Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542024-03-13T18:33:02.981707ZWerkzeugClassifying invariant \(\sigma\)-ideals with analytic base on good Cantor measure spaceshttps://zbmath.org/1528.031952024-03-13T18:33:02.981707Z"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Rałowski, Robert"https://zbmath.org/authors/?q=ai:ralowski.robert"Żeberski, Szymon"https://zbmath.org/authors/?q=ai:zeberski.szymonSummary: Let \(X\) be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel \(\sigma\)-additive measure \(\mu\) which is good in the sense that for any clopen subsets \(U\), \(V\subset X\) with \(\mu (U)<\mu (V)\) there is a clopen set \(W\subset V\) with \(\mu (W)=\mu (U)\). We study \(\sigma\)-ideals with Borel base on \(X\) which are invariant under the action of the group \(\mathcal{H}_\mu (X)\) of measure-preserving homeomorphisms of \((X,\mu )\), and show that any such \(\sigma \)-ideal \(\mathcal {I}\) is equal to one of seven \(\sigma\)-ideals: \(\{\emptyset \}\), \([X]^{\leq \omega }\), \(\mathcal E\), \(\mathcal {M}\cap \mathcal N\), \(\mathcal {M}\), \(\mathcal N\), or \([X]^{\leq \mathfrak{c}}\). Here \([X]^{\leq \kappa }\) is the ideal consisting of subsets of cardinality \(\leq \kappa\) in \(X\), \(\mathcal {M}\) is the ideal of meager subsets of \(X\), \(\mathcal N=\{A\subset X:\mu (A)=0\}\) is the ideal of null subsets of \((X,\mu )\), and \(\mathcal E\) is the \(\sigma \)-ideal generated by closed null subsets of \((X,\mu )\).On ``ternary'' density points in Cantor spacehttps://zbmath.org/1528.031982024-03-13T18:33:02.981707Z"Frankowska, Marta"https://zbmath.org/authors/?q=ai:frankowska.marta"Nowik, Andrzej"https://zbmath.org/authors/?q=ai:nowik.andrzejSummary: We prove inclusions between the density topology on the real line, the density topology on the Cantor set and (defined in this article) ``ternary'' density topology on the Cantor set via standard ``almost injection'' \(\phi : 2^{\mathbb{N}} \to \mathbb{R}\). We also prove some properties of the ``ternary'' density topology.Menger-bounded groups and axioms about filtershttps://zbmath.org/1528.032012024-03-13T18:33:02.981707Z"He, Jialiang"https://zbmath.org/authors/?q=ai:he.jialiang"Tsaban, Boaz"https://zbmath.org/authors/?q=ai:tsaban.boaz"Zhang, Shuguo"https://zbmath.org/authors/?q=ai:zhang.shuguoSummary: A topological group \(G\) is \textit{Menger-bounded} if, for each sequence \(U_1, U_2, \ldots\) of open sets, there are finite sets \(F_1, F_2, \ldots\) such that \(G = \bigcup_n F_n \cdot U_n\). It is \textit{Scheepers-bounded} if all of its finite powers are Menger-bounded. A notorious open problem asks whether, consistently, every product of two Menger-bounded subgroups of the Baer-Specker group \(\mathbb{Z}^{\mathbb{N}}\) is Menger-bounded. We prove that the same assertion for Scheepers-bounded groups is equivalent to the set-theoretic axiom NCF (Near Coherence of Filters). We also show that Menger-bounded \textit{sets} are not productive, and that the preservation of Scheepers-bounded subsets of \([ \mathbb{N} ]^\omega\) by finite-to-one quotients is equivalent to nonexistence of rapid filters.On the structure of acyclic binary relationshttps://zbmath.org/1528.032022024-03-13T18:33:02.981707Z"Alcantud, José Carlos R."https://zbmath.org/authors/?q=ai:alcantud.jose-carlos-rodriguez"Campión, María J."https://zbmath.org/authors/?q=ai:campion.maria-jesus"Candeal, Juan C."https://zbmath.org/authors/?q=ai:candeal.juan-carlos"Catalán, Raquel G."https://zbmath.org/authors/?q=ai:catalan.raquel-g"Induráin, Esteban"https://zbmath.org/authors/?q=ai:indurain.estebanSummary: We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.
For the entire collection see [Zbl 1481.68020].Independent families and some notions of finitenesshttps://zbmath.org/1528.032042024-03-13T18:33:02.981707Z"Hall, Eric"https://zbmath.org/authors/?q=ai:hall.eric-joseph"Keremedis, Kyriakos"https://zbmath.org/authors/?q=ai:keremedis.kyriakosIn the paper under review, the authors investigate the deductive strength of the following well-known theorems relative to the axiom of choice (\(\mathrm{AC}\)) and weaker forms of \(\mathrm{AC}\):
\begin{itemize}
\item The \textit{Fichtenholz-Kantorovich-Hausdorff theorem} (\(\mathrm{FKHT}\)): ``For every infinite set \(X\), \(\mathrm{Id}(X)\)'', where \(\mathrm{Id}(X)\) is ``\(X\) has an independent family of size \(|\wp(X)|\)'';
\item the \textit{Hewitt-Marczewski-Pondiczery theorem} (\(\mathrm{HMPT}\)): ``For every infinite set \(X\), the Cantor cube \(2^{\wp(X)}\) has a dense set of size \(|X|\)'';
\item the \textit{strong Hewitt-Marczewski-Pondiczery theorem} (\(\mathrm{SrmHMPT}\)): ``For every set \(k\) and every family \(\{\langle X_{i},\tau_{i}\rangle:i\in I\}\) of topological spaces such that \(|I|\leq |2^{k}|\) and each \(X_{i}\) has a dense subset of size \(\leq |k|\), the product space \(\prod_{i\in I}X_{i}\) has a dense set of size \(\leq |k|\)''.
\end{itemize}
The authors also determine the relative strengths in \(\mathrm{ZF}\) (Zermelo-Fraenkel set theory minus the \(\mathrm{AC}\)) between \(\neg\mathrm{Id}(X)\) and some of the notions of finiteness studied in \textit{A. Lévy}'s paper [Fundam. Math. 46, 1--13 (1958; Zbl 0089.00702)]. Typical results are:
\begin{itemize}
\item[1.] \(\mathrm{FKHT}\) is not a theorem of \(\mathrm{ZF}+\mathrm{BPI}\) (where \(\mathrm{BPI}\) denotes the Boolean prime ideal theorem: ``Every Boolean algebra has a prime ideal'').
\item[2.] \(\mathrm{FrmKHT}\) is equivalent to \(\mathrm{HMPT}\).
\item[3.] \(\mathrm{ZF}+\mathrm{BPI}+\mathrm{FKHT}\) is consistent with the existence of an infinite, Dedekind-finite set of reals (i.e., an infinite set of reals with no countably infinite subsets). Thus, \(\mathrm{FKHT}\) is not equivalent to \(\mathrm{AC}\) in \(\mathrm{ZF}\).
\item[4.] \(\mathrm{SHMPT}\) is equivalent to \(\mathrm{ArmC}\).
\item[5.] \(\mathrm{ZF}\) \(+\) ``Every Dedekind-finite set is finite'' is consistent with \(\neg\mathrm{FKHT}\).
\item[6.] For any infinite set \(X\), ``\(|X\times X|=|X|\)'' implies \(\mathrm{Id}(X)\), which in turn implies ``\(\wp(X)\) is Dedekind-infinite'' and ``\(|[\wp(X)]^{<\omega}|=|\wp(X)|\)'' (where \([\wp(X)]^{<\omega}\) denotes the set of finite subsets of \(\wp(X)\)).
\item[7.] There exists a model of \(\mathrm{ZF}\) in which there is an infinite set \(X\) with no infinite, linearly orderable subsets (such a set \(X\) is called \textit{\(\Delta_{3}\)-finite} according to \textit{J. Truss} [Fundam. Math. 84, 187--208 (1974; Zbl 0292.02049)]), and for which \(\mathrm{Id}(X)\) is true in the model. Hence, in \(\mathrm{ZF}\), ``\(X\) is \(\Delta_{3}\)-finite'' does not imply \(\neg\mathrm{Id}(X)\).
\item[8.] \(\mathrm{ZF}\) is consistent with the existence of an infinite set \(X\) such that \(2|X|=|X|\) and \(|\wp(X)|\neq |[\wp(X)]^{<\omega}|\).
\item[9.] In \(\mathrm{ZF}\), it is not provable that for every infinite set \(X\), \(2|X|=|X|\) implies \(\mathrm{Id}(X)\).
\end{itemize}
In connection with the above, I include Levy's terminology for certain notions of finiteness addressed in this paper. A set \(X\) is called: \textit{III-finite} if \(\wp(X)\) is Dedekind-finite; \textit{IV-finite} if \(X\) is Dedekind-finite; \textit{V-finite} if \(|X|<2|X|\) or \(|X|=0\); \textit{VI-finite} if \(|X|<|X|^{2}\) or \(|X|\leq 1\). Note also that \(\Delta_{3}\)-finiteness implies IV-finiteness.
Reviewer: Eleftherios Tachtsis (Karlovassi)Sheaves and dualityhttps://zbmath.org/1528.060142024-03-13T18:33:02.981707Z"Gehrke, Mai"https://zbmath.org/authors/?q=ai:gehrke.mai"v. Gool, Samuel J."https://zbmath.org/authors/?q=ai:van-gool.samuel-jSummary: It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.On the commutativity of the powerspace constructionshttps://zbmath.org/1528.060172024-03-13T18:33:02.981707Z"de Brecht, Matthew"https://zbmath.org/authors/?q=ai:de-brecht.matthew"Kawai, Tatsuji"https://zbmath.org/authors/?q=ai:kawai.tatsujiSummary: We investigate powerspace constructions on topological spaces, with a particular focus on the category of quasi-Polish spaces. We show that the upper and lower powerspaces commute on all quasi-Polish spaces, and show more generally that this commutativity is equivalent to the topological property of consonance. We then investigate powerspace constructions on the open set lattices of quasi-Polish spaces, and provide a complete characterization of how the upper and lower powerspaces distribute over the open set lattice construction.Pulling and pushing certain ideals in function ringshttps://zbmath.org/1528.060182024-03-13T18:33:02.981707Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Stephen, Dorca Nyamusi"https://zbmath.org/authors/?q=ai:stephen.dorca-nyamusiSummary: Let \(X\) be a Tychonoff space. Associated with every subset \(S\) of \textit{\( \beta\) X} are the ideals
\[
\boldsymbol{M}^S = \{f \in C(X) \mid S \subseteq \operatorname{cl}_{\beta X} Z(f) \} \text{ and } \boldsymbol{O}^S = \{f \in C(X) \mid S \subseteq \operatorname{int}_{\beta X} \operatorname{cl}_{\beta X} Z(f) \}
\]
of the ring \(C(X)\), where \(Z(f)\) denotes the zero-set of \(f\). We show that \(\langle C(f) [ \boldsymbol{O}^K] \rangle = \boldsymbol{O}^{( \beta f )^{- 1} [ K ]}\) for any continuous map \(f : X \to Y\) and every closed subset \(K\) of \textit{\( \beta Y\)}, where \(\beta f : \beta X \to \beta Y\) is the Stone extension of \(f\) and \(C(f) : C(Y) \to C(X)\) is the ring homomorphism \(g \mapsto g \circ f\). On the other hand, \(C ( f )^{- 1} [ \boldsymbol{M}^S] = \boldsymbol{M}^{( \beta f ) [ S ]}\) for every subset \(S\) of \textit{\( \beta\) X} if and only if \(f\) is a WN-map, in the sense of \textit{R. G. Woods} [J. Lond. Math. Soc., II. Ser. 7, 453--461 (1974; Zbl 0271.54005)]. These results (and others) are corollaries of more general ones obtained in pointfree function rings.Maximal \(d\)-subgroups and ultrafiltershttps://zbmath.org/1528.060232024-03-13T18:33:02.981707Z"Bhattacharjee, Papiya"https://zbmath.org/authors/?q=ai:bhattacharjee.papiya"McGovern, Warren Wm."https://zbmath.org/authors/?q=ai:mcgovern.warren-wmSummary: We study the space \(\operatorname{Max}_d(G)\) of maximal \(d\)-subgroups of a lattice-ordered group, paying specific attention to archimedean \(\ell\)-groups with weak order unit. For such an object \((G,u)\), \(\operatorname{Max}_d(G)\) lays at a level in between the space of minimal prime subgroups and the Yosida space of \((G,u)\). Theorem 5.10 gives the appropriate generalization of a quasi \(F\)-space to \(\mathbf{W}\)-objects which avoids a discussion of \(o\)-complete \(\ell\)-groups.From Freudenthal's spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectrahttps://zbmath.org/1528.060242024-03-13T18:33:02.981707Z"Ball, Richard N."https://zbmath.org/authors/?q=ai:ball.richard-n"Marra, Vincenzo"https://zbmath.org/authors/?q=ai:marra.vincenzo"McNeill, Daniel"https://zbmath.org/authors/?q=ai:mcneill.daniel-k"Pedrini, Andrea"https://zbmath.org/authors/?q=ai:pedrini.andreaSummary: We use a landmark result in the theory of Riesz spaces -- Freudenthal's 1936 spectral theorem -- to canonically represent any Archimedean lattice-ordered group \(G\) with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate \(G\)-indexed zero-dimensional compactification \(w_{G}Z_{G}\) of its space \(Z_{G}\) of \textit{minimal} prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of \(G\) on its space \(X_{G}\) of \textit{maximal} ideals, and the well-known continuous surjection of \(Z_{G}\) onto \(X_{G}\). We then establish our main result by showing that the inclusion-minimal extension of this representation of \(G\) that separates the points of \(Z_{G}\) -- namely, the sublattice subgroup of \(\mathrm{C}(Z_{G})\) generated by the image of \(G\) along with all characteristic functions of clopen (closed and open) subsets of \(Z_{G}\) which are determined by elements of \(G\) -- is precisely the classical projectable hull of \(G\). Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.On a theorem of Anderson and Chunhttps://zbmath.org/1528.130022024-03-13T18:33:02.981707Z"Aliabad, Ali Rezaie"https://zbmath.org/authors/?q=ai:aliabad.ali-rezaie"Farrokhpay, Farimah"https://zbmath.org/authors/?q=ai:farrokhpay.farimah"Siavoshi, Mohammad Ali"https://zbmath.org/authors/?q=ai:siavoshi.m-aAll rings are commutative unital ring. A ring \(R\) is called strongly associate if for each \(a,b\in R\) whenever \(Ra=Rb\) implies that \(a=ub\) for some unit \(u\) of \(R\). \(R\) is called strongly regular associate if whenever \(Ra=Rb\), for \(a,b\in R\), then there exist regular (non-zerodivisor) elements \(r,s\in R\) such that \(a=rb\) and \(b=ra\) (and therefore \(a\) and \(b\) are (strongly) associate in classical quotient of \(R\)). A ring \(R\) has stable range \(1\) if whenever \(a, b\in R\) and \(Ra+Rb=R\), then there exists \(x\in R\), such that \(a+bx\) is a unit in \(R\). Finally a ring \(R\) has regular range \(1\), if whenever \(a, b\in R\) and \(Ra+Rb=R\), then there exists \(x\in R\) such that \(a+bx\) is regular in \(R\) (and there for is a unit in classical quotient ring of \(R\)). The authors proved that the polynomial ring always has regular range \(1\) and each regular range \(1\) ring is a strongly regular associate. Finally the characterized when the ring \(C(X)\) is strongly regular associate or has stable range \(1\).
Reviewer: Alborz Azarang (Ahvāz)Bures-Wasserstein minimizing geodesics between covariance matrices of different rankshttps://zbmath.org/1528.150272024-03-13T18:33:02.981707Z"Thanwerdas, Yann"https://zbmath.org/authors/?q=ai:thanwerdas.yann"Pennec, Xavier"https://zbmath.org/authors/?q=ai:pennec.xavierThe authors consider the set of all positive semidefinite \(n\times n\) matrices (known to statisticians as \textit{covariance matrices}), for any integer \(n\). They equip this set with the Bures-Wasserstein metric, a particular Riemannian metric which is invariant under orthogonal change of basis. They compute the domain of the exponential map, the logarithms, horizontal lifts, and the minimizing geodesics between any two points. They overcome significant and inevitable technicalities, since the space of covariance matrices of given size \(n\) is not a manifold, but is naturally stratified by rank.
Reviewer: Benjamin McKay (Cork)Isomorphisms of semirings of continuous binary relations on topological spaceshttps://zbmath.org/1528.160412024-03-13T18:33:02.981707Z"Vechtomov, E. M."https://zbmath.org/authors/?q=ai:vechtomov.e-m|vechtomov.evgenii-mikhailovichIn this short note, the author proves the following result:
Theorem 1. For any two topological spaces \(X\) and \(Y\), each isomorphism between the semirings \(CR(X)\) and \(CR(Y)\) is induced by a unique homeomorphism between \(X\) and \(Y\).
Reviewer: Kıvanç Ersoy (Berlin)Some network-type properties of the space of G-permutation degreehttps://zbmath.org/1528.180102024-03-13T18:33:02.981707Z"Kočinac, Lj. D. R."https://zbmath.org/authors/?q=ai:kocinac.ljubisa-d-r"Mukhamadiev, F. G."https://zbmath.org/authors/?q=ai:mukhamadiev.farkhod-g"Sadullaev, A. K."https://zbmath.org/authors/?q=ai:sadullaev.a-k"Meyliev, Sh. U."https://zbmath.org/authors/?q=ai:meyliev.sh-uThe study of the influence of normal, weakly normal and seminormal functors to topological and geometric properties of topological spaces, in particular to the cardinal properties (density, weak density, local density, tightness, set tightness, T-tightness, functional tightness, mini-tightness), has been developed in recent investigations in literature.
For example, in [\textit{R. B. Beshimov} et al., Filomat 36, No. 1, 187--193 (2022; \url{doi:10.2298/FIL2201187B})] it was proved that the exponential functor of finite degree preserves the functional tightness and minimal tightness of compact sets, while in [\textit{L. D. R. Kočinac} et al., Mathematics 10, No. 18, 3341 (2022; \url{doi:10.3390/math10183341})] a similar investigation (related to the T-tightness, set tightness, functional tightness, mini-tightness) was done for the functor \({SP^n}_G\) of \(G\)-permutation degree. The authors mention here that tightness-type properties of function spaces with the compact-open topology have been studied previously in [\textit{L. D. R. Kočinac}, Appl. Gen. Topol. 4, No. 2, 255--261 (2003; Zbl 1055.54007)].
In this paper, they study the behavior of the network-type properties of topological spaces under the influence of the functor of \(G\)-permutation degree. They prove that this functor preserves the network, cs-network, cs*-network, cn-network and ck-network of topological spaces.
Recall that the concept of functor of \(G\)-permutation degree was first introduced by \textit{V. V. Fedorchuk} [Russ. Math. Surv. 36, No. 3, 211--233 (1981; Zbl 0495.54008); translation from Usp. Mat. Nauk 36, No. 3(219), 177--195 (1981)] and \textit{V. V. Fedorchuk} and \textit{V. V. Filippov} [Topologiya giperprostranstv i ee prilozheniya (Russian). Moskva: Znanie (1989; Zbl 1081.54500)].
Reviewer: Cenap Özel (Jeddah)Representing structured semigroups on étale groupoid bundleshttps://zbmath.org/1528.201032024-03-13T18:33:02.981707Z"Bice, Tristan"https://zbmath.org/authors/?q=ai:bice.tristan-matthewSummary: We examine a semigroup analogue of the Kumjian-Renault representation of \(C^*\)-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as `slice-sections' of groupoid bundles.Graev ultrametrics and free products of Polish groupshttps://zbmath.org/1528.220022024-03-13T18:33:02.981707Z"Slutsky, Konstantin"https://zbmath.org/authors/?q=ai:slutsky.konstantinSummary: We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups \(G\) and \(H\) can be embedded into a Polish group \(T\) in such a way that the subgroup of \(T\) generated by \(G\) and \(H\) is isomorphic to the free product \(G * H\).Definable continuous mappings and Whyburn's conjecturehttps://zbmath.org/1528.260112024-03-13T18:33:02.981707Z"Dinh, Sĩ Tiệp"https://zbmath.org/authors/?q=ai:dinh.si-tiep"Phạm, Tien-Son"https://zbmath.org/authors/?q=ai:pham-tien-son.In this paper, the authors consider the problem of finding necessary and sufficient conditions for a continuous mapping to be open. They deal with the case of a definable continuous mapping \(f :\Omega \to \mathbb{R}^n\), where \(\Omega\) is a definable connected open set in \(\mathbb{R}^n\).
Definable mappings are related with o-minimal structures in \(\mathbb{R}^n\) that are sequences of Boolean algebras of subsets of \(\mathbb{R}^n\), \(D := (D_n)_{n\in N}\), such that for each \(n\in N\):
\begin{itemize}
\item[a)] If \(X\in D_m\) and \(Y\in D_n\), then \(X \times Y\in D_{m+n}\).
\item[b)] If \(X\in D_(n+1)\), then \(\pi(X)\in D_n\), where \(\pi: \mathbb{R}^{n+1} \to \mathbb{R}^n\) is the projection on the \(n\) first coordinates.
\item[c)] \(D_n\) contains all algebraic subsets of \(\mathbb{R}^n\).
\item[d)] Each set belonging to \(D_1\) is a finite union of points and intervals.
\end{itemize}
A set belonging to \(D\) is said to be definable with respect to this structure and definable mappings in \(D\) are mappings whose graphs are definable sets in \(D\).
For a mapping of an open subset of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), denote: \(D_f\) the set of points at which \(f\) is differentiable, \(R_f\) the set of points \(x\) such that \(f\) is of class \(C^1\) in a neihborhood of \(x\) and the Jacobian \(Jf(x)\) is nonzero and \(B_f\) the set of points at which \(f\) fails to be a local homeomorphism.
The result proved by the authors is the following:
Let \(f :\Omega\to \mathbb{R}^n\) be a definable continuous mapping, where \(\Omega\) is a definable connected open set in \(\mathbb{R}^n\). Then the following conditions are equivalent:
\begin{itemize}
\item[i)] The mapping \(f\) is open.
\item[ii)] The fibers of \(f\) are finite and the Jacobian \(Jf\) does not change sign on \(D_f\).
\item[iii)] The fibres of \(f\) are finite and the Jacobian \(Jf\) does not change sign on \(R_f\).
\item[iv)] The fibres of \(f\) are finite and the set \(B_f\) has dimension at most \(n-2\).
\end{itemize}
As an application, they prove that Whyburn's conjecture is true for definable mappings. Writing \(B_r^n\) and \(S_r^n\) for the closed ball and the sphere of radius \(r\) centered at the origin, respectively, one has:
Let \(f : B_r^n \to B_s^n\) be a definable surjective open continuous mapping such that \(f^{-1}(S_s^{n-1}) = S_r^{n-1}\) and the restriction of \(f\) to \(S_r^{n-1}\) is a homeomorphism. Then \(f\) is a homeomorphism.
Reviewer: Julià Cufí (Bellaterra)Minkowski dimension for measureshttps://zbmath.org/1528.280072024-03-13T18:33:02.981707Z"Falconer, Kenneth J."https://zbmath.org/authors/?q=ai:falconer.kenneth-j"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-m"Käenmäki, Antti"https://zbmath.org/authors/?q=ai:kaenmaki.anttiLet \(X\) be a compact metric space and \(\mathcal{M}\) be the set of fully supported finite Borel measures on \(X\). In the article, the authors introduce the notion of the upper \(\overline{\dim}_M(\mu)\) and the lower \(\underline{\dim}_M(\mu)\) Minkowski dimension for a measure \(\mu\) and also consider other notions of dimension for measures such as:
the upper and lower packing dimensions \(\overline{\dim}_P(\mu)\), \(\underline{\dim}_P(\mu)\);
the density dimension \(\dim_\Theta(\mu)\);
the \(L^q\)-dimension \(\dim_{L^q}(\mu)\) for \({-\infty<q<1}\);
the Assouad dimension \(\dim_A(\mu)\);
the Assouad spectrum \(\dim^\theta_A(\mu)\) for \({0<\theta<1}\);
the lower spectrum \(\dim^\theta_L(\mu)\) for \({0<\theta<1}\) and
the Frostman dimension \(\dim_F(\mu).\)
The article discusses the connections between them and between their dimension counterparts for sets. Particularly, the following is presented:
\[
\overline{\dim}_M(X) = \min\{\overline{\dim}_M(\mu) : \mu \in\mathcal{M}\},
\]
\[
\underline{\dim}_M(X) = \min\{\underline{\dim}_M(\mu) : \mu \in\mathcal{M}\};
\]
\[
\dim_\Theta(\mu)\leq\overline{\dim}_M(\mu)\text{ with an example for strict inequality};
\]
\[
\dim_P(X)=\min\{\dim_\Theta(\mu) : \mu\in\mathcal{M}\}\text{ for X with doubling property};
\]
\[\overline{\dim}_P(\mu)\leq{\dim}_{L^q}(\mu)\leq\overline{\dim}_M(\mu)\leq{\dim}^\theta_A(\mu)\leq\min\left\{{\dim}_A(\mu),\frac{\overline{\dim}_M(\mu)}{1-\theta}\right\};
\]
\[
\overline{\dim}_M(\mu)=\sup\limits_{-\infty<q<1}{\dim}_{L^q}(\mu)=\lim\limits_{q\to-\infty}{\dim}_{L^q}(\mu);
\]
\[
{\dim}^\theta_L(\mu)\leq{\dim}_F(\mu).
\]
Reviewer: Ivan Podvigin (Novosibirsk)Erdős properties of subsets of the Mahler set \(S\)https://zbmath.org/1528.280282024-03-13T18:33:02.981707Z"Chalebgwa, Taboka Prince"https://zbmath.org/authors/?q=ai:chalebgwa.taboka-prince"Morris, Sidney A."https://zbmath.org/authors/?q=ai:morris.sidney-aSummary: Erdős proved that every real number is the sum of two Liouville numbers. A set \(W\) of complex numbers is said to have the Erdős property if every real number is the sum of two members of \(W\). Mahler divided the set of all transcendental numbers into three disjoint classes \(S\), \(T\) and \(U\) such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set \(U\) and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if \(m\in [0,\infty)\), then there exist \(2^{\mathfrak{c}}\) dense subsets \(W\) of \(S\) each of Lebesgue measure \(m\) such that \(W\) has the Erdős property and no two of these \(W\) are homeomorphic. It is also proved that there are \(2^{\mathfrak{c}}\) dense subsets \(W\) of \(S\) each of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are \(2^{\mathfrak{c}}\) dense subsets \(W\) of \(S\) such that every complex number is the sum of two members of \(W\) and such that no two of these \(W\) are homeomorphic.Functional and differential inequalities and their applications to control problemshttps://zbmath.org/1528.340142024-03-13T18:33:02.981707Z"Benarab, S."https://zbmath.org/authors/?q=ai:benarab.sarra"Zhukovskaya, Z. T."https://zbmath.org/authors/?q=ai:zhukovskaya.zukhra-tagurovna|zhukovskaya.zukhra-tagirovna"Zhukovskiy, E. S."https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-s"Zhukovskiy, S. E."https://zbmath.org/authors/?q=ai:zhukovskiy.s-eSummary: We study boundary value and control problems using methods based on results on operator equations in partially ordered spaces. Sufficient conditions are obtained for the existence of a coincidence point for two mappings acting from a partially ordered space into an arbitrary set, an estimate for such a point is found, and corollaries about a fixed point for a mapping that acts in a partially ordered space and is not monotone are derived. The established results are applied to the study of functional and differential equations. For the Nemytskii operator in the space of measurable vector functions, sufficient conditions for the existence of a fixed point are obtained and it is shown that these conditions do not follow from the classical fixed point theorems. Assertions on the existence and estimates of the solution of the Cauchy problem are proved, and the solutions are given to a periodic boundary value problem and a control problem for systems of ordinary differential equations of the first order unsolved for the derivative of the desired vector function.Branched coverings of the sphere having a completely invariant continuum with infinitely many Wada lakeshttps://zbmath.org/1528.370202024-03-13T18:33:02.981707Z"Iglesias, J."https://zbmath.org/authors/?q=ai:iglesias.jorge"Portela, A."https://zbmath.org/authors/?q=ai:portela.aldo"Rovella, A."https://zbmath.org/authors/?q=ai:rovella.alvaro"Xavier, J."https://zbmath.org/authors/?q=ai:xavier.juliana-cSummary: We construct a family of smooth branched coverings of degree 2 of the sphere \(S^2\) having a completely invariant indecomposable continuum \(K\) and infinitely many Wada Lakes.On the statistical convergence of nested sequences of setshttps://zbmath.org/1528.400022024-03-13T18:33:02.981707Z"Albayrak, H."https://zbmath.org/authors/?q=ai:albayrak.huseyin"Babaarslan, F."https://zbmath.org/authors/?q=ai:babaarslan.funda"Ölmez, Ö."https://zbmath.org/authors/?q=ai:olmez.oznur"Aytar, S."https://zbmath.org/authors/?q=ai:aytar.salihSummary: In this paper, we show that Wijsman convergence and statistical Wijsman convergence are equivalent to each other if we choose the sequences of sets as monotone. Then, we show that every statistical Wijsman convergent monotone sequence of sets is not only Hausdorff convergent but also statistical Hausdorff convergent to the same set.Smooth approximation of mappings with rank of the derivative at most 1https://zbmath.org/1528.410432024-03-13T18:33:02.981707Z"Goldstein, Paweł"https://zbmath.org/authors/?q=ai:goldstein.pawel"Hajłasz, Piotr"https://zbmath.org/authors/?q=ai:hajlasz.piotrSummary: It was conjectured that if \(f\in C^1(\mathbb{R}^n, \mathbb{R}^n)\) satisfies \(\mathrm{rank}Df\leq m < n\) everywhere in \(\mathbb{R}^n\), then \(f\) can be uniformly approximated by \(C^\infty\)-mappings \(g\) satisfying \(\mathrm{rank}Dg\leq m\) everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when \(m = 1\). More precisely, if \(m = 1\), our result yields an almost-uniform approximation of locally Lipschitz mappings \(f: \Omega\rightarrow\mathbb{R}^n\), satisfying \(\mathrm{rank}Df\leq 1\) a.e., by \(C^\infty\)-mappings \(g\) with \(\mathrm{rank}Dg\leq 1\), provided \(\Omega\subset\mathbb{R}^n\) is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees (\(\mathbb{R}\)-trees).On the transfinite symmetric strong diameter two propertyhttps://zbmath.org/1528.460072024-03-13T18:33:02.981707Z"Ciaci, Stefano"https://zbmath.org/authors/?q=ai:ciaci.stefanoA Banach space \(X\) has the symmetric strong diameter two property (SSD2P) if, for every \(x_1^*,x_2^*,\ldots,x_n^* \in S_{X^*}\) and \(\varepsilon > 0\), there are \(x_1,\ldots,x_n,y \in B_X\) such that \(\|y\| \ge 1 - \varepsilon\), \(x_i \pm y \in B_X\) and \(x_i(x_i) \ge 1 - \varepsilon\) for all \(1 \le i \le n\).
In this paper, the author studies generalizations of the SSD2P where, instead of a set of finite cardinality we start with a larger set. These generalizations with slightly different notation first appeared in [\textit{A. Avilés} et al., Stud. Math. 271, No. 1, 39--63 (2023; Zbl 1528.46004)].
Let \(r \in (0,1)\), \(B \subset B_X\) and \(A \subset S_{X^*}\). We say that \(B\) \(r\)-norms \(A\) if, for every \(x^* \in A\), there is an \(x \in B\) such that \(x^*(x) \ge r\). If this holds for all \(r \in (0,1)\), \(B\) norms \(A\).
For a Banach space \(X\) and an infinite cardinal \(\kappa\), we say that \(X\) has the SSD2P\(_\kappa\) if, for every set \(A\subset S_{X^*}\) of cardinality \(< \kappa\) and \(\varepsilon > 0\), there are \(B \subset B_X\) which \((1-\varepsilon)\)-norms \(A\), and \(y \in B_X\) with \(\|y\| \ge 1 - \varepsilon\) satisfying \(B \pm y \in B_X\).
A \(1\)-norming and attaining version of the SSD2P\(_\kappa\) is also studied: \(X\) has the \(1\)-ASSD2P\(_\kappa\) if, for every set \(A \subset S_{X^*}\) of cardinality \(< \kappa\), there are \(B \subset S_X\) which norms \(A\), and \(y \in S_X\) satisfying \(B\pm y \subset S_X\).
Let \(X\) and \(Y\) be Banach spaces and \(\kappa > \aleph_0\). It is shown that \(X\) or \(Y\) has the SSD2P\(_\kappa\) if and only if \(X \oplus_\infty Y\) has the SSD2P\(_\kappa\). General \(\infty\)-sums are also studied. It is also shown that, if \(X\) and \(Y\) have the SSD2P\(_\kappa\), then the projective tensor product \(X \hat{\otimes}_\pi Y\) has the SSD2P\(_\kappa\). However, there are characterizations of the SSD2P which cannot be transferred to the transfinite setting and the proof of the latter result is different from the proof for the SSD2P. Positive results are given for \(C_0(X)\) spaces where \(X\) is a locally compact Hausdorff space, but even in this case the situation is not entirely clear.
Reviewer: Vegard Lima (Kristiansand)Generalized Sierpinski functions and fragmentable compact spaces.https://zbmath.org/1528.460112024-03-13T18:33:02.981707Z"Matsuda, Minoru"https://zbmath.org/authors/?q=ai:matsuda.minoru(no abstract)On the property (C) of Corson and other sequential properties of Banach spaceshttps://zbmath.org/1528.460132024-03-13T18:33:02.981707Z"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Poveda, Alejandro"https://zbmath.org/authors/?q=ai:poveda.alejandroThis paper deals with several properties of Banach spaces whose definition involves the analysis of their convex subsets. A Banach space is said to have \textit{Corson's property (C)} if for every family of closed convex subsets with empty intersection, there exists a countable subfamily that already has empty intersection. The authors consider (at least) two more properties, analogous to the topological characterization of closed sets and points in the closure by means of convergent sequences, but involving convex subsets. A space has \textit{property \(\mathcal E\)} if for convex subsets one can characterize belonging to the weak\(^*\)-closure by means of being the weak\(^*\)-limit of a sequence in the set; on the other hand, a space has \textit{property \(\mathcal E'\)} if weak\(^*\)-sequentially closed convex subsets of the dual ball are weak\(^*\)-closed (thus it is not hard to see that property \(\mathcal E'\) is a weakening of property \(\mathcal E\)).
After the introduction (the first section), the second section of the present paper contains the proof that property \(\mathcal E'\) implies Corson's property (C); this section also features a proof of a result (Theorem~3 in the paper), stemming from the PhD thesis of the first author, that property \(\mathcal E'\) implies that the dual ball is weak\(^*\)-block compact. Finally, the third section contains one of the core results of the paper, namely that under \(\mathsf{PFA}\) Corson's property (C) implies that the unit ball of the dual, equipped with the weak\(^*\) topology, has countable tightness. It remains open whether such an implication holds in \(\mathsf{ZFC}\) alone.
At the end of the paper, putting all of the results therein together with a result of \textit{Z.~Balogh} [Proc. Am. Math. Soc. 105, No.~3, 755--764 (1989; Zbl 0687.54006)], the conclusion is that four conditions are equivalent under \(\mathsf{PFA}\): Corson's property (C), property \(\mathcal E\), the unit ball of the dual having countable tightness under the weak\(^*\) topology, and having a weak\(^*\) sequential dual ball. It is worth noting that this combines with a result of \textit{C.~Brech} [Construções genéricas de espaços de Asplund \(C(K)\). Universidade de São Paulo and Université Paris VII (PhD Thesis) (2008)], claiming the consistency that property (C) does not imply property \(\mathcal E\) -- but establishing, in fact, that property (C) does not imply property \(\mathcal E'\). Therefore one may conclude that the statement that property (C) implies property \(\mathcal E'\) is independent of \(\mathsf{ZFC}\).
Reviewer: David J. Fernández-Bretón (Ciudad de México)Compact Hölder retractions and nearest point mapshttps://zbmath.org/1528.460182024-03-13T18:33:02.981707Z"Medina, Rubén"https://zbmath.org/authors/?q=ai:medina.rubenSummary: In this paper, an \(\alpha\)-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive \(\alpha < 1\). This constitutes a positive solution to a Hölder version of a question raised by \textit{G.~Godefroy} and \textit{N.~Ozawa} [Proc. Am. Math. Soc. 142, No.~5, 1681--1687 (2014; Zbl 1291.46013)].
In fact, compact convex sets are found to be absolute \(\alpha\)-Hölder retracts under certain assumption of flatness.Universality theorems for asymmetric spaceshttps://zbmath.org/1528.460212024-03-13T18:33:02.981707Z"Alimov, A. R."https://zbmath.org/authors/?q=ai:alimov.alexey-rSummary: Spaces with asymmetric metric and asymmetric norm are considered. It is shown that any metrizable separable asymmetrically normed linear space \((X,\Vert\cdot|)\) can be isometrically isomorphically imbedded, as an affine linear manifold, into the classical space \(C[0,1]\) with uniform norm \(\Vert\cdot \Vert_C\). A similar result is obtained for spaces of density \(\mathfrak{a}\). For spaces with asymmetric metric, it is shown that each such space of density \(\mathfrak{a}\) is isometric to a part of the space \(C([0,1]^{\mathfrak{a}})\) with the asymmetric seminorm \(p(f)=\Vert f_+ \Vert_C\), where \(f_+ (t)=\max\{f(t),0\}\).Fuzzy multivalued mappings and stability of fixed point setshttps://zbmath.org/1528.470072024-03-13T18:33:02.981707Z"Bhandari, Samir Kumar"https://zbmath.org/authors/?q=ai:bhandari.samir-kumar"Chandok, Sumit"https://zbmath.org/authors/?q=ai:chandok.sumitSummary: In this article, we introduce multivalued fuzzy contraction mappings and use the fuzzy nearest point property to demonstrate the existence of a fixed point. We derive a result by defining fuzzy fixed point set stability for multivalued fuzzy contractions. An example is also included to demonstrate the usability of the main result.Multi-valued graph contraction principle with applicationshttps://zbmath.org/1528.470082024-03-13T18:33:02.981707Z"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://zbmath.org/authors/?q=ai:petrusel.gabriela"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: The aim of this paper is to present a generalization of Nadler's fixed point principle for the case of multi-valued graph contractions. Also, a strict fixed point theorem is obtained. Connections to some variational analysis concepts are discussed and applications to generalized coupled fixed point problems are given.Fixed point theory and variational principles in metric spaceshttps://zbmath.org/1528.540012024-03-13T18:33:02.981707Z"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Sahu, D. R."https://zbmath.org/authors/?q=ai:sahu.daya-ramThe book under review is devoted to metric aspects of fixed point theory, an important part of contemporary mathematics lying at the interface of nonlinear functional analysis and topology and finding numerous interesting applications in the theory of differential equations, optimization, game theory, mathematical economics and many other sciences.
The first chapter is introductory. It contains basic definitions, notions and facts from the theory of metric spaces. The subject of the second chapter is the Banach Contraction Principle and some of its versions. A characterization of the completeness of a space in terms of the Banach Contraction Principle is presented. Some extensions, such as the Boyd-Wong fixed point theorem for \(\psi\)-contractions, a fixed point theorem for weakly contraction maps and Caristi's fixed point theorem are considered.
The third chapter deals with set-valued mappings. Here various concepts of continuity for set-valued maps are discussed. The set-valued fixed point theory is presented with Nadler's generalization of the Banach fixed point theorem, fixed point results for directional contractions, dissipative maps, \(\Psi\)-contractions and weak contractions.
The fourth chapter starts with various versions of Ekeland's variational principle. The applications to Banach's, Caristi's, Clarke's and other fixed point theorems for single-valued and set-valued maps are considered. The equivalence of the Ekeland's variational principle with Takahashi's minimization theorem and the Caristi-Kirk theorem is demonstrated.
In the fifth chapter the authors describe the following equilibrium problem: to find an element \(\overline{x}\) of a set \(K\) such that \(F(\overline{x},y) \geq 0\) for all \(y \in K\), where \(F \colon K \times K \to \mathbb{R}\) is a function such that \(F(x,x) = 0\) for all \(x \in K\). The equilibrium (or extended) version of Ekeland's variational principle is studied and several equivalent versions such as the extended Takahashi's minimization theorem, the Caristi-Kirk fixed point theorem and others are presented. The concept of weak sharp solutions for equilibrium problems is discussed.
The contents of the sixth chapter are applications of the Banach contraction principle and its versions to systems of linear equations, differential equations, second order two-point boundary value problems and to various classes of integral equations.
Endowed with examples and exercises, the book may serve as a good introduction to the subject for graduate and postgraduate students and it can be useful also for researchers working in the sphere of nonlinear analysis and its applications.
Reviewer: Valerii V. Obukhovskij (Voronezh)Large strongly anti-Urysohn spaces existhttps://zbmath.org/1528.540022024-03-13T18:33:02.981707Z"Juhász, István"https://zbmath.org/authors/?q=ai:juhasz.istvan"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharon"Soukup, Lajos"https://zbmath.org/authors/?q=ai:soukup.lajos"Szentmiklóssy, Zoltán"https://zbmath.org/authors/?q=ai:szentmiklossy.zoltanRecall that a topological space \(X\) is called Urysohn if for any \(x,y\in X\) with \(x\neq y\), there are open sets \(U,V\subseteq X\) such that \(x\in U\), \(y\in V\), and \(\overline{U}\cap\overline{V}=\emptyset\). A Hausdorff space \(X\) is called anti-Urysohn if for any nonempty open sets \(U,V\subseteq X\), \(\overline{U}\cap\overline{V}\neq\emptyset\). A Hausdorff space \(X\) is called strongly anti-Urysohn if it has at least two non-isolated points and any two infinite closed subsets intersect. In [\textit{I. Juhász} et al., Topology Appl. 213, 8--23 (2016; Zbl 1352.54004)], it was asked (a) if there is a strongly anti-Urysohn space in ZFC, and (b) if it is consistent that there is a strongly anti-Urysohn space of cardinality \(>\mathfrak c\). In the paper under review, the authors answer these questions affirmatively.
Reviewer: Akira Iwasa (Big Spring)Projections of inverse systems.https://zbmath.org/1528.540032024-03-13T18:33:02.981707Z"Chiba, Keiko"https://zbmath.org/authors/?q=ai:chiba.keiko(no abstract)Remarks on special refinements.https://zbmath.org/1528.540042024-03-13T18:33:02.981707Z"Chiba, Keiko"https://zbmath.org/authors/?q=ai:chiba.keiko"Hayata, Tetsuya"https://zbmath.org/authors/?q=ai:hayata.tetsuya(no abstract)A generalization of de Vries duality to closed relations between compact Hausdorff spaceshttps://zbmath.org/1528.540052024-03-13T18:33:02.981707Z"Abbadini, Marco"https://zbmath.org/authors/?q=ai:abbadini.marco"Bezhanishvili, Guram"https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Carai, Luca"https://zbmath.org/authors/?q=ai:carai.lucaIn his well-known paper published in 1936, Stone established the famous Stone duality which states that the category \(\mathsf{Stone}\) of Stone spaces (zero-dimensional compact Hausdorff spaces) and continuous maps is dually equivalent to the category \(\mathsf{BA}\) of boolean algebras and boolean homomorphisms. In [\textit{H. de Vries}, Compact spaces and compactifications. An algebraic approach. University of Amsterdam (PhD thesis) (1962)], de Vries generalized Stone duality to a duality for the category \(\mathsf{KHaus}\) of compact Hausdorff spaces and continuous maps. The objects of the dual category \(\mathsf{DeV}\) are complete boolean algebras equipped with a proximity relation, known as de Vries algebras. The morphisms of \(\mathsf{DeV}\) are functions satisfying certain conditions. A major drawback of \(\mathsf{DeV}\) is that composition of morphisms is not the usual function composition.
In this paper, the authors propose an alternative approach to de Vries duality, where morphisms between de Vries algebras become certain relations and composition is the usual relation composition. In this way, Stone duality generalizes to an equivalence between the categories \(\mathsf{Stone}^{\mathrm{R}}\) of Stone spaces and closed relations and \(\mathsf{BA}^{\mathrm{S}}\) of boolean algebras and subordination relations. This equivalence is in fact an equivalence of allegories, hence self-dual categories. Splitting equivalences in \(\mathsf{Stone}^{\mathrm{R}}\) yields a category that is equivalent to the category \(\mathsf{KHaus}^{\mathrm{R}}\) of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in \(\mathsf{BA}^{\mathrm{S}}\) yields a category that is equivalent to the category \(\mathsf{DeV}^{\mathrm{S}}\) of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields the equivalence between \(\mathsf{KHaus}^{\mathrm{R}}\) and \(\mathsf{DeV}^{\mathrm{S}}\), thus resolving a problem recently raised in [\textit{G. Bezhanishvili} et al., Appl. Categ. Struct. 27, No. 6, 663--686 (2019; Zbl 1437.54021)]. The equivalence between \(\mathsf{KHaus}^{\mathrm{R}}\) and \(\mathsf{DeV}^{\mathrm{S}}\) is further restricted to an equivalence between the category \(\mathsf{KHaus}\) of compact Hausdorff spaces and continuous functions and the wide subcategory \(\mathsf{DeV}^{\mathrm{F}}\) of \(\mathsf{DeV}^{\mathrm{S}}\) whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is the usual relation composition.
Reviewer: Xiaoquan Xu (Zhangzhou)Semi-linearity on spaces of set-valued functionshttps://zbmath.org/1528.540062024-03-13T18:33:02.981707Z"Apreutesei, Gabriela"https://zbmath.org/authors/?q=ai:apreutesei.gabriela"Croitoru, Anca"https://zbmath.org/authors/?q=ai:croitoru.ancaSummary: The aim of this paper is to offer a general theoretical frame for clustering validation problems. For this purpose we consider a near metric \(d_1\) and a nonnegative symmetrical application \(d_2\) on some spaces \(\mathcal{M}\) of set-valued functions and compare them. We also study the semi-linearity of the induced topologies \(\tau_1\) and \(\tau_2\) respectively, as well as the translated topology \(\tau_3\) of \(\tau_1\) on these spaces of set-valued functions and present sufficient conditions and characteristic properties for \(\tau_3\) to be semi-linear. We formulate some conditions which give approximation properties for addition on \(\mathcal{M}\times \mathcal{M}\) and for multiplication by scalars in special points of \(\mathbb{R}\times\mathcal{M}\).Directional derivatives for set-valued maps based on set convergenceshttps://zbmath.org/1528.540072024-03-13T18:33:02.981707Z"Durea, Marius"https://zbmath.org/authors/?q=ai:durea.mariusSummary: We explore the possibility to define and to meaningfully apply some new concepts of directional derivative which incorporate in their construction set convergences to set-optimization problems. We connect these new constructions with other directional derivatives for set-valued maps and we emphasize the flexibility and the potential applicability of this new approach. In this vein, we indicate a possible axiomatic perspective that allows one to significantly increase the number and (maybe) the efficiency of these derivatives when applied to concrete problems.Expandabilities and covering properties of inverse limits.https://zbmath.org/1528.540082024-03-13T18:33:02.981707Z"Chiba, Keiko"https://zbmath.org/authors/?q=ai:chiba.keiko(no abstract)Locally compact, \( \omega_1\)-compact spaceshttps://zbmath.org/1528.540092024-03-13T18:33:02.981707Z"Nyikos, Peter"https://zbmath.org/authors/?q=ai:nyikos.peter-j"Zdomskyy, Lyubomyr"https://zbmath.org/authors/?q=ai:zdomskyy.lyubomyrLet \(X\) be a topological Hausdorff space. The space \(X\) is \(\omega_1\)-compact if every closed discrete subspace of \(X\) is countable. The space \(X\) is \(\sigma\)-countably compact if \(X\) is expressible as a countable union of some of its countably compact subspaces.
In this article, it is clarified that the statement ``Every locally compact, \(\omega_1\)-compact Hausdorff space of size \(\leq\aleph_1\) is \(\sigma\)-countably compact'' is independent of ZFC. Consistent examples of Hausdorff, locally compact, \(\omega_1\)-compact spaces of size \(\aleph_1\) which fail to be \(\sigma\)-countably compact are given. For instance, it is shown that every Souslin tree with the interval topology is a Hausdorff, locally compact, locally countable, \(\omega_1\)-compact and hereditarily collectionwise normal (so also hereditarily normal) space of size \(\aleph_1\) which is not \(\sigma\)-countably compact. In ZFC, \(\clubsuit\) implies that there is a topology \(\tau\) on \(\omega_1\), finer than the usual order topology and such that the \(\tau\)-relative topology on the set of all countable limit ordinals is the usual order topology, \((\omega_1, \tau)\) is a locally compact, monotonically normal, \(\omega_1\)-compact space which is not \(\sigma\)-countably compact.
On the other hand, among other relevant results, the authors show that, in ZFC, the conjunction of the \(P\)-Ideal Dichotomy Axiom (in abbreviation, PID) and \(\mathfrak{b}>\aleph_1\) implies that every Hausdorff, locally compact, \(\omega_1\)-compact, normal space of size \(\aleph_1\) is \(\sigma\)-countably compact. Furthermore, the conjunction of PID and \(\min\{\mathfrak{b}, \mathfrak{s}\}>\aleph_1\) implies that every Hausdorff, locally compact, \(\omega_1\)-compact space of weight \(\aleph_1\) is \(\sigma\)-countably compact. In ZFC, PID implies that every Hausdorff, locally compact, \(\omega_1\)-compact normal space of size \(<\mathfrak{b}\) is countably paracompact.
Suppose that \(X\) is a Hausdorff, locally compact, \(\omega_1\)-compact space in a model \(\mathcal{M}\) of ZFC. The authors prove that each of the following conditions \((i)\)-\((iii)\) implies that it is true in \(\mathcal{M}\) that \(X\) is \(\sigma\)-\(\omega\)-bounded (so also \(\sigma\)-countably compact) and is either Lindelöf or contains a copy of \(\omega_1\): \((i)\) \(X\) is monotonically normal and PID is true in \(\mathcal{M}\); \((ii)\) \(X\) is hereditarily \(\omega_1\)-strongly collectionwise Hausdorff and either the Proper Forcing Axiom (in abbreviation, PFA) is true in \(\mathcal{M}\) or \(\mathcal{M}\) is a PFA\((S)[S]\) model; \((iii)\) \(X\) is hereditarily normal in \(\mathcal{M}\) or \(\mathcal{M}\) is an \(MM(S)[S]\) model. Moreover, the authors deduce that it is true in \(\mathcal{M}\) that if the space \(X\) is normal, hereditarily \(\omega_1\)-strongly collectionwise Hausdorff and either PFA holds in \(\mathcal{M}\) or \(\mathcal{M}\) is a PFA\((S)[S]\) model, then \(X\) is countably paracompact.
It is explained that, in every \(MM(S)[S]\) model, the following statements are both true: (a) every Hausdorff, locally compact Dowker space of cardinality \(\leq\aleph_1\) includes both a copy of \(\omega_1\) and an uncountable closed discrete subspace; (b) every Hausdorff, locally compact, hereditarily normal Dowker space contains an uncountable closed discrete subspace.
The authors discuss the least cardinality among the cardinalities of Hausdorff, locally compact, \(\omega_1\)-compact spaces which fail to be \(\sigma\)-countably compact. It is announced that there does exist a consistent example (constructed under \(\square_{\aleph_1}\) by the first author) of a Hausdorff, locally countable, normal, \(\omega\)-bounded (hence countably compact) space of cardinality \(\aleph_2\). Apart from other questions, all relevant to the above-mentioned main results of the paper, the following open problem is posed: Is there a Hausdorff, normal, locally countable, countably compact space of cardinality greater than \(\aleph_2\)?
Finally, the authors show that PFA implies that if a Hausdorff, locally compact, locally countable space \(X\) is a quasi-perfect preimage of the space of irrationals, then \(X\) is not normal. The authors ask if there is a ZFC example of a scattered, countably compact, \(T_3\)-space that can be mapped continuously onto \([0, 1]\).
Reviewer: Eliza Wajch (Siedlce)Remarks on Suslin number of inverse limits.https://zbmath.org/1528.540102024-03-13T18:33:02.981707Z"Chiba, Keiko"https://zbmath.org/authors/?q=ai:chiba.keiko(no abstract)On set star-Menger spaceshttps://zbmath.org/1528.540112024-03-13T18:33:02.981707Z"Singh, Sumit"https://zbmath.org/authors/?q=ai:singh.sumit.1|singh.sumitA space \(X\) is said to have the \textit{set star-Menger} (abbreviated as \textbf{set-SM}) property, if for each nonempty set \(A\subseteq X\) and each sequence \(\langle\mathcal U_n : n \in \omega\rangle\) of collections of open sets in \(X\) such that cl\((A)\subseteq \bigcup\mathcal U_n\) there is a sequence \(\langle \mathcal V_n :n\in \omega\rangle \) such that for each \(n \in\omega\), \(\mathcal V_n\) is a finite subset of \(\mathcal U_n\) and \(A \subseteq\bigcup\{ St(\bigcup\mathcal V_n,\mathcal U_n):n\in\omega\}\); A space is said to have the \textit{set strongly star-Menger} (abbreviated as \textbf{set-SSM}) property, if for each nonempty set \(A\subseteq X\) and each sequence \(\langle\mathcal U_n : n \in \omega\rangle\) of collections of open sets in \(X\) such that cl\((A)\subseteq \bigcup\mathcal U_n\) there is a sequence \(\langle \mathcal F_n :n\in \omega\rangle \) of finite subsets of cl\((A)\) such that \(A\subseteq \bigcup\{St(F_n,\mathcal U_n):n\in\omega\}\). A product space \(X \times Y\) is said to be \textit{rectangular set star-Menger} (abbreviated as rectangular \textbf{set-SM}) if it satisfies the defining conditions for the set star-Menger property for subsets of the product space \(X\times Y\) of the form \(A = B \times C\), where \(B\) and \(C\) are nonempty subsets of \(X\) and \(Y,\) respectively. The main results of this paper are as follows: (1) The product of a \textbf{set-SM} space and a compact space is rectangular \textbf{set-SM}; and (2) If \(\omega_1<\mathfrak d\) there exists a Tychonoff, pseudocompact, \textbf{set-SM} space which is not \textbf{set-SSM}. The first result gives a partial answer to Problem 3.9 posed by \textit{L. D. R. Kočinac} et al. [Math. Slovaca 72, No. 1, 185--196 (2022; Zbl 1491.54024)], while the second result gives a consistent positive answer to Problem 1 of the same paper.
Reviewer: Richard G. Wilson (Ciudad de México)Uniqueness and rigidity of the second symmetric product of standard universal dendriteshttps://zbmath.org/1528.540122024-03-13T18:33:02.981707Z"Maya, David"https://zbmath.org/authors/?q=ai:maya-escudero.davidGiven a metric continuum \(X\), let \(F_{n}(X)\) be the hyperspace of nonempty subsets of \(X\) with at most \(n\) points, endowed with the Hausdorff metric. A continuum \(X\) has unique hyperspace \(F_{n}(X)\) provided the following implication holds: if \(Y\) is a continuum satisfying that \(F_{n}(X)\) is homeomorphic to \(F_{n}(Y)\), then \(X\) is homeomorphic to \(Y\).
A dendrite is a locally connected metric continuum without simple closed curves. An interesting problem on hyperspaces is to determine whether each dendrite \(D\) has unique hyperspace \(F_{n}(D)\). There are several partial answers to this problem. It is known that the answer is positive for dendrites with closed set of end-points.
In the paper under review, the author works with dendrites on the other extreme. A universal dendrite \(D_{m}\) (\(m\geq 3\)) is characterized by the following properties: each ramification point in \(D_{m}\) has order \(m\) and for each arc \(A\) in \(D_{m}\), the set of ramification points of \(D_{m}\) that belong to \(A\) is dense in \(A\). These dendrites \(D_{m}\) have dense set of end-points, so \(D_{m}\) is very far from having closed set of end-points.
Using unicoherence techniques the author of this paper proves that for each \(m\geq 5\), \(D_{m}\) has unique hyperspace \(F_{2}(D_{m})\). He also proves that if \(m\geq 5\), then each homeomorphism \(h\) from \(F_{2}(D_{m})\) onto \(F_{2}(D_{m})\) satisfies \(h(F_{1}(D_{m}))=F_{1}(D_{m})\).
As we can see, the problem of the uniqueness of the hyperspace \(F_{n}(D)\), for dendrites \(D\), still offers many possibilities of research. For more information about related problems see [\textit{A. Illanes}, Quest. Answers Gen. Topology 30, No. 1, 21--44 (2012; Zbl 1275.54006)].
Reviewer: Alejandro Illanes (Ciudad de México)Capacities and analytic setshttps://zbmath.org/1528.540132024-03-13T18:33:02.981707Z"Dellacherie, Claude"https://zbmath.org/authors/?q=ai:dellacherie.claudeFor the entire collection see [Zbl 1465.03026].Zero preservation for a family of multivalued functionals, and applications to the theory of fixed points and coincidenceshttps://zbmath.org/1528.540142024-03-13T18:33:02.981707Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevna"Zakharyan, Yu. N."https://zbmath.org/authors/?q=ai:zakharyan.yu-nSummary: A theorem on the zero existence preservation for a parametric family of multivalued \(( \alpha, \beta )\)-search functionals on an open subset of a metric space is proved. Several corollaries on the existence preservation for preimages of a closed subspace, for coincidence points, and for common fixed points under the action of a parametric family (a number of families) of mappings are obtained. The notion of a Zamfirescu-type pair of mappings is introduced, and a coincidence theorem for such pairs of mappings is obtained. In addition, a theorem on the coincidence existence preservation for a parametric family of such pairs of mappings is obtained. The obtained results imply several well-known theorems.Erratum to: ``Zero preservation for a family of multivalued functionals, and applications to the theory of fixed points and coincidences''https://zbmath.org/1528.540152024-03-13T18:33:02.981707Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevna"Zakharyan, Yu. N."https://zbmath.org/authors/?q=ai:zakharyan.yu-nErratum to the authors' paper [ibid. 102, No. 1, 272--275 (2020; Zbl 1528.54014)].Common fixed points for \((\psi -\varphi)\)-weak contractions type in \(b\)-metric spaceshttps://zbmath.org/1528.540162024-03-13T18:33:02.981707Z"Morales, José R."https://zbmath.org/authors/?q=ai:morales-medina.jose-roberto"Rojas, Edixon M."https://zbmath.org/authors/?q=ai:rojas.edixon-mSummary: The aim of this paper is to prove the existence and uniqueness of common fixed points for a pair of \((\psi -\varphi)\)-weak contractive self-maps in the setting of \(b\)-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, \(R\)-weakly commuting and \(R\)-weakly commuting of types \((A_T), (A_S)\) and \((A_P)\). Also, we will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on \(b\)-metric spaces endowed with a convex structure.\(\perp_{\psi F}\)-contractions and some fixed point results on generalized orthogonal setshttps://zbmath.org/1528.540172024-03-13T18:33:02.981707Z"Touail, Youssef"https://zbmath.org/authors/?q=ai:touail.youssef"El Moutawakil, Driss"https://zbmath.org/authors/?q=ai:el-moutawakil.drissSummary: In this paper, we define the notion of generalized orthogonal sets by extending orthogonal sets. Also, we introduce the concept of \(\perp_{\psi F}\)-contractions, which generalizes \(\perp_F\) contractions. For such contractions, some fixed point theorems are proved. Moreover, an application to a differential equation is given.Some local fixed point theorems and applications to open mapping principles and continuation resultshttps://zbmath.org/1528.540182024-03-13T18:33:02.981707Z"Truşcă, Radu"https://zbmath.org/authors/?q=ai:trusca.raduSummary: The purpose of this article is to present, under weaker assumptions, some local fixed point theorems for Ćirić-Reich-Rus, Chatterjea and Berinde type generalized contractions. Then, as applications we will obtain open mapping theorems and continuation principles for these classes of mappings.Geometric progressions in distance spaces; applications to fixed points and coincidence pointshttps://zbmath.org/1528.540192024-03-13T18:33:02.981707Z"Zhukovskiy, Evgeny S."https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-sThe author investigates which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings defined on spaces \(X\) endowed with the generalized distance \(\rho_X\). This holds if each geometric progression with \(ratio < 1\) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete \(f\)-quasimetric space \(X\), if the distance \(\rho X\) satisfies this inequality \(\rho_X(x, z) \leq \rho_X(x, y) + (\rho_X(y, z))^{\eta}, x, y, z\in X\), for some \(\eta \in (0, 1)\).
In the last part of the paper the author considers the following function \(f(r_1,r_2)=\max \{r_1^{\eta},r_2^{\eta}\}\) where \(\eta \in (0, 2^{-1}]\). He shows that for any \(\gamma > 0\) there exists an \(f\)-quasimetric space containing a geometric progression with ratio \(\gamma\) which is not a Cauchy sequence. For \(f\)-quasimetric spaces the ``zero-one law'' is discussed, which means that either each geometric progression with \(ratio< 1\) is a Cauchy sequence or, for any \(\gamma \in (0, 1)\), there exists a geometric progression with ratio \(\gamma\) that is not Cauchy.
Reviewer: Monica-Felicia Bota (Cluj-Napoca)Efficient generation of \(W\) Entangled states among superconducting qubits via Lie-algebra-based transformshttps://zbmath.org/1528.810992024-03-13T18:33:02.981707Z"Zhou, Yuanyuan"https://zbmath.org/authors/?q=ai:zhou.yuanyuan"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian.7|zhang.qian.2|zhang.qian.3|zhang.qian.17"Hao, Yongle"https://zbmath.org/authors/?q=ai:hao.yongle|hao.yongle.1"Zhao, Huitao"https://zbmath.org/authors/?q=ai:zhao.huitao"Zhou, Chongyun"https://zbmath.org/authors/?q=ai:zhou.chongyunSummary: An efficient scheme is proposed to construct an evolution operator by using Lie-algebra-based transforms for generating tripartite and quadripartite \(W\) states among superconducting qubits. We consider four superconducting qubits coupled to a microwave cavity in the circuit quantum electrodynamics, where one plays the role of auxiliary qubit while the other three ones are systematic qubits. By adjusting the coupling strength between the qubits and the cavity, the complicated superconducting circuit system can be reduced to an effective system with three quantum states. Then by using Lie-algebra-based transforms, an evolution operator can be designed on the Hilbert subspace of the effective system, and further the time-dependent Rabi frequencies of external classical fields can be engineered to generate the tripartite and quadripartite \(W\) states.Dominant energy condition and dissipative fluids in general relativityhttps://zbmath.org/1528.830422024-03-13T18:33:02.981707Z"Faraoni, Valerio"https://zbmath.org/authors/?q=ai:faraoni.valerio"Mokkedem, El Mokhtar Z. R."https://zbmath.org/authors/?q=ai:mokkedem.el-mokhtar-z-rSummary: Existing literature implements the Dominant Energy Condition for dissipative fluids in general relativity. It is pointed out that this condition fails to forbid superluminal flows, which is what it is ultimately supposed to do. Tilted perfect fluids, which formally have the stress-energy tensor of imperfect fluids, are discussed for comparison.Dressed black holes in the new tensor-vector-scalar theoryhttps://zbmath.org/1528.830702024-03-13T18:33:02.981707Z"Bernardo, Reginald Christian"https://zbmath.org/authors/?q=ai:bernardo.reginald-christian-s"Chen, Che-Yu"https://zbmath.org/authors/?q=ai:chen.che-yuSummary: As incarnations of gravity in its prime, black holes are arguably the best target for us to demystify gravity. Keeping in mind the prominent role black holes play in gravitational wave astronomy, it becomes a must for a theory to possess black hole solutions with only measurable departures from their general relativity counterparts. In this paper, we present black holes in a tensor-vector-scalar representation of relativistic modified Newtonian dynamics. We find that the theory allows Schwarzschild and nearly-Schwarzschild black holes as solutions, while the nontrivial scalar and vector fields generally diverge at the event horizon. Whether this is a physical pathology or not poses a challenge for these solutions, and by extension, the model. However, even if it is, this pathology could be overcome when the black hole hair vanishes.Conformal cyclic cosmology, gravitational entropy and quantum informationhttps://zbmath.org/1528.831372024-03-13T18:33:02.981707Z"Eckstein, Michał"https://zbmath.org/authors/?q=ai:eckstein.michalSummary: We inspect the basic ideas underlying Roger Penrose's Conformal Cyclic Cosmology from the perspective of modern quantum information. We show that the assumed loss of degrees of freedom in black holes is not compatible with the quantum notion of entropy. We propose a unitary version of Conformal Cyclic Cosmology, in which quantum information is globally preserved during the entire evolution of our universe, and across the crossover surface to the subsequent aeon. Our analysis suggests that entanglement with specific quantum gravitational degrees of freedom might be at the origin of the second law of thermodynamics and the quantum-to-classical transition at mesoscopic scales.On the asymptotic assumptions for Milne-like spacetimeshttps://zbmath.org/1528.831472024-03-13T18:33:02.981707Z"Ling, Eric"https://zbmath.org/authors/?q=ai:ling.eric"Piubello, Annachiara"https://zbmath.org/authors/?q=ai:piubello.annachiaraSummary: Milne-like spacetimes are a class of hyperboloidal FLRW spacetimes which admit continuous spacetime extensions through the big bang, \(\tau =0\). In a previous paper [\textit{E. Ling}, Found. Phys. 50, No. 5, 385--428 (2020; Zbl 1436.83049)], it was advocated that the existence of this big bang extension could have applications to fundamental problems in cosmology, which illustrates the physical importance of such extensions. By definition, the scale factor for a Milne-like spacetime satisfies the past asymptotic assumption \(a(\tau) = \tau + o(\tau^{1+\varepsilon})\) as \(\tau \rightarrow 0\) for some \(\varepsilon > 0\). The existence of the big bang extension follows from writing the metric in conformal Minkowskian coordinates and using the past asymptotic assumption of the scale factor. This asymptotic assumption implies \(a(\tau) = \tau + o(\tau)\) as \(\tau \rightarrow 0\). In this paper, we show that \(a(\tau) = \tau + o(\tau)\) is not sufficient to achieve a big bang extension, but it is necessary (provided its derivative converges as \(\tau \rightarrow 0)\). We also show that the \(\varepsilon\) in \(a(\tau) = \tau + o(\tau^{1+\varepsilon}\)) is not necessary to achieve a big bang extension.