complex analogue \(C_\mathcal{P}(X, \mathbb{C})\) of \(C_\mathcal{P}(X)\) consisting of all functions with support (closure of the set of points where the functions are non-zero) in \(\mathcal{P}\) to be a prime ideal in \(C(X, \mathbb{C})\).
Also found are some estimates for certain parameters for zero divisor graphs [\textit{F. Azarpanah} and \textit{M. Motamedi}, Acta Math. Hung. 108, No. 1--2, 25--36 (2005; Zbl 1092.54007)] of an intermediate ring in \(\Sigma (X, C)\).On the cardinality of non-isomorphic intermediate rings of \(C(X)\)https://zbmath.org/1472.540082021-11-25T18:46:10.358925Z"Bose, B."https://zbmath.org/authors/?q=ai:bose.benjamin|bose.bedanta|bose.bella"Acharyya, S. K."https://zbmath.org/authors/?q=ai:acharyya.sudip-kumar|acharyya.s-kThe authors study ``intermediate rings'', which are subrings of the ring of real valued continuous functions which contains the ring of bounded real valued continuous functions.
To each intermediate ring \(A(X)\), it is associated a subspace \(\nu_A(X)\) of the Stone-Čech compactification of \(X\), which is an analogue of the Hewitt realcompactification. Also, they consider \([A(X)]\), the class of intermediate rings with homeomorphic subspaces \(\nu_A(X)\).
Some results in this context are obtained, for example: for a locally compact, non-compact but realcompact space \(X\), each class \([A(X)]\) has cardinality greater than \(2^c\); for a first countable noncompact realcompact space \(X\), there exist at least \(2^c\) intermediate subrings of \([A(X)]\), no two of which are isomorphic.Abundance of isomorphic and non isomorphic \(C\)-type intermediate ringshttps://zbmath.org/1472.540092021-11-25T18:46:10.358925Z"Bose, Bedanta"https://zbmath.org/authors/?q=ai:bose.bedanta"Acharyya, Sudip Kumar"https://zbmath.org/authors/?q=ai:acharyya.sudip-kumarAuthors' abstract: For a nonpseudocompact space \(X\), the family \(\Sigma (X)\) of all intermediate subrings of \(C(X)\) which contain \(C^*(X)\) contains at least \(2^c\) many distinct rings. We show that if in addition \(X\) is first countable and realcompact, then there are at least \(2^c\) many \(C\)-type intermediate rings in \(\Sigma (X)\) no two of which are pairwise isomorphic. With the special case \(X = \mathbb{N}\), it is shown that there exists a family containing \(c\)-many pairwise isomorphic \(C\)-type intermediate rings in \(\Sigma (\mathbb{N})\).On the sum of \(z^\circ\)-ideals in two classes of subrings of \(C(X)\)https://zbmath.org/1472.540102021-11-25T18:46:10.358925Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Parsinia, Mehdi"https://zbmath.org/authors/?q=ai:parsinia.mehdiLet \(X\) be a Tychonoff space, \(C(X)\) be the ring of real-valued continuous functions on \(X\), \(I\) be an ideal of a subring \(A(X)\) of \(C(X)\) and \(C^*(X)\subseteq A(X)\). Then \(I\) is called a \(z^\circ\)-ideal if for every \(a \in I,~ P_a \subseteq I\), where \(P_a\) denotes the intersection of all minimal prime ideals of \(A(X)\) that contain \(a\). Generally, the goal of this paper is to consider sums of \(z^\circ\)-ideals in intermediate rings of \(C(X)\). Recall that \(X\) is called an \(F\)-space if every cozero-set in \(X\) is \(C^*\)-embedded. Algebraically, \(X\) is an \(F\)-space if and only if every finitely generated ideal of \(C(X)\) is principal. Recall that if every dense cozero-set is \(C^*\)-embedded in \(X\), then \(X\) is called a quasi \(F\)-space.
In this paper it is proved that \(X\) is a quasi \(F\)-space if and only if the sum of any two \(z^\circ\)-ideals in \(A(X)\) (any intermediate ring) is a \(z^\circ\)-ideal or all of \(A(X)\) (the whole intermediate ring). Let \(X\) be an almost \(P\)-space. Then, \(X\) is an \(F\)-space if and only if for each ideal \(I\) in \(C(X)\), such that \(I+R\) generates the topology on \(X\), the sum of any two \(z^\circ\)-ideals in \(I+\mathbb{R}\) is a \(z^\circ\)-ideal in \(I+\mathbb{R}\) or all of \(I+\mathbb{R}\).Extending the realm of Horvath spaceshttps://zbmath.org/1472.540112021-11-25T18:46:10.358925Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: A KKM space is an abstract convex space satisfying the abstract form of the KKM theorem and its open-valued version. In this article we introduce a typical subclass of KKM spaces called the Horvath spaces including \(c\)-spaces due to Horvath. We show that hyperbolic metric spaces, metric spaces with continuous midpoints, metric spaces with global nonpositive curvature (NPC) and convex hull finite property (CHFP), certain Riemannian manifolds, and \(\mathbb{B}\)-spaces are relatively new examples of Horvath spaces. Many of their properties are introduced by following our previous works [the author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 12, 4352--4364 (2008; Zbl 1163.47044); Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 4, 1028--1042 (2010; Zbl 1214.47042)].Cardinal invariants of dually CCC spaceshttps://zbmath.org/1472.540122021-11-25T18:46:10.358925Z"Xuan, Wei-Feng"https://zbmath.org/authors/?q=ai:xuan.weifeng"Song, Yan-Kui"https://zbmath.org/authors/?q=ai:song.yankuiThe notion of the class \( P^* \) dual to a topological property \( P \) was introduced by [\textit{O. T. Alas} et al., Topol. Proc. 30(30), 25--38 (2006; Zbl 1127.54009)]. A neighborhood assignment for a space \( (X,\tau) \) is a function \( \phi:X \rightarrow \tau \) with \( x \in \phi(x) \); a set \( Y \subset X \) is a kernel of \( \phi \) if \( \phi(Y)=\{\phi(y): y \in Y \} \) covers \( X \). The class \( P^* \) consists of those spaces \( X \) such that each neighborhood assignment admits a kernel \( Y \) with property \( P \). In case \( P^*=P \), \( P \) is self-dual.
In this paper, the authors consider the dual of the countable chain condition (CCC), weakly Lindelöf and separable. They show that:
\begin{enumerate}
\item The cardinality of a dually CCC (resp., dually weakly Lindelöf) first countable Hausdorff (resp., normal) space is at most \( 2^\mathfrak{c} \).
\item Assuming \( 2^{<\mathfrak{c}}=\mathfrak{c} \), the extent of a normally dually CCC space with \( \chi (X) \leq \mathfrak{c} \) is at most \( \mathfrak{c} \), where \( \chi (X) \) is a character of \( X \).
\item The extent of a dually separable Hausdorff space with a \( G_\delta^* \)-diagonal is at most \( \mathfrak{c} \).
\item The cardinality (resp., cellularity) of a dually separable regular (resp., dually CCC Hausdorff) space with a \( G_\delta \)-diagonal is at most \( \mathfrak{c} \).\end{enumerate}Resolvability of pseudocompact spaces at a pointhttps://zbmath.org/1472.540132021-11-25T18:46:10.358925Z"Lipin, A. E."https://zbmath.org/authors/?q=ai:lipin.anton-evgenevichSummary: A topological space \(X\) is called resolvable at a point \(x_0\) if \(X\setminus \{x_0\}\) contains two disjoint subsets \(A\), \(B\) such that \(x_0\in \overline{A}\), \(x_0\in \overline{B} \). In this paper we prove that if a regular topological space \(X\) is irresolvable at some non-isolated point \(x_0 \in X\), then \(X\) contains an infinite discrete in \(X\) family \(\mathfrak{{W}}=\{W_\alpha \}\) of non-empty open subsets of \(X\). Therefore, every feebly compact regular space is resolvable at any non-isolated point. Consequently, every pseudocompact space is resolvable at any non-isolated point.
For the entire collection see [Zbl 1467.34001].Toward a quasi-Möbius characterization of invertible homogeneous metric spaceshttps://zbmath.org/1472.540142021-11-25T18:46:10.358925Z"Freeman, David"https://zbmath.org/authors/?q=ai:freeman.david-mandell|freeman.david-f|freeman.david-l|freeman.david-j"Le Donne, Enrico"https://zbmath.org/authors/?q=ai:le-donne.enricoThe paper under review contributes to the ongoing metric characterization of boundaries of rank-one symmetric spaces of non-compact type. These spaces posses boundaries at infinity, metric spaces equipped with visual distances. In the following a boundary of a rank-one symmetric space is abbreviated BROSS.
In the present paper it is conjectured that:
Conjecture 1.1. A metric space is bi-Lipschitz equivalent to some BROSS if and only if it is locally compact, connected, uniformly bi-Lipschitz homogeneous and quasi-invertible.
The article presents a series of theorems that work towards resolving this conjecture. First in terms of Möbius homogenity, then coarse versions in terms of uniformly strongly quasi-Möbius homogenity, and finally two results pertaining to unbounded, proper and disconnected metric spaces are presented.
The following terminology is used to state the theorems. For a set \(X\) and \(p \in X\) let
\[
X_p := X \setminus \{p\} \quad\text{and}\quad \hat{X}:=X\cup\{\infty\}.
\]
A metric space \((X,d)\) is called \emph{invertible} if it is unbounded and admits a homeomorphism \(\tau_p:X_p\to X_p\) (called \emph{inversion at p}) such that for \(x,y \in X_p\):
\[
d(\tau_p(x), \tau_p(y)) = \frac{d(x,y)}{d(x,p)d(y,p)},
\]
and \(\tau_p\) admits a continuous extension to \(\infty \in \hat{X}\). Furthermore define \(\text{Inv}_p(X) := (\hat{X},i_p)\) and \(\text{Sph}_p(X):=(\hat{X},s_p)\), where
\[
i_p(x,y):= \frac{d(x,y)}{d(x,p)d(y,p)},\quad i_p(x,\infty):= \frac{1}{d(x,p)},
\]
\[
s_p(x,y):= \frac{d(x,y)}{(1+d(x,p))(1+d(y,p))},\quad s_p(x,\infty):= \frac{1}{1+d(x,p)}.
\]
\(i_p\) respectively \(s_p\) are metrics if and only if \(X\) is a Ptolemy space. Furthermore a space admits a metric inversion if and only if \(\text{Inv}_p(X)\) is isometric to \(X\).
A homeomorphism \(f:X \to Y\) between (quasi-)metric spaces is called \emph{Möbius} if it preserves the cross-ratio for any quadruples \((a,b,c,d)\) of distinct points in \(X\):
\[
\frac{d(f(a),f(c))d(f(b),f(d))}{d(f(a),f(d))d(f(b),f(c))} = \frac{d(a,c)d(b,d)}{d(a,d)d(b,c)}.
\]
The group of all Möbius self-homeomorphisms of \(X\) is denoted \(\text{Möb}(X)\). A metric space \(X\) is called \emph{\(2\)-point Möbius homogeneous} if for every two pairs \(\{x,y\}, \{a,b\}\) of distinct points in \(X\), there exists an \(f \in \text{Möb}(X)\) with \(f(x) = a\) and \(f(y) = b\).
Coarse versions in the definitions (quasi-invertible, quasi-Möbius, etc.) are established by requiring only bi-Lipschitz equivalence instead of equality in the defining statements.
Given a metric space \((X,d)\) and \(\alpha \in \ ]0,1]\), \((X,d^\alpha)\) is called the \emph{\(\alpha\)-snowflake of \((X,d)\)}.
The main results then are:
Theorem 1.2. Suppose \(X\) is an unbounded, locally compact, complete, and connected metric space. The following statements are equivalent:
\begin{enumerate}
\item \(X\) is Möbius homeomorphic to some BROSS.
\item \(X\) is isometric to some BROSS.
\item \(X\) is isometrically homogeneous and invertible.
\item The sphericalized space \(\text{Sph}_p(X)\) is \(2\)-point Möbius homogeneous, for some (and hence any) \(p \in X\).
\end{enumerate}
This is similar to the main result in [\textit{S. Buyalo} and \textit{V. Schroeder}, Geom. Dedicata 172, 1--45 (2014; Zbl 1362.53061)] but in the present theorem the involution is not required to be fixed point free and does not assume the presence of a Ptolemy circle.
Theorem 1.2. results in
Corollary 1.3. Suppose \(X\) is an unbounded, locally compact, and connected metric space. There exists \(n \in \mathbb{N}\) and \(\alpha \in \ ]0,1]\) such that \(X\) is isometric to \((\mathbb{R}^n, |\cdot|^\alpha)\) if and only if the space \(\text{Sph}_p(X)\) is \(3\)-point Möbius homogeneous, for some/any \(p \in X\).
The consequences of (\(1\)-point) Möbius homogenity are also explored:
Theorem 1.4. Let \(X\) be a compact and quasi-convex metric space of finite topological dimension. If \(X\) is Möbius homogeneous, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Corollary 1.5. Let \(X\) be the boundary of a CAT(\(-1\))-space. If \(X\) is Möbius homogeneous, of finite topological dimension, and connected by Möbius circles, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Coarse versions of the results are then presented:
Proposition 1.6. A proper and unbounded metric space \(X\) is uniformly bi-Lipschitz homogeneous and quasi-invertible if and only if, for some \(p \in X\), the quasi-sphericalized space \(\text{sph}_p(X)\) (the metric space bi-Lipschitz to \(\text{Sph}_p(X)\)) is \(2\)-point uniformly strongly quasi-Möbius homogeneous.
Proposition 1.7. A homeomorphism \(f:X \to Y\) between proper and unbounded metric spaces is strongly quasi-Möbius if and only if it is bi-Lipschitz. Furthermore, \(f\) is Möbius if and only if \(f\) is a similarity.
The article's main contribution towards Conjecture 1.1. is the following
Theorem 1.8. If \(X\) is an unbounded locally compact metric space that is uniformly bi-Lipschitz homogeneous, quasi-invertible, and contains an non-degenerate curve, then \(X\) is path connected, locally path connected, proper, and Ahlfors regular. Furthermore,
\begin{enumerate}
\item if in addition \(X\) contains a cut point, then \(X\) is bi-Lipschitz homeomorphic to \((\mathbb{R}, |\cdot|^\alpha)\), for some \(\alpha \in \ ]0,1]\);
\item if instead \(X\) contains no cut points, then \(X\) is linearly locally connected. Moreover,
\begin{enumerate}
\item if in addition \(X\) contains a non-degenerate rectifiable curve, then \(X\) is annularly quasi-convex;
\item if instead all rectifiable curves in \(X\) are degenerate, then, for some \(\alpha \in \ ]0,1[\), the space \(X\) is bi-Lipschitz homeomorphic to an \(\alpha\)-snowflake.
\end{enumerate}
\end{enumerate}
There is no quasi-Möbius analogue to Corollary 1.3.:
Proposition 1.9. The sphericalized Heisenberg group \(\text{Sph}_e(\mathbb{H}_{\mathbb{C}}^1)\) is \(3\)-point uniformly strongly quasi-Möbius homogeneous. Equivalently, there exists \(L \geq 1\) such that, given any \(x,y \in \mathbb{H}_{\mathbb{C}}^1 \setminus \{e\}\), there exists a \((\lambda, L)\)-quasi-dilation \(f: \mathbb{H}_{\mathbb{C}}^1 \to \mathbb{H}_{\mathbb{C}}^1\) such that \(f(e) = e, f(x)=y\), and \(\lambda = \rho(e,y)/\rho(e,x)\).
Results pertaining to unbounded, proper, and disconnected metric spaces are presented:
Theorem 1.10. Suppose \(X\) is disconnected, unbounded, locally compact, and isometrically homogeneous. If \(X\) is invertible, then there exists \(s>1\) and a positive integer \(N \geq 2\) such that \(X\) is bi-Lipschitz homeomorphic to \((C_N, \rho_s)\).
Here \(C_N\) is the parabolic visual boundary of the \((N+1)\)-regular tree equipped with the path distance with edge length \(1\). \(\rho_s\) is the parabolic visual distance with parameter \(s\).
Theorem 1.11. Suppose \(X\) is a disconnected, unbounded, and locally compact metric space. There exists \(s > 1\) and a positive integer \(N \geq 2\) such that \(X\) is isometric to \((C_N, \rho_s)\) if and only if \(\hat{X}\) is \(3\)-point Möbius homogeneous.
Theorem 1.12. Suppose \(X\) is a disconnected, unbounded, locally compact, and uniformly bi-Lipschitz homogeneous metric space. If \(X\) is quasi-invertible, then \(X\) is quasi-Möbius homeomorphic to \((C_2, \rho_2)\).The localic compact interval is an Escardó-Simpson interval objecthttps://zbmath.org/1472.540152021-11-25T18:46:10.358925Z"Vickers, Steven"https://zbmath.org/authors/?q=ai:vickers.stevenSummary: The locale corresponding to the real interval \([ - 1, 1]\) is an interval object, in the sense of Escardó and Simpson, in the category of locales. The map \(c : 2^\omega \to [-1, 1]\), mapping a stream \(s\) of signs \(\pm 1\) to \({\displaystyle \sum_{i = 1}^\infty s_i 2^{- i}}\), is a proper localic surjection; it is also expressed as a coequalizer. The proofs are valid in any elementary topos with natural numbers object.All projections of a typical Cantor set are Cantor setshttps://zbmath.org/1472.540162021-11-25T18:46:10.358925Z"Frolkina, Olga"https://zbmath.org/authors/?q=ai:frolkina.olga-dIn this interesting and well-written paper the author studies the dimensionality of projections of a typical Cantor set lying in a Euclidean space. It is known due to classical examples of Antoine and (in a more general setting) of Borsuk that in any Euclidean space there exist Cantor sets such that each of their projections onto \(m\)-dimensional planes have maximal dimension \(m\) (\(m\) is fixed here). Further examples of this type have been constructed by \textit{J. Cobb} [Fundam. Math. 144, No. 2, 119--128 (1994; Zbl 0821.54020)], and by the author, e.g. [Topology Appl. 157, No. 4, 745--751 (2010; Zbl 1186.54031)]. In the paper under review, however, the author shows that these examples are untypical in the following sense: for each Euclidean space there exists a \(G_\delta\) (in the Vietoris topology on the set of all non-empty compacta) set of Cantor sets such that each projection of any element of this set to any non-trivial plane is a Cantor set. Besides being interesting in its own right, this result provides a partial answer to the question of Cobb of whether the Cantor sets that raise dimension under all projections are more ``common''.
In the proof of the main theorem, the author uses the completeness of the space of all compacta and the standard technique for such statements of obtaining the desired set as a complement to some closed nowhere dense (or \(F_\sigma\) meager) subsets. Namely, it is shown that the set of all Cantor sets with trivial projections to some non-trivial plane is closed and nowhere dense, and that the set of all Cantor sets whose (non-trivial) projections to some non-trivial plane have isolated points is meager and \(F_\sigma\). These two statements, combined with the fact (proved by the author) that the set of all Cantor sets with all projections being zero-dimensional contains a dense \(G_\delta\) subset, yields the desired result.The shore point existence problem is equivalent to the non-block point existence problemhttps://zbmath.org/1472.540172021-11-25T18:46:10.358925Z"Anderson, Daron"https://zbmath.org/authors/?q=ai:anderson.daronThe main result of the paper under review is the following (see Theorem 1): Every Hausdorff continuum has two or more shore points if and only if every Hausdorff continuum has two or more non-block points if and only if every Hausdorff continuum is coastal at each point. This result combined with an earlier result of the same author [Topology Appl. 218, 42--52 (2017; Zbl 1360.54026)] shows that, under Near Coherence of Filters (NCF), the Stone-Čech remainder of the half line gives an example of a continuum without shore points. Note that by a result of \textit{R. L. Moore} [Proc. Natl. Acad. Sci. USA 9, 101--106 (1923; JFM 49.0143.01)] every metric continuum has at least two shore points. The paper also contains a new characterization of shore points in continua (Theorem 2): Let \(p\) be a point in a continuum \(X\), \(C_p\) be its composant in \(X\), and \(C(X)\) be the hyperspace of subcontinua of \(X\). The point \(p\) is a shore point if and only if there exists a net of subcontinua in \(\{K\in C(X): K\subset C_p\setminus\{p\}\}\) tending to \(X\) in the Vietoris topology.Markov-like set-valued functions on finite graphs and their inverse limitshttps://zbmath.org/1472.540182021-11-25T18:46:10.358925Z"Imamura, Hayato"https://zbmath.org/authors/?q=ai:imamura.hayatoSummary: We introduce Markov-like functions on finite graphs and define the notation of the same pattern between those Markov-like functions. Then we show that two generalized inverse limits with Markov-like bonding functions on finite graphs having the same pattern are homeomorphic. This result gives a generalization of our recent result [\textit{H. Imamura}, Glas. Mat., III. Ser. 53, No. 2, 385--401 (2018; Zbl 1432.54034)].On the \(\cap\)-structure spaceshttps://zbmath.org/1472.540192021-11-25T18:46:10.358925Z"Hashemi, Jamal"https://zbmath.org/authors/?q=ai:hashemi.jamalSummary: The family \(\mathcal{M}_{X}\subseteq\mathcal{P}(X)\) is called an \(\cap\)-structure, when it is closed under the arbitrary intersection. This concept has been studied and considered in algebra, specially in lattices. Using this concept, we define a quasi topological structure which is called \(\cap\)-structure space. By studying this space, we attempt to explain some algebraic concepts through this structure.Common fixed points of Presić operators via simulation functionshttps://zbmath.org/1472.540202021-11-25T18:46:10.358925Z"Alecsa, Cristian Daniel"https://zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: In this paper, using the concept of simulation function, we present existence and uniqueness results for coincidence and common fixed points related to some Presić type operators in the framework of partially ordered metric spaces. Then, some corollaries unifying several existing theorems in the fixed point literature are presented and several examples are given.New type of proximal contractions via implicit simulation functionshttps://zbmath.org/1472.540212021-11-25T18:46:10.358925Z"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Houmani, Hassan"https://zbmath.org/authors/?q=ai:houmani.hassan"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyabSummary: In this paper, we introduce new types of proximal contraction mappings by using a new concept of implicit function, namely the implicit simulation function. Further, we prove best proximity point theorems for such mappings in complete metric spaces. As consequences of our proximal contraction conditions and related results, we also obtain some new fixed point theorems.A generalization of Caristi's fixed point theorem in the variable exponent weighted formal power series spacehttps://zbmath.org/1472.540222021-11-25T18:46:10.358925Z"Bakery, Awad A."https://zbmath.org/authors/?q=ai:bakery.awad-a"El Dewaik, M. H."https://zbmath.org/authors/?q=ai:el-dewaik.m-hSummary: Suppose \((p_n)\) be sequence of positive reals. By \(\mathscr{H}_w((p_n))\), we represent the space of all formal power series \(\sum_{n = 0}^\infty a_n z^n\) equipped with \(\sum_{n = 0}^\infty | \lambda a_n /(n+1)|^{p_n}<\infty \), for some \(\lambda>0\). Various topological and geometric behavior of \(\mathscr{H}_w((p_n))\) and the prequasi ideal constructs by \(s\)-numbers and \(\mathscr{H}_w((p_n))\) have been considered. The upper bounds for \(s\)-numbers of infinite series of the weighted \(n\)-th power forward shift operator on \(\mathscr{H}_w((p_n))\) with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi's fixed point theorem in \(\mathscr{H}_w((p_n))\).Some fixed point theorems for weakly subsequentially continuous and compatible of type (E) mappings with an applicationhttps://zbmath.org/1472.540232021-11-25T18:46:10.358925Z"Beloul, Said"https://zbmath.org/authors/?q=ai:beloul.saidSummary: In this paper, we will establish some fixed point results, for two pairs of self mappings satisfying generalized contractive condition, by using a new concept as weak subsequential continuity, with compatibility of type (E) in metric spaces, as an application the existence of unique solution for a system of functional equations arising in system programming is proved.Common fixed point of multivalued graph contraction in metric spaceshttps://zbmath.org/1472.540242021-11-25T18:46:10.358925Z"Hadian Dehkordi, Masoud"https://zbmath.org/authors/?q=ai:hadian-dehkordi.masoud"Ghods, Masoud"https://zbmath.org/authors/?q=ai:ghods.masoudSummary: In this paper, we introduce the \((G\)-\(\psi)\) contraction in a metric space by using a graph. Let \(F, T\) be two multivalued mappings on \(X\). Among other things, we obtain a common fixed point of the mappings \(F, T\) in the metric space \(X\) endowed with a graph \(G\).Fixed point of generalized weak contraction in \(b\)-metric spaceshttps://zbmath.org/1472.540252021-11-25T18:46:10.358925Z"Iqbal, Maryam"https://zbmath.org/authors/?q=ai:iqbal.maryam"Batool, Afshan"https://zbmath.org/authors/?q=ai:batool.afshan"Ege, Ozgur"https://zbmath.org/authors/?q=ai:ege.ozgur"de la Sen, Manuel"https://zbmath.org/authors/?q=ai:de-la-sen.manuelSummary: In this manuscript, a class of generalized \((\psi,\alpha,\beta)\)-weak contraction is introduced and some fixed point theorems in the framework of \(b\)-metric space are proved. The result presented in this paper generalizes some of the earlier results in the existing literature. Further, some examples and an application are provided to illustrate our main result.An implicit relation approach in metric spaces under \(w\)-distance and application to fractional differential equationhttps://zbmath.org/1472.540262021-11-25T18:46:10.358925Z"Jain, Reena"https://zbmath.org/authors/?q=ai:jain.reena"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh|kumar.santosh.4|kumar.santosh.1|kumar.santosh.3|kumar.santosh.2Summary: The purpose of this work is to introduce a new class of implicit relation and implicit type contractive condition in metric spaces under \(w\)-distance functional. Further, we derive fixed point results under a new class of contractive condition followed by three suitable examples. Next, we discuss results about weak well-posed property, weak limit shadowing property, and generalized \(w\)-Ulam-Hyers stability of the mappings of a given type. Finally, we obtain sufficient conditions for the existence of solutions for fractional differential equations as an application of the main result.Existence of metric fixed points for generalized contractive type mappingshttps://zbmath.org/1472.540272021-11-25T18:46:10.358925Z"Latif, Abdul"https://zbmath.org/authors/?q=ai:latif.abdul"Bin Dehaish, Buthinah A."https://zbmath.org/authors/?q=ai:bin-dehaish.buthinah-abdullatif"Al Rwaily, Asma"https://zbmath.org/authors/?q=ai:al-rwaily.asmaSummary: In this paper, we establish some new results on the existence of metric fixed points for generalized nonlinear contractive mappings. In support of our results some examples are also presented. Our results either improve or generalize many known fixed point results of metric fixed point theory.Fixed point theorems in fuzzy metric spaces for mappings with some contractive type conditionshttps://zbmath.org/1472.540282021-11-25T18:46:10.358925Z"Patir, Bijoy"https://zbmath.org/authors/?q=ai:patir.bijoy"Goswami, Nilakshi"https://zbmath.org/authors/?q=ai:goswami.nilakshi"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayanIn this paper, different fixed point theorems on fuzzy metric spaces with several contractive type mappings and by using an altering distance function are proved. Examples are given to validate the results. The obtained results generalize the results of \textit{N. Wairojjana} et al. [Fixed Point Theory Appl. 2015, Paper No. 69, 19 p. (2015; Zbl 1338.54231)].Fixed points, coupled fixed points and best proximity points for cyclic operatorshttps://zbmath.org/1472.540292021-11-25T18:46:10.358925Z"Petruçsel, Adrian"https://zbmath.org/authors/?q=ai:petrucsel.adrian"Petruçsel, Gabriela"https://zbmath.org/authors/?q=ai:petrucsel.gabrielaSummary: In this paper, we present some fixed point and coupled fixed point theorems for cyclic mappings satisfying to an orbital contraction condition. Some applications to best proximity theory are given. Our results generalize some recent theorems in the literature.Combinatorial equivalences of the Brouwer fixed point theoremhttps://zbmath.org/1472.540302021-11-25T18:46:10.358925Z"Tsai, Feng-Sheng"https://zbmath.org/authors/?q=ai:tsai.feng-sheng"Lee, Shyh-Nan"https://zbmath.org/authors/?q=ai:lee.shyh-nan"Hsu, Sheng-Yi"https://zbmath.org/authors/?q=ai:hsu.sheng-yi"Shih, Mau-Hsiang"https://zbmath.org/authors/?q=ai:shih.mauhsiang|shih.mau-hsiangSummary: Two combinatorial equivalences of the Brouwer fixed point theorem are presented.Periodic and fixed points for Caristi-type \(G\)-contractions in extended \(b\)-gauge spaceshttps://zbmath.org/1472.540312021-11-25T18:46:10.358925Z"Zikria, Nosheen"https://zbmath.org/authors/?q=ai:zikria.nosheen"Samreen, Maria"https://zbmath.org/authors/?q=ai:samreen.maria"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Yeşilkaya, Seher Sultan"https://zbmath.org/authors/?q=ai:yesilkaya.seher-sultanSummary: In this paper, we introduce extended \(b\)-gauge spaces and the extended family of generalized extended pseudo-\(b\)-distances. Moreover, we define the sequential completeness and construct the Caristi-type \(G\)-contractions in the framework of extended \(b\)-gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.Convenient pretopologies on \(\mathbb{Z}^2\)https://zbmath.org/1472.540322021-11-25T18:46:10.358925Z"Šlapal, J."https://zbmath.org/authors/?q=ai:slapal.josefThis is an excellent, ground-breaking paper in digital topology, that includes complete basic definitions of key terms and a number of important results in terms of Jordan curves in the digital plane \(\mathbb{Z}^2\). This paper focuses on pretopologies on the digital plane \(\mathbb{Z}^2\) convenient for studying and processing digital images. It introduces a natural graph on the vertex sets \(\mathbb{Z}^2\), whose cycles are eligible for Jordan curves in the digital plane and solves the problem of finding those cycles that are Jordan curves.
Let \(2^X\) denote the collection of subsets in a nonvoid set \(X\) and the empty set is denoted by \(\emptyset\). Recall that a topology \(p\) on a nonvoid set \(X\) is the Kuratowski closure operator on \(p:2^X\to 2^X\) that satisfies the following axioms:
\begin{enumerate}
\item[(i)] \(p\emptyset = \emptyset\),
\item[(ii)] \(A\subseteq pA\ \mbox{for all}\ A\subseteq X\),
\item[(iii)] \(p(A\cup B) = pA \cup pB\ \mbox{for all}\ A,B\subseteq X\),
\item[(iv)] \(ppA = pA\ \mbox{for all}\ A\subseteq X\).
\end{enumerate}
The pair \((X,p)\) is a pretopological space, provided \(p\) satisfies axioms (i)--(iii), but not necessarily axiom (iv). Also recall that a \emph{digital simple closed curve} in a pretopological space \((\mathbb{Z}^2,p)\) is a nonempty subset \(C\subseteq \mathbb{Z}^2\) such that there are exactly two points of \(C\) adjacent to \(x\) in the connectedness graph of \(p\), for every \(x\in C\). A digital simple closed curve \(C\) in \((\mathbb{Z}^2,p)\) is a digital Jordan curve, provided \(C\) separates the space into precisely two components.
For a planar digital image, let \(A_4\) denote the 4-adjaceny graph on \(\mathbb{Z}^2, (x,y)\in \mathbb{Z}^2\); we have
\[
A_4(x,y) = \left\{\left(x+i,y+j\right): i,j\in \left\{-1,0,1\right\}, ij=0,i+j\neq 0\right\}.
\]
It is observed (and demonstrated) that the basic pretopologies possess a rich variety of Jordan curves useful in structuring the digital plane \(\mathbb{Z}^2\). This paper introduces \(sd\)-pretopology (Definition 3.2) as well as Alexandroff pretopologies \(u\) and \(v\) on \(\mathbb{Z}^2\), leading to a number of important companion results, e.g.,
Theorem 4.4, p. 49
Let \(D\) be a simple closed curve in \((\mathbb{Z}^2,u)\) having more than four points and such that every pair of different points \(z_1,z_2\in D\) with both coordinates even satisfies \(A_4(z_1)\cap A_4(z_2)\subseteq D\). Then \(D\) is a Jordan curve in \((\mathbb{Z}^2,u)\).
Theorem 4.5, p. 50
Let \(D\) be a simple closed curve in \((\mathbb{Z}^2,u)\) such that, for every point \(z\in D\) with both coordinates odd, \(A_4(z)\cap D\neq \emptyset\). Then \(D\) is a Jordan curve in \((\mathbb{Z}^2,v)\).Contraction principle for trajectories of random walks and Cramér's theorem for kernel-weighted sumshttps://zbmath.org/1472.600532021-11-25T18:46:10.358925Z"Vysotsky, Vladislav"https://zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: In 2013 \textit{A. A. Borovkov} and \textit{A. A. Mogulskii} [Theory Probab. Appl. 57, No. 1, 1--27 (2013; Zbl 1279.60037); translation from Teor. Veroyatn. Primen. 57, No. 1, 3--34 (2012)] proved a weaker-than-standard ``metric'' large deviations principle (LDP) for trajectories of random walks in \(\mathbb{R}^d\) whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cramér theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in \(\mathbb{R}^d\).A recent result about random metric spaces explains why all of us have similar learning potentialhttps://zbmath.org/1472.621622021-11-25T18:46:10.358925Z"Servin, Christian"https://zbmath.org/authors/?q=ai:servin.christian"Kosheleva, Olga"https://zbmath.org/authors/?q=ai:kosheleva.olga-m"Kreinovich, Vladik"https://zbmath.org/authors/?q=ai:kreinovich.vladik-yaSummary: In the same class, after the same lesson, the amount of learned material often differs drastically, by a factor of ten. Does this mean that people have that different learning abilities? Not really: experiments show that among different students, learning abilities differ by no more than a factor of two. This fact have been successfully used in designing innovative teaching techniques, techniques that help students realize their full learning potential. In this paper, we deal with a different question: how to explain the above experimental result. It turns out that this result about learning abilities -- which are, due to genetics, randomly distributed among the human population -- can be naturally explained by a recent mathematical result about random metrics.
For the entire collection see [Zbl 1467.62007].Doubly-symmetric periodic orbits in the spatial Hill's lunar problem with oblate secondary primaryhttps://zbmath.org/1472.700272021-11-25T18:46:10.358925Z"Xu, Xingbo"https://zbmath.org/authors/?q=ai:xu.xingboSummary: In this article we consider the existence of a family of doubly-symmetric periodic orbits in the spatial circular Hill's lunar problem, in which the secondary primary at the origin is oblate. The existence is shown by applying a fixed point theorem to the equations with periodical conditions expressed in Poincaré-Delaunay elements for the double symmetries after eliminating the short periodic effects in the first-order perturbations of the approximated system.Fuzzy-genetic approach to epidemiologyhttps://zbmath.org/1472.922162021-11-25T18:46:10.358925Z"Hathiwala, Minakshi Biswas"https://zbmath.org/authors/?q=ai:hathiwala.minakshi-biswas"Chauhan, Jignesh Pravin"https://zbmath.org/authors/?q=ai:chauhan.jignesh-pravin"Hathiwala, Gautam Suresh"https://zbmath.org/authors/?q=ai:hathiwala.gautam-sureshSummary: Mathematics has been used in life ever since the very existence of human beings. Mathematical model involving fuzzy systems has become a powerful tool for better modeling and control of processes in different fields of medical science. Conventional epidemics transmission models are used to interpret the spread of epidemic without immunity. There are major communicable diseases, which can be treated with genetically based intrusions. To understand the cure of these diseases, it is equally important to understand the molecular evolution as it reflects the study of inherent dissipative processes that are complicated in nature, particularly for systems where genetic properties play a noteworthy role. While analyzing the complex networks and description of evolutionary processes, concepts such as neighborhood, similarity, connectedness, or continuity of change appear spontaneously. Characteristically, these concepts are topological in nature. The principal sources for natural selection acting on genetic disparity originate from mutation and/or recombination. The recombination can be treated as a binary operator \(R\) from \(C\times C\) to power set of \(C,P(C)\) on the set of all chromosomes \(C\). If a and b are two chromosomes, then the recombination set \(R(a,b)\) is composed of those chromosomes which are attained by recombining a and b with the help of a certain class of crossover operators. The recombination space can be studied in fuzzy situation by allotting different possibilities of occurrence to each recombinant. Considering fuzzy setting, fuzzy pretopology, in its natural course, is generated in the set of chromosomes. In this chapter, the fuzzy topological features of the recombination space are analyzed. Various crossover models are spontaneously produced in the recombination space, and this can be structured using fuzzy pretopology. In this context, we explore the property of connectedness in fuzzy recombination space in unequal crossover model. Besides that, we examine the property of compactness and separation axioms for the same model.
For the entire collection see [Zbl 1461.92002].