Recent zbMATH articles in MSC 03Ehttps://www.zbmath.org/atom/cc/03E2021-04-16T16:22:00+00:00WerkzeugTowards a descriptive theory of cb\(_{0}\)-spaces.https://www.zbmath.org/1456.030772021-04-16T16:22:00+00:00"Selivanov, Victor"https://www.zbmath.org/authors/?q=ai:selivanov.victor-lSummary: The paper tries to extend some results of the classical Descriptive Set Theory to as many countably based \(T_{0}\)-spaces (cb\(_{0}\)-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case of \(k\)-partitions. In particular, we investigate the difference hierarchy of \(k\)-partitions and the fine hierarchy closely related to the Wadge hierarchy.A minimal Kurepa tree with respect to club embeddings.https://www.zbmath.org/1456.030742021-04-16T16:22:00+00:00"Ramandi, Hossein Lamei"https://www.zbmath.org/authors/?q=ai:ramandi.hossein-lameiThe author establishes the consistency with GCH of the existence of a Kurepa tree \(T\) such that (a) given any downward-closed Kurepa subtree \(U\) of \(T\), there is, for some closed unbounded subset \(C\) of \(\omega_1\), a one-to-one function \(f : T \vert C \rightarrow U \vert C\) that is level- and order- preserving, and (b) \(T\) has no Aronszajn subtrees.
Reviewer: Pierre Matet (Caen)Dirac operator on the quantum fuzzy four-sphere \(S_{q F}^4\).https://www.zbmath.org/1456.811712021-04-16T16:22:00+00:00"Lotfizadeh, M."https://www.zbmath.org/authors/?q=ai:lotfizadeh.mSummary: \(q\)-deformed fuzzy Dirac and chirality operators on quantum fuzzy four-sphere \(S_{q F}^4\) are studied in this article. Using the \(q\)--deformed fuzzy Ginsparg-Wilson algebra, the \(q\)--deformed fuzzy Dirac and chirality operators in an instanton and no-instanton sector are studied. In addition, gauged Dirac and chirality operators in both cases have also been constructed. It has been shown that in each step, our results have a correct commutative limit in the limit case when \(q \rightarrow 1\) and the noncommutative parameter \(l\) tends to infinity.
{\copyright 2021 American Institute of Physics}A comparison of concepts from computable analysis and effective descriptive set theory.https://www.zbmath.org/1456.030762021-04-16T16:22:00+00:00"Gregoriades, Vassilios"https://www.zbmath.org/authors/?q=ai:gregoriades.vassilios"Kispéter, Tamás"https://www.zbmath.org/authors/?q=ai:kispeter.tamas"Pauly, Arno"https://www.zbmath.org/authors/?q=ai:pauly.arno-mSummary: Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the two disciplines has so far been neglected, a situation this paper is meant to remedy.
As the role of the Cauchy completion is relevant for both effective approaches to Polish spaces, we consider the interplay of effectivity and completion in some more detail.Marcus-Wyse topological rough sets and their applications.https://www.zbmath.org/1456.682222021-04-16T16:22:00+00:00"Han, Sang-Eon"https://www.zbmath.org/authors/?q=ai:han.sang-eonSummary: The aim of this paper is to establish two new types of rough set structures associated with the Marcus-Wyse (MW, for brevity) topology, such as an M-rough set and an MW-topological rough set. The former focuses on studying the rough set theoretic tools for 2-dimensional Euclidean spaces and the latter contributes to the study of the rough set structures for digital spaces in \(\mathbb{Z}^2\), where \(\mathbb{Z}\) is the set of integers. These two rough set structures are related to each other via an M-digitization. Thus, these can successfully be used in the field of applied science, such as digital geometry, image processing, deep learning for recognizing digital images, and so on. For a locally finite covering approximation (LFC, for short) space \((U, \mathbf{C})\) and a subset \(X\) of \(U\), we firstly introduce a new neighborhood system on \(U\) related to \(X\). Next, we formulate the lower and upper approximations with respect to \(X\), where all of the sets \(U\) and \(X(\subseteq U)\) need not be finite and the covering \( \mathbf{C}\) is locally finite. Actually, the notion of M-digitization of a 2-dimensional Euclidean space plays an important role in developing an M-rough and MW-topological rough set structures. Further, we prove that M-rough set operators have a duality between them. However, each of MW-topological rough set operators need not have the property as an interior or a closure from the viewpoint of MW-topology.Well ordering groups with no monotone arithmetic progressions.https://www.zbmath.org/1456.030732021-04-16T16:22:00+00:00"Károlyi, Gyula"https://www.zbmath.org/authors/?q=ai:karolyi.gyula"Komjáth, Péter"https://www.zbmath.org/authors/?q=ai:komjath.peterSummary: Károlyi-Kós and Ardal-Brown-Jungić proved that every vector space over \(\mathbb {Q}\) has an ordering with no monotone three term arithmetic progression (3-AP). We show that every solvable group has a well ordering with no monotone 6-AP, and each hypoabelian group has an ordering omitting monotone 5-APs. Finally, we prove that every group has a well ordering with no infinite monotone AP.Computational method for fuzzy arithmetic operations on triangular fuzzy numbers by extension principle.https://www.zbmath.org/1456.681942021-04-16T16:22:00+00:00"Gerami Seresht, Nima"https://www.zbmath.org/authors/?q=ai:gerami-seresht.nima"Fayek, Aminah Robinson"https://www.zbmath.org/authors/?q=ai:fayek.aminah-robinsonSummary: Fuzzy arithmetic operations are applied to mathematical equations that include fuzzy numbers, which are commonly used to represent non-probabilistic uncertainty in different applications. Although there are two mathematical approaches available in the literature for implementing fuzzy arithmetic (i.e., the \(\alpha\)-cut approach, and the extension principle approach), the existing computational methods are mainly focused on implementing the \(\alpha\)-cut approach due to its simplicity. However, this approach causes overestimation of uncertainty in the resulting fuzzy numbers, a phenomenon that reduces the interpretability of the results. This overestimation can be reduced by implementing fuzzy arithmetic using the extension principle; however, existing computational methods for implementing the extension principle approach are limited to the use of min and drastic product t-norms. Using the min t-norm produces the same result as the \(\alpha\)-cuts and interval calculations approach, and the drastic product t-norm is criticized for producing resulting fuzzy numbers that are highly sensitive to the changes in the input fuzzy numbers. This paper presents original computational methods for implementing fuzzy arithmetic operations on triangular fuzzy numbers using the extension principle approach with product and Lukasiewicz t-norms. These computational methods contribute to the different applications of fuzzy arithmetic; they reduce the overestimation of uncertainty, as compared to the \(\alpha\)-cut approach, and they reduce the sensitivity of the resulting fuzzy numbers to changes in the input fuzzy numbers, as compared to the extension principle approach using drastic product t-norm.On binary relations induced from overlap and grouping functions.https://www.zbmath.org/1456.030822021-04-16T16:22:00+00:00"Qiao, Junsheng"https://www.zbmath.org/authors/?q=ai:qiao.junshengSummary: In this paper, firstly, we introduce the binary relation \(\preceq_O\) derived from an overlap function \(O\). And then, we show that \(\preceq_O\) does not satisfy the reflexivity, anti-symmetry and transitivity naturally, and investigate the conditions under which \(\preceq_O\) can become a reflexive, anti-symmetric or transitive relation, respectively. In particular, we obtain a necessary and sufficient condition for the binary relation \(\preceq_O\) becoming a partial order on the unit interval \([0, 1]\). Finally, we give an analogous discussion for the binary relations induced from grouping functions.Compactness properties defined by open-point games.https://www.zbmath.org/1456.540092021-04-16T16:22:00+00:00"Dorantes-Aldama, A."https://www.zbmath.org/authors/?q=ai:dorantes-aldama.alejandro"Shakhmatov, D."https://www.zbmath.org/authors/?q=ai:shakhmatov.dmitri-bThe authors study topological properties related to sequences in an abstract space, by considering suitable topological games.
Consider that it is given a topological property \(S\) about sequences in a topological space (properties such as, for example, ``to contain a convergent subsequence'' or ``to have an accumulation point'').
The authors introduce an open-point game for two players 1 and 2 on a topological space \(X\), such that in the \(n\)-th
move, Player 1 chooses a non-empty open set (say \(U_n\)) in the topology given a priori on \(X\), and Player 2 responds by selecting a point that belongs to that open set \(U_n\).
Player 2 wins the game if the sequence of points so constructed satisfies the property \(S\) considered a priori on the given topological space \(X\).
Otherwise, it is Player 1 who wins.
The (non-)existence of regular or stationary winning strategies in that game for both players defines new compactness properties of the underlying space \(X\).
The authors investigate in depth those properties. They
construct suitable examples for this analysis, for an arbitrary property \(S\) sandwiched between sequential compactness and countable compactness.
Reviewer: Esteban Induraín (Pamplona)Trice-valued fuzzy sets: mathematical model for three-way decisions.https://www.zbmath.org/1456.030802021-04-16T16:22:00+00:00"Horiuchi, Kiyomitsu"https://www.zbmath.org/authors/?q=ai:horiuchi.kiyomitsu"Šešelja, Branimir"https://www.zbmath.org/authors/?q=ai:seselja.branimir"Tepavčević, Andreja"https://www.zbmath.org/authors/?q=ai:tepavcevic.andrejaSummary: Under the idea to develop a mathematical model for three-way decisions, the aim of the paper is to study trice-valued fuzzy sets, i.e., mappings from a set to a structure called a trice. A trice is a triple semilattice satisfying roundabout absorption laws, suitable for representing multi-dimensional orders, which appear in complex movements in a plane or in a space. Our approach is cutworthy, namely, we investigate cuts of such fuzzy sets and prove theorems of decomposition and synthesis. This new notion provides a possibility to capture vague triangular situations. Therefore, a motivation for our research is to provide a new algebraic and order-theoretic model for three-way decisions, as this topic has been introduced recently for solving particular human problems and for information processing.Normal measures on a tall cardinal.https://www.zbmath.org/1456.030782021-04-16T16:22:00+00:00"Apter, Arthur W."https://www.zbmath.org/authors/?q=ai:apter.arthur-w"Cummings, James"https://www.zbmath.org/authors/?q=ai:cummings.jamesThe authors show that (a) the least measurable may be tall and carry any specified number of normal measures, and (b) the least measurable limit of tall cardinals may carry any specified number of normal measures. For (a) they use a strong cardinal, and for (b) a measurable limit of strong cardinals.
Reviewer: Pierre Matet (Caen)Forking independence from the categorical point of view.https://www.zbmath.org/1456.030592021-04-16T16:22:00+00:00"Lieberman, Michael"https://www.zbmath.org/authors/?q=ai:lieberman.michael-j"Rosický, Jiří"https://www.zbmath.org/authors/?q=ai:rosicky.jiri"Vasey, Sebastien"https://www.zbmath.org/authors/?q=ai:vasey.sebastienThe model-theoretic notion of forking introduced by Shelah for classes of models axiomatized by stable first-order theories is a generalization of linear independence in vector spaces and algebraic independence in fields. In fact, (non)forking can be seen as a commutative diagram of embeddings also known as an amalgam.
In this paper, the broad model-theoretic framework of abstract elementary classes (AECs) is used. A \(\mu\)-AEC is simply an accessible category with all morphisms monomorphisms. The authors describe when a category has a `stable independence notion' -- a class of distinguished commutative squares that itself forms an accessible category -- and show that this is a purely category-theoretic axiomatization of forking in a \(\mu\)-AEC. This generalizes a result of [\textit{W. Boney} et al., Ann. Pure Appl. Logic 167, No. 7, 590--613 (2016; Zbl 1400.03060)] that characterized stable forking in the framework of AECs but depended on set-representations of their objects. The category \(\mathcal K_{\mathrm{reg}}\) of regular monomorphisms in a locally presentable coregular category \(\mathcal K\) that has effective unions, in the sense of Barr, is shown to have a stable independence notion, thus showing that forking occurs in both Grothendieck toposes and Grothendieck abelian categories.
Assuming a large cardinal axiom the authors also characterize when a stable independence notion exists in a \(\mu\)-AEC. Thus, it is established that model-theoretic stability is invariant under equivalence of categories.
Reviewer: Amit Kuber (Kanpur)On the \(c_0\)-extension property.https://www.zbmath.org/1456.460192021-04-16T16:22:00+00:00"Correa, Claudia"https://www.zbmath.org/authors/?q=ai:correa.claudiaThis paper investigates the \(c_0\)-extension property introduced in [\textit{C. Correa} and \textit{D. V. Tausk}, J. Funct. Anal. 266, No. 9, 5765--5778 (2014; Zbl 1329.46008)]. More precisely, we say that a Banach space \(X\) has the \(c_0\)-extension property if every \(c_0\)-valued
bounded operator defined on a closed subspace of \(X\) admits a \(c_0\)-valued bounded extension defined on \(X\). It is proven that a sufficient condition for a Banach space to have this property is that its closed dual unit ball is weak-star monolithic. Among other results it is also shown that the existence of a Corson compactum \(K\) such that \(C(K)\) does not have the \(c_0\)-extension property is independent of the axioms of ZFC.
Reviewer: Elói M. Galego (São Paulo)Game theoretic approach to shadowed sets: a three-way tradeoff perspective.https://www.zbmath.org/1456.682122021-04-16T16:22:00+00:00"Zhang, Yan"https://www.zbmath.org/authors/?q=ai:zhang.yan.5"Yao, JingTao"https://www.zbmath.org/authors/?q=ai:yao.jingtaoSummary: Three-way approximations can be constructed by shadowed sets based on a pair of thresholds. The determination and interpretation of the thresholds is one of the key issues for applying three-way approximations. We apply a principle of tradeoff with games in order to determine the thresholds of three-way approximations in the shadowed set context. The changes of the elevation and reduction errors with the alteration of thresholds are examined and analyzed. The proposed game-theoretic shadowed sets (GTSS) aim to determine the thresholds of three-way approximations according to a principle of tradeoff with games. GTSS employ game theoretic approaches to formulate games between the elevation and reduction errors. A repetition learning mechanism is adopted to gradually reach balanced threshold pairs by repeatedly formulating games and finding the equilibria between the errors. The shadowed set based three-way approximations defined by the resulting thresholds represent a tradeoff between the elevation and reduction errors. Feasibility study and effectiveness analysis of GTSS is conducted with an experimental data set.The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (extended abstract).https://www.zbmath.org/1456.030752021-04-16T16:22:00+00:00"Gregoriades, Vassilios"https://www.zbmath.org/authors/?q=ai:gregoriades.vassiliosSummary: In this article we treat a notion of continuity for a multi-valued function \(F\) and we compute the descriptive set-theoretic complexity of the set of all \(x\) for which \(F\) is continuous at \(x\). We give conditions under which the latter set is either a \(G_\delta\) set or the countable union of \(G_\delta\) sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of \(F\) is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function \(F\) whose graph is a Borel set and the set of points of continuity of \(F\) is not a Borel set. Finally we give some analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of \textit{M. Ziegler} [Ann. Pure Appl. Logic 163, No. 8, 1108--1139 (2012; Zbl 1259.03059)].
For the entire collection see [Zbl 1391.03010].Intuitionistic fuzzy relations compatible with the group \(Z_n\).https://www.zbmath.org/1456.030792021-04-16T16:22:00+00:00"Emam, E. G."https://www.zbmath.org/authors/?q=ai:emam.e-gIn the paper the concept of compatibility of fuzzy relations with some multiplicative semi-group is extended to the case of intuitionistic fuzzy relations. The mentioned compatibility of intuitionistic fuzzy relations is studied in the case of the well-known abelian group \(Z_n\) with the sum modulo \(n\) (which is isomorphic to some multiplicative abelian group). As a result the concept of \(Z_n\)-compatiblity of intuitionistic fuzzy relations is considered and several related properties are provided. Namely, a characterization of \(Z_n\)-compatible intuitionistic fuzzy relation is presented, the number of elements in \(Z_n\)-compatible intuitionistic fuzzy relation is determined, preservation of \(Z_n\)-compatiblity by basic operations on intuitionistic fuzzy relations (sum, intersection, converse relation, the complement) as well as some other operations are considered. Moreover, the symmetry of the composition of two \(Z_n\)-compatible intuitionistic fuzzy relations is proved. Finally, two relations related to any intuitionistic fuzzy relation are distinguished and their properties in the context of \(Z_n\)-compatiblity are delivered.
Reviewer: Urszula Bentkowska (Rzeszów)Mathematics for computer science. Basic concepts, structures and their applications. 3rd expanded and updated edition.https://www.zbmath.org/1456.680022021-04-16T16:22:00+00:00"Berghammer, Rudolf"https://www.zbmath.org/authors/?q=ai:berghammer.rudolfThis is the third edition of this textbook. For a review of the first edition see [Zbl 1309.00001].
The second edition added two new chapters and an
elaborate appendix giving a formal and complete introduction to the natural numbers based on Peano structures. With these two chapters, important aspects of computer science are addressed, namely those that concern the programming of algorithms. Keywords for the first of these chapters are program specification and verification, Hoare-calculus, loop-invariants and program construction; the second one focuses on generic programming, which is exemplified by several graph-theoretic algorithms, thus also extending concepts of graph theory from a former chapter. The third edition finally adds solutions to all exercises except those for the new chapters, which are available online.
The presentation of the topics is always detailed and in full mathematical rigour, starting with motivating examples and naive concepts, which are
then gradually conducted to full formalization. The reader will find many topics and detailed proofs not often found in other introductory
textbooks. The book is especially valuable for those students of computer science who are interested in a rigorous mathematical foundation.
Reviewer: Dieter Riebesehl (Lüneburg)Constructive forcing, CPS translations and witness extraction in interactive realizability.https://www.zbmath.org/1456.031092021-04-16T16:22:00+00:00"Aschieri, Federico"https://www.zbmath.org/authors/?q=ai:aschieri.federicoSummary: In Interactive realizability for second-order Heyting Arithmetic with \textsf{EM}\(_1\) and \textsf{SK}\(_1\) (the excluded middle and Skolem axioms restricted to \(\Sigma_1^0\)-formulas), realizers are written in a classical version of Girard's System \textsf{F}. Since the usual reducibility semantics does not apply to such a system, we introduce a constructive forcing/reducibility semantics: though realizers are not computable functionals in the sense of Girard, they can be forced to be computable. We apply this semantics to show how to extract witnesses for realizable \(\Pi_2^0\)-formulas. In particular, a constructive and efficient method is introduced. It is based on a new `(state-extending-continuation)-passing-style translation' whose properties are described with the constructive forcing/reducibility semantics.Shadowed numbers and their standard and multidimensional arithmetic.https://www.zbmath.org/1456.030812021-04-16T16:22:00+00:00"Landowski, Marek"https://www.zbmath.org/authors/?q=ai:landowski.marekSummary: A shadowed set was introduced by W. Pedrycz as a concept of modeling vagueness. There are methods and algorithms for obtaining a shadowed set on the basis of a fuzzy set; a shadowed number can also be obtained from a fuzzy number. This article presents definitions of a shadowed number and two concepts of its arithmetic. The first arithmetic is called standard shadowed arithmetic (SSA) and relies on standard interval arithmetic (SIA), while the second is called multidimensional RDM shadowed arithmetic (RDMSA) and is based on multidimensional relative distance measure interval arithmetic (RDMIA). This paper presents the basic properties of operations on shadowed numbers with SSA and RDMSA. It also provides examples that show the difference between the results obtained with SSA and those obtained with multidimensional RDMSA. RDMSA introduces a multidimensional approach to the concept of uncertainty calculation results. Theories and examples presented in this paper will help to develop three-way decision methods and models, and can be applied in granular computing.