Recent zbMATH articles in MSC 54Chttps://zbmath.org/atom/cc/54C2023-11-13T18:48:18.785376ZWerkzeugStability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaceshttps://zbmath.org/1521.030692023-11-13T18:48:18.785376Z"Khanaki, Karim"https://zbmath.org/authors/?q=ai:khanaki.karimSummary: We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, \textit{Talagrand's stability }, and explain the relationship between this property and the NIP in continuous logic. Using a result of \textit{J. Bourgain} et al. [Am. J. Math. 100, 845--886 (1978; Zbl 0413.54016)], we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula \(\varphi(x,y)\) has the strict order property if and only if there is a convergent sequence of continuous functions on the space of \(\phi\)-types such that its limit is not continuous. We deduce from this a theorem of \textit{S. Shelah} [Ann. Math. Logic 3, 271--362 (1971; Zbl 0281.02052)] and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.On \(p\)-adic semi-algebraic continuous selectionshttps://zbmath.org/1521.031032023-11-13T18:48:18.785376Z"Thamrongthanyalak, Athipat"https://zbmath.org/authors/?q=ai:thamrongthanyalak.athipatSummary: Let \(E \subseteq \mathbb{Q}_p^n\) and \(T\) be a set-valued map from \(E\) to \(\mathbb{Q}_p^m\). We prove that if \(T\) is \(p\)-adic semi-algebraic, lower semi-continuous and \(T (x)\) is closed for every \(x \in E\), then \(T\) has a \(p\)-adic semi-algebraic continuous selection. In addition, we include three applications of this result. The first one is related to \textit{C. Fefferman}'s and \textit{J. Kollár}'s [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)] question on existence of \(p\)-adic semi-algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of \(p\)-adic semi-algebraic continuous extensions of continuous functions. The other application is on the characterization of right invertible \(p\)-adic semi-algebraic continuous functions under the composition.Model-theoretic properties of dynamics on the Cantor sethttps://zbmath.org/1521.031072023-11-13T18:48:18.785376Z"Eagle, Christopher J."https://zbmath.org/authors/?q=ai:eagle.christopher-james|eagle.christopher-j"Getz, Alan"https://zbmath.org/authors/?q=ai:getz.alanThis paper uses the tools of continuous first-order logic to study dynamical systems of the form \(\langle X,s\rangle\), where \(X\) is a Cantor set and \(s\) is a homeomorphism on \(X\). By the Gel'fand duality, the category of compact Hausdorff spaces and continuous maps is contravariantly equivalent to the category of commutative unital \(C^*\)-algebras and unit-preserving algebra homomorphisms. This allows continuous model theory to be applied to structures of the form \(\langle C,\sigma\rangle\), where \(C=C(X)\), is the \(C^*\)-algebra of all continuous complex-valued functions on \(X\) (the Gel'fand dual of \(X\)), and \(\sigma=C(s)\) is the isomorphism on \(C\) induced by \(S\). The main results are the following:
\begin{itemize}
\item[Theorem 1.] The theory of \(\langle C,\sigma\rangle\) does not have a model companion.
\item[Theorem 2.] If two odometers on \(X\) are elementarily equivalent, then they are topologically conjugate.
\item[Theorem 3.] Being a generic homeomorphism is not axiomatizable, but it is expressible as an omitting types property. If \(s\) is the generic homeomorphism of the Cantor set \(X\), then \(\langle C,\sigma\rangle\) is the prime model of its theory.
\end{itemize}
Reviewer: Paul Bankston (Milwaukee)Kirszbraun's theorem via an explicit formulahttps://zbmath.org/1521.470872023-11-13T18:48:18.785376Z"Azagra, Daniel"https://zbmath.org/authors/?q=ai:azagra.daniel"Le Gruyer, Erwan"https://zbmath.org/authors/?q=ai:le-gruyer.erwan-y"Mudarra, Carlos"https://zbmath.org/authors/?q=ai:mudarra.carlosThe problem of extending maps from a smaller to a larger set, keeping their properties, is of utmost importance and has an impact on many application-oriented questions. Thus, \textit{M. D. Kirszbraun} [Fundam. Math. 22, 77--108 (1934; Zbl 0009.03904)] proved that a Lipschitz map \(f: E\to\mathbb{R}^n\) admits an extension from an arbitrary subset \(E\subseteq\mathbb{R}^m\) to a map \(F: \mathbb{R}^m\to\mathbb{R}^n\) with the same minimal Lipschitz constant. Subsequently, this result was generalized to maps between two Hilbert spaces by \textit{F. A. Valentine} [Am. J. Math. 67, 83--93 (1945; Zbl 0061.37507)]. However, these results have the flaw that they are not constructive and not transparent.
In this interesting paper, the authors give an explicit formula for the extension in the Kirszbraun-Valentine theorem and discuss related questions.
Reviewer: Jürgen Appell (Würzburg)Fixed point theorems for some generalized multi-valued nonexpansive mappings in Hadamard spaceshttps://zbmath.org/1521.470892023-11-13T18:48:18.785376Z"Klangpraphan, Chayanit"https://zbmath.org/authors/?q=ai:klangpraphan.chayanit"Panyanak, Bancha"https://zbmath.org/authors/?q=ai:panyanak.banchaSummary: We show that a condition on mappings introduced by \textit{N. Bunlue} and \textit{S. Suantai} [Afr. Mat. 30, No. 3--4, 483--494 (2019; Zbl 1438.47092)] is weaker than the notion of nonexpansive mappings and stronger than the notion of quasi-nonexpansive mappings. We also obtain the demiclosed principle as well as a fixed point theorem for the class of mappings satisfying this condition. A~convergence theorem of the Ishikawa iteration for discontinuous quasi-nonexpansive mappings is also given.\(ps\)-\(ro\) fuzzy strongly \(\alpha\)-irresolute functionhttps://zbmath.org/1521.540032023-11-13T18:48:18.785376Z"Chettri, Pankaj"https://zbmath.org/authors/?q=ai:chettri.pankaj"Chettri, Anamika"https://zbmath.org/authors/?q=ai:chettri.anamikaSummary: The prime objective of this paper is to introduce and characterize a new type of function in a fuzzy topological spaces called ps-ro fuzzy strongly \(\alpha\)-irresolute function. The interrelations of this function with the parallel existing allied concepts are established. The independence of ps-ro fuzzy strongly \(\alpha\)-irresolute and well known concept of fuzzy strongly \(\alpha\)-irresolute function motivate authors to explore it. Also, this function is found to be stronger than ps-ro fuzzy continuity, ps-ro fuzzy semicontinuity, ps-ro fuzzy precontinuity and ps-ro fuzzy \(\alpha\)-continuity. Further, several characterizations of these functions along with different conditions for their existence are obtained.The Vietoris functor and modal operators on rings of continuous functionshttps://zbmath.org/1521.540042023-11-13T18:48:18.785376Z"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Carai, L."https://zbmath.org/authors/?q=ai:carai.luca"Morandi, P. J."https://zbmath.org/authors/?q=ai:morandi.patrick-jIn this paper, the authors introduce an endofunctor \(\mathcal{H}\) on the category \(\textbf{bal}\) of bounded archimedean \(\textit{l}\)-algebras and show that there is a dual adjunction between the category \(\textbf{Alg}(\mathcal{H})\) of algebras for \(\mathcal{H}\) and the category \(\textbf{Coalg}(\mathcal{V})\) of coalgebras for the Vietoris endofunctor \(\mathcal{V}\) on the category of compact Hausdorff spaces. They also introduce an endofunctor \(\mathcal{H}^u\) on the reflective subcategory of \(\textbf{bal}\) consisting of uniformly complete objects of \(\textbf{bal}\) and show that Gelfand duality lifts to a dual equivalence between \(\textbf{Alg}(\mathcal{H}^u)\) and \(\textbf{Coalg}(\mathcal{V})\). Finally, some known results are generalized.
Reviewer: Sami Lazaar (Sidi Daoued)On function spaces equipped with Isbell topology and Scott topologyhttps://zbmath.org/1521.540072023-11-13T18:48:18.785376Z"Xu, Xiaoquan"https://zbmath.org/authors/?q=ai:xu.xiaoquan"Bao, Meng"https://zbmath.org/authors/?q=ai:bao.meng"Zhang, Xiaoyuan"https://zbmath.org/authors/?q=ai:zhang.xiaoyuanThe concept of sobriety of a topological space may be relaxed by requiring only specific irreducible subsets to be (unique) point-closures. The authors of this paper use the framework of irreducible subset systems H to study H-sober spaces (in particular sober spaces, well-filtered spaces and \(d\)-spaces) in relation to their function spaces. For a \(T_0\) space \(X\), it is proved that the following three conditions are equivalent: (1) the Scott space \(\Sigma \mathcal O(X)\) of the lattice of all open sets of \(X\) is H-sober; (2) for every H-sober space \(Y\), the function space \(\mathbb{C}(X, Y)\) of all continuous mappings from \(X\) to \(Y\) equipped with the Isbell topology is H-sober; (3) for every H-sober space \(Y\), the Isbell topology on \(\mathbb{C}(X, Y)\) has property S with respect to H, the latter property stating that if a continuous mapping \(g\) has compatible preimages with an H-set \(\mathscr{F}\) in a certain sense, then \(g\) and \(\mathscr{F}\) have the same closure. Further investigations concern function spaces equipped with the pointwise convergence topology, the compact-open topology and the Scott topology. The authors detect several open questions in this area; to name just one: for a \(T_0\) space \(X\) and a sober space \(Y\), is the function space \(\mathbb{C}(X, Y)\) endowed with the compact-open topology sober?
Reviewer: Alexander Vauth (Lübbecke)Function space of continuous maps from \([0, 1]\) to tree. IIhttps://zbmath.org/1521.540082023-11-13T18:48:18.785376Z"Yang, Hanbiao"https://zbmath.org/authors/?q=ai:yang.hanbiao"Yang, Lin"https://zbmath.org/authors/?q=au:Yang, Lin"Wen, Zhaoying"https://zbmath.org/authors/?q=ai:wen.zhaoying"Lin, Wenhui"https://zbmath.org/authors/?q=ai:lin.wenhui"Jin, Yingying"https://zbmath.org/authors/?q=ai:jin.yingyingAuthors' abstract: For a finite tree \(T\) and its endpoint \textsf{T}, we can define a natural partial order \(\leq\) such that \textsf{T} is the top element in \(T\). For \(X = [0, 1]\) and a continuous map \(f : X \to T\), let \(\downarrow\negthickspace f = \{(x, t) \mid t \leq f(x)\}\) and \(\downarrow\negthickspace \mathrm{C}(X, T) = \{\downarrow\negthickspace f \mid f \text{ is continuous from } X \text{ to } T\}\). We investigate the subspace \(\downarrow\negthickspace \mathrm{C}(X, T)\) of the family of all non-empty closed sets in \(X \times T\) with the Hausdorff distance. Let \(\overline{\downarrow\negthickspace \mathrm{C} (X, T)}\) be the closure of \(\downarrow\negthickspace \mathrm{C}(X, T)\) in \(\operatorname{Cld}(X \times T)\). Let \(\mathcal{E}_v\) be the family of all edges in \(T\) with the upper vertex \(v\), \(V_v\) be the set of lower vertex of all edges in \(\mathcal{E}_v\). For a pair of vertices \(v\) and \(u \in V_v\), define \(T [v, u] = \widehat{vu} \;\cup \downarrow\negthickspace u\). For \(A \in \operatorname{Cld}(X \times T)\) and \(x \in X\), let \(A(x) = \{t \in T \mid (x, t) \in A\}\). In the paper, we show
\(\overline{\downarrow\negthickspace \mathrm{C} (X, T)} = \{A \in \operatorname{Cld}(X \times T) \mid \text{ for all } x \in X\), there exists \(a(x) \in A(x)\) such that either \(A(x) = \downarrow\negthickspace a(x)\) or \(a(x)\) is a vertex and \(A(x) = T [a(x), u]\) for some \(u \in V_{a(x)}\}\).
For Part I of this paper see [\textit{H. Yang} et al., Topology Appl. 301, Article ID 107543, 10 p. (2021; Zbl 1479.54039)].
Reviewer: Sandip Jana (Kolkata)On localization of the Menger propertyhttps://zbmath.org/1521.540102023-11-13T18:48:18.785376Z"Alam, Nur"https://zbmath.org/authors/?q=ai:alam.nur"Chandra, Debraj"https://zbmath.org/authors/?q=ai:chandra.debrajSummary: In this paper we introduce and study the local version of the Menger property, namely locally Menger property (or, locally Menger space). We explore some preservation like properties in this space. We also discuss certain situations where this local property behaves somewhat differently from the classical Menger property. Some observations about the character of a point, network weight and weight in this space are also investigated carefully. We also introduce the notion of Menger generated space (in short, MG-space) and make certain investigations in these spaces. Several topological observations on the decomposition and the remainder of locally Menger spaces are also discussed.The hyperspace of a semi-Eberlein compact space is semi-Eberleinhttps://zbmath.org/1521.540132023-11-13T18:48:18.785376Z"Rojas-Hernández, R."https://zbmath.org/authors/?q=ai:rojas-hernandez.reynaldo"Tenorio, J. F."https://zbmath.org/authors/?q=ai:tenorio.jesus-f"Yescas-Aparicio, C."https://zbmath.org/authors/?q=ai:yescas-aparicio.cLet $T$ be a set. Denote by $c_0([0,1]^T)$ the subspace of $[0,1]^T$ consisting of functions from $T$ to $[0,1]$ taking only a finite subset of $T$ to the half-open interval $(\varepsilon,1]$, for each $\varepsilon>0$. Similarly, denote by $\Sigma[0,1]^T$ the subspace of $[0,1]^T$ (called the \textbf{$\Sigma$-product} of $[0,1]^T$) consisting of functions $f:T\rightarrow [0,1]$ mapping at most a countable number of points of $T$ to the half-open interval $(0,1]$.
A compact space $X$ is \textbf{Eberlein} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow c_0([0,1]^T)$, i.e., $X$ is homeomorphic to a subspace of $c_0([0,1]^T)$. A compact space $X$ is \textbf{semi-Eberlein} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow [0,1]^T$ such that the preimage $f^{-1}(c_0([0,1]^T))$ is dense in $X$. A compact space $X$ is \textbf{Corson} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow\Sigma[0,1]^T$. A compact space $X$ is \textbf{Valdivia} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow[0,1]^T$ such that the preimage $f^{-1}(\Sigma[0,1]^T)$ is dense in $X$.
Given a space $X$, let $K(X)$ denote the \textbf{(hyper)space} of compact subsets of $X$ as a metric space with respect to the Hausdorff distance and let $C_p(X)$ denote the space of real-valued continuous functions on $X$ with respect to the topology of poinwise convergence (i.e., the standard product topology on $\mathbb{R}^X$). Let $\omega$ denote the first infinite ordinal and let $\omega_1$ denote the first uncountable ordinal as a space with respect to the standard \textbf{order topology} (the topology generated by intervals of the form $(a,b)=\{x\in\omega_1:a<x<b\}$, for $a,b\in\omega_1$).
This paper (i) establishes that if $X$ is a semi-Eberlein compact space then so is $K(X)$, in particular, answering an open question, Question 1.3, from [\textit{C. Islas} and \textit{D. Jardon}, Open Math. 13, 188--195 (2015; Zbl 1331.54008)]. Other major results of this paper include (ii) characterization of the transfer of certain structures between $X$ and $C_p(X)$ (answering an open question, Question 5.9, from [\textit{F. Casarrubias-Segura} et al., J. Math. Anal. Appl. 451, No. 2, 1154--1164 (2017; Zbl 1397.54035)]) and (iii) characterization of metrizable subspaces of $\omega_1$ (answering an open question, Question 4.7, from [\textit{V. V. Tkachuk}, Topology Appl. 209, 289--300 (2016; Zbl 1346.54004)]).
{\flushleft \textbf{A note from the authors:}} According to the first sentence on page 9 of this paper, it was proved in [Casarrubias-Segura et al., loc. cit.] that ``\textit{$X$ admits a (full) $c$-skeleton if and only if $C_p(X)$ admits a (full) $q$-skeleton}''. Indeed, by [loc. cit., Proposition 5.4], ``\textit{if $X$ admits a (full) $c$-skeleton then $C_p(X)$ admits a (full) $q$-skeleton}''. Conversely, by [loc. cit., Proposition 5.3], ``\textit{if $X$ admits a (full) $q$-skeleton then $C_p(X)$ admits a (full) $c$-skeleton}''. Applying this to $C_p(X)$ we get ``\textit{if $C_p(X)$ admits a (full) $q$-skeleton then $C_p(C_p(X))$ admits a (full) $c$-skeleton}''. By a comment after [loc. cit., Definition 5.2], subspaces inherit $c$-skeletons and so, since $X$ can be embedded in $C_p(C_p(X))$, it follows that ``\textit{if $C_p(X)$ admits a (full) $q$-skeleton then $X$ admits a (full) $c$-skeleton}''.
Reviewer: Earnest Akofor (Bambili)Approximate fixed points and fixed points for multi-valued almost \(E\)-contractionshttps://zbmath.org/1521.540262023-11-13T18:48:18.785376Z"Hoc, Nguyen Huu"https://zbmath.org/authors/?q=ai:hoc.nguyen-huuSummary: In this paper, we introduce the concept of multi-valued almost \(E\)-contractions. We then present some approximate fixed point and fixed point results for such mappings in metric spaces. Our results generalize and improve several well-known results in literature. We also provide several illustrative examples to compare our findings with some earlier results. An application to homotopy theory is given.Solvability of functional equations via fixed point theorem of \(L\)-fuzzy set-valued mapshttps://zbmath.org/1521.540352023-11-13T18:48:18.785376Z"Zia, Iqra"https://zbmath.org/authors/?q=ai:zia.iqra"Shagari, Mohammed Shehu"https://zbmath.org/authors/?q=ai:shagari.mohammed-shehu"Azam, Akbar"https://zbmath.org/authors/?q=ai:azam.akbarSummary: In this paper, a fixed point theorem for \(L\)-fuzzy set-valued map defined on a complete metric space is established by taking a general contractive inequality. It is shown that a few known significant metric fixed point theorems in the framework of fuzzy set-valued and multivalued mappings can be easily derived from our main result. A~nontrivial example is provided to validate the hypotheses of our established idea. From an application point of view, we study solvability conditions of a class of functional equations arising in dynamic programming by using the techniques of \(L\)-fuzzy set-valued maps.Elements of higher homotopy groups undetectable by polyhedral approximationhttps://zbmath.org/1521.550112023-11-13T18:48:18.785376Z"Aceti, John K."https://zbmath.org/authors/?q=ai:aceti.john-k"Brazas, Jeremy"https://zbmath.org/authors/?q=ai:brazas.jeremyThe main purpose of shape theory is to study spaces that do not have a polyhedral homotopy type (spaces with nontrivial local structure). It is common to consider polyhedral expansions of such spaces (such as Čech expansions) which serve as polyhedral approximations. For a pointed space \((X,x_0)\) an inverse limit of the homotopy pro-group pro-\(\pi_n(X,x_0)\), obtained by acting on its Čech expansion with the homotopy group functor \(\pi_{k}\), is the natural algebraic invariant in shape theory called \(n\)-th Čech (or shape) homotopy group \(\pi_n(X,x_0)\). The kernel \(\ker(\Psi_n)\) of the canonical homomorphism \(\Psi_n:\pi_n(X,x_0)\to \check{\pi}_n(X,x_0)\) is an algebraic measure of the \(n\)-dimensional homotopic deviation of the pointed space \((X,x_0)\) from its polyhedral approximation, i.e. it provides information on those elements of \(\pi_n(X,x_0)\) which cannot be detected by polyhedral approximation of \(X\).
\textit{E. H. Spanier} [Algebraic topology. New York etc.: McGraw-Hill Book Company (1966; Zbl 0145.43303)] characterized the semilocal simple connectivity of a topological space in terms of the vanishing of a certain subgroup of its fundamental group and these groups are named in [\textit{H. Fischer} et al., Topology Appl. 158, No. 3, 397--408 (2011; Zbl 1219.54028)] as the Spanier group. Higher dimensional analogs of Spanier groups (\(n\)-Spanier groups, \(\pi_n^{S_{p}}(X,x_0))\) were introduced, very recently, in [\textit{A. A. Bahredar} et al., Filomat 35, No. 9, 3169--3182 (2021; \url{doi:10.2298/FIL2109169B})] as subgroups of the \(n\)-th homotopy group \(\pi_n(X,x_0)\). In [\textit{J. Brazas} and \textit{P. Fabel}, Rocky Mt. J. Math. 44, No. 5, 1415--1444 (2014; Zbl 1306.57004), Theorem 6.1] it was proved that \(\pi_1^{S_{p}}(X,x_0)=\ker(\Psi_1)\) if \(X\) is paracompact, Hausdorff and locally path connected.
In this paper it is proved that \(\pi_{n}^{S_{p}}(X,x_0)=\ker(\Psi_{n})\), \(n>1\), provided that \(X\) is paracompact, Hausdorff and \(LC^{n-1} \). Furthmore, applying Spanier groups, the authors provide a generalization of Kozlowski-Segal's theorem [\textit{G. Kozlowski} and \textit{J. Segal}, Fundam. Math. 99, 213--225 (1978; Zbl 0396.55008); \textit{H. Fischer} and \textit{A. Zastrow}, ibid. 197, 167--196 (2007; Zbl 1137.55006)] which gives conditions ensuring that \(\Psi_n\) is an isomorphism.
Reviewer: Nikola Koceić-Bilan (Split)Probing the nuclear deformation with three-particle asymmetric cumulant in RHIC isobar runshttps://zbmath.org/1521.814272023-11-13T18:48:18.785376Z"Zhao, Shujun"https://zbmath.org/authors/?q=ai:zhao.shujun"Xu, Hao-jie"https://zbmath.org/authors/?q=ai:xu.hao-jie"Liu, Yu-Xin"https://zbmath.org/authors/?q=ai:liu.yuxin"Song, Huichao"https://zbmath.org/authors/?q=ai:song.huichaoSummary: \(_{44}^{96}\mathrm{Ru}+_{44}^{96}\mathrm{Ru}\) and \(_{40}^{96}\mathrm{Zr}+_{40}^{96}\mathrm{Zr}\) collisions at \(\sqrt{ s_{_{\operatorname{NN}}}} = 200\) GeV provide unique opportunities to study the geometry and fluctuations raised from the deformation of the colliding nuclei. Using iEBE-VISHNU hybrid model, we predict \(\operatorname{ac}_2 \{3 \}\) ratios between these two collision systems and demonstrate that the ratios of \(\operatorname{ac}_2 \{3 \} \), as well as the ratios of the involving flow harmonics and event-plane correlations, are sensitive to quadrupole and octupole deformations, which could provide strong constrains on the shape differences between \(^{96}\mathrm{Ru}\) and \(^{96}\mathrm{Zr}\). We also study the nonlinear response coefficients \(\chi_{4 , 22} \), which show insensitivity to the deformation effect.