Recent zbMATH articles in MSC 54Chttps://zbmath.org/atom/cc/54C2024-03-13T18:33:02.981707ZWerkzeugPulling 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.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)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)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.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.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)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.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)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)].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)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.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.