Recent zbMATH articles in MSC 03https://zbmath.org/atom/cc/032024-02-15T19:53:11.284213ZWerkzeugTailoring symbolic clothes for ancient logicshttps://zbmath.org/1526.030012024-02-15T19:53:11.284213Z"Glashoff, Klaus"https://zbmath.org/authors/?q=ai:glashoff.klausWestern studies of ancient Indian logics in the nineteenth and early twentieth century explained their findings using the traditional logic of categorical syllogisms inherited by way of the Middle Ages ultimately from Aristotle. As knowledge of classical logic, the propositional and predicate calculi pioneered by Gottlob Frege in the late nineteenth century, spread (in less idiosyncratic notations) during the twentieth century, students of Indian logics began to appeal to it. As syllogistic had done, this lens introduced presuppositions. This paper describes the two approaches and also the attempts to explain syllogistic itself in Fregean terms, highlighting what is introduced by tendentious presuppositions from the logic being used, and suggests that students of ancient Indian logics avail themselves of the wider range of formal logical approaches now available, including axiom-free natural deduction.
The presentation is weakened by a number of typographical errors (including the misspelling of Begriffsschrift on p. 113), by the choice of \(x\) and \(y\) as place-holders for terms in syllogisms on p. 110 forcing the use of the Greek letter \(\xi\) as an indexial (e.g. on p. 113) whereas \(x\) is used as an indexial in other examples, and by the unexplained use of Frege's original notation on p. 114. The identification of the traditional syllogistic with Aristotle's own system introduces obscurity into the account of existential import (Aristotle himself not only gave examples of categorical terms (p. 110) but stipulated general conditions for them. He required a term to be exemplified, unlike for example ``goat-stag'' (\(\tau\rho\alpha\gamma\varepsilon\lambda\alpha\varphi o\zeta\), or as we might say ``unicorn''), An. Post. ii 7, 92b 5--9, thereby resolving questions of existential import, just as he required it to be capable of occurring both as a subject and as a predicate, thereby excluding individual proper names, An. Pr. i 27, 43a 39. Both these requirements were dispensed with in traditional syllogistic.) The statement that the zenith of traditional logic was in in the seventeenth and eighteenth centuries (p. 111) betrays a surprising ignorance of medieval logic.
Reviewer: Jim Mackenzie (Sydney)Indeterminateness and ``the'' universe of sets: multiversism, potentialism, and pluralismhttps://zbmath.org/1526.030022024-02-15T19:53:11.284213Z"Barton, Neil"https://zbmath.org/authors/?q=ai:barton.neilVarious set-theoretic conjectures -- if true -- would privilege some special philosophical view about the universe of sets. Those views are similar and were developed in parallel with similar views about the physical universe: universism, multiversism, potentialism and pluralism. The subject of this study is how such views are implied or are less or more compatible with various unsolved problems. This situation is also similar with that discussed in the philosophy of physics, because there one also considers various hypotheses which influence our view about the physical universe.
The paper does not lack mathematical rigor. The development of philosophical views about the universe of sets is presented with motivations coming from the set theoretical research -- like the question if ZF is consistent -- and was strongly influenced by the development of various proof techniques, like forcing. There is a trade-off in science and culture. So many mathematical programs are motivated by special philosophical views, like the multiverse axiomatisations motivated by anti-universism. Finally, also a kind of pluralism is to be considered, at the level of theories and not at the level of models: to tolerate multiple competing theories of sets, and not necessarily identify one as privileged.
For the entire collection see [Zbl 1505.03005].
Reviewer: Mihai Prunescu (București)The classification of dp-minimal and dp-small fieldshttps://zbmath.org/1526.030032024-02-15T19:53:11.284213Z"Johnson, Will"https://zbmath.org/authors/?q=ai:johnson.willThe paper under review presents further progress in the classification programme of NIP fields, which originally goes back to [\textit{S. Shelah}, Isr. J. Math. 204, 1--83 (2014; Zbl 1371.03043)]. The two following conjectures (Conjectures~1.8 and 1.9) are currently at the core of this programme:
Henselianity conjecture: If \((K,v)\) is an NIP valued field, then \(v\) is Henselian.
``Shelah'' conjecture: If \(K\) is an NIP field, then at least one of the following holds:
\begin{itemize}
\item \(K\) is separably closed.
\item \(K\) is real closed.
\item \(K\) is finite.
\item \(K\) admits a definable henselian valuation.
\end{itemize}
An approach towards the conjectures that has proven fruitful in recent years is to consider certain refinements of the property NIP. First, there are well-studied ``minimality'' notions, such as (weak) o-minimality, strong minimality and \(C\)-minimality, for which (versions of) the conjectures above have already been verified. All of these three tameness conditions imply VC-minimality; and, moreover, we have the following implications:
\[
\text{VC-minimal} \Rightarrow \text{dp-small} \Rightarrow \text{dp-minimal} \Rightarrow \text{strongly dependent} \Rightarrow \text{NIP}
\]
(see page 469 for further details). The class of dp-minimal structures ``can be regarded as the simplest type of NIP structure'' (page 467).
Theorem~1.2 establishes the following instance of Shelah's conjecture: If \(K\) is a dp-minimal field, then at least one of the following holds:
\begin{itemize}
\item \(K\) is algebraically closed.
\item \(K\) is real closed.
\item \(K\) is finite.
\item \(K\) admits a non-trivial definable Henselian valuation.
\end{itemize}
As a consequence of Theorem~1.2, a classification of dp-minimal fields is given in Theorem~1.3, where further distinctions are made depending on the characteristic of \(K\).
With view to the Henselianity conjecture, dp-minimal \textit{valued} fields are considered in \S~8: Theorems~1.5 and 1.6 show that for a dp-finite valued field \((K,v)\), its valuation \(v\) is Henselian. Moreover, distinguishing between the cases that the residue field \(Kv\) is finite or infinite, a complete characterisation of dp-minimal valued fields in terms of the residue fields and value groups is established.
Several further results also allow additional structure on the valued field, its value group and its residue field. Also in this context, Theorem~1.7 addresses dp-small structures. By this theorem, any infinite dp-small field \(K\) is either real closed or algebraically closed, solving the ``VC-minimal fields conjecture'' of [\textit{V. Guingona}, Arch. Math. Logic 53, No. 5--6, 503--517 (2014; Zbl 1354.03047)].
Reviewer: Lothar Sebastian Krapp (Konstanz)Space-bounded OTMs and \(\mathrm{REG}^\infty \)https://zbmath.org/1526.030042024-02-15T19:53:11.284213Z"Carl, Merlin"https://zbmath.org/authors/?q=ai:carl.merlinThe notions of automata and Turing machines may be extended to machines whose tapes are indexed by the class of ordinals (see [the author, Ordinal computability. An introduction to infinitary machines. Berlin: De Gruyter (2019; Zbl 07046234)]), and this article presents an infinitary version of the classical result of [\textit{R. E. Stearns} et al., in: IEEE Conf. Record VI. Ann. Sympos. Circuit theory logic design 179--190 (1965), Russian translation in Probl. Mat. Logiki. Slozn. Algoritm. Klassy Vychisl. Funkcii, 301--319 (1970; Zbl 0229.02033)] that LOGLOG\-SPACE languages are regular.
In this article, the notions of ``deterministic finite automata'' and ``nondeterministic finite automata with \(\epsilon\)-moves'' are extended to obtain ``deterministic ordinal automata'' (DOA) and ``nondeterministic ordinal automata with \(\epsilon\)-moves'' (\(\epsilon\)-NOA). Considering an (infinitary) word to be a map from an ordinal to a finite alphabet, and thus an (infinitary) language to be a set of (infinitary) words, the DOAs and the \(\epsilon\)-NOAs accept the same languages, that set of languages being denoted REG\({}^{\infty}\).
Meanwhile, an ``ordinal Turing machine'' (OTM) is like the usual Turing machine, except that its tape squares could be indexed by the ordinals rather than the natural numbers. On the ordinals, the space bound on an OTM would be a function \(f\) : On \(\to\) On such that for any ordinal \(\alpha\) and any input \(x\) of length \(\alpha\), the squares of the work tape used in the computation would all be within the first \(f(\alpha)\) squares. In particular, an OTM is ``strictly space-bounded'' if \(f(\alpha) <\) card\((\alpha)\) for all sufficiently large \(\alpha \in\) On.
Several classical results, such as the Myhill-Nerode theorem, extend to REG\({}^{\infty}\), and several innocent-looking languages turn out not to be in REG\({}^{\infty}\). The primary result is that the language accepted by a strictly space-bound OTM is in REG\(^{\infty}\).
Reviewer: Gregory Loren McColm (Tampa)Forcing constructions and countable Borel equivalence relationshttps://zbmath.org/1526.030052024-02-15T19:53:11.284213Z"Gao, Su"https://zbmath.org/authors/?q=ai:gao.su"Jackson, Steve"https://zbmath.org/authors/?q=ai:jackson.steve|jackson.steve-c"Krohne, Edward"https://zbmath.org/authors/?q=ai:krohne.edward"Seward, Brandon"https://zbmath.org/authors/?q=ai:seward.brandon-mThe authors prove a number of results about countable Borel equivalence relations. These results involve multiple different topics on countable Borel equivalence relationships, and their commonality is that their proofs all use forcing constructions and arguments.
Let \(E\) be an equivalence relation on a Polish space \(X\). We say \(E\) is \textit{Borel} if it is a Borel subset of \(X\times X\). \(E\) is \textit{countable} if each \(E\)-equivalence class is countable. For \(x\in X\), we let \([x]_E\), or \([x]\) for brevity, denote the \(E\)-equivalence class of \(x\). A set \(S\subseteq X\) is a \textit{complete section} if \(S\) meets every \(E\)-equivalence class.
Let \(\Gamma\) be a countable discrete group, \(X\) a Polish space, and let \(\Gamma\curvearrowright X\) be a Borel action of \(\Gamma\) on \(X\), we say \(X\) is a \textit{Borel \(\Gamma\)-space}. In particular, if the action \(\Gamma\curvearrowright X\) is continuous, we say \(X\) is a \textit{Polish \(\Gamma\)-space}. The \textit{orbit equivalence relation} \(E_\Gamma^X\) defined by \[E_\Gamma^X=\{(x,y)\in X\times X:\exists g\in\Gamma\,(g\cdot x=y)\}.\] It is well known that, every countable Borel equivalence relation is of the form \(E_\Gamma^X\) for some Borel \(\Gamma\)-space. Let \(A\subseteq\Gamma\) and \(x\in X\), we denote \(A\cdot x=\{g\cdot x:g\in A\}\). Let \(e_\Gamma\) be the identity element of \(\Gamma\), the \textit{free part} of \(X\) is \[F(X)=\{x\in X:\forall g\in\Gamma\,(e_\Gamma\ne g\Rightarrow g\cdot x\ne x)\}.\] If \(F(X)=X\), we say that the action is \textit{free}.
The following are results presented in this article.
Theorem. Let \(\Gamma\) be a countable group, \(X\) a compact Polish \(\Gamma\)-space. Let \((S_n)_{n\in\mathbb N}\) be a sequence of Borel complete sections of \(E_\Gamma^X\). If \((A_n)_{n\in\mathbb N}\) is any sequence of finite subsets of \(\Gamma\) such that every finite subset of \(\Gamma\) is contained in some \(A_n\), then there is an \(x\in X\) such that \(A_n\cdot x\cap S_n\ne\emptyset\) for infinitely may \(n\).
For a countable group \(\Gamma\), the \textit{Bernoulli (left) shift} of \(\Gamma\) is the action \(\Gamma\curvearrowright 2^\Gamma\) defined by \((g\cdot x)(h)=x(g^{-1}h)\) for \(x\in 2^\Gamma\) and \(g,h\in\Gamma\). The orbit equivalence relation of this action denoted by \(E_\Gamma\), and the free part of this action denoted by \(F(2^\Gamma)\).
Theorem. If \(B\subseteq F(2^\Gamma)\) is a Borel complete section, then there is \(x\in F(2^\Gamma)\) and finite \(T\subseteq\Gamma\) such that \(T\cdot y\cap B\ne\emptyset\) for any \(y\in[x]\).
Moreover, in the case of \(\Gamma={\mathbb Z}^d\), we put \(\|(g_1,\ldots,g_d)\|=|g_1|+\cdots+|g_d|\).
Theorem. Let \(d\ge 1\). If \(B\subseteq F(2^{{\mathbb Z}^d})\) is a Borel complete section, then there is \(x\in F(2^{{\mathbb Z}^d})\) and finite \(T\subseteq\{g\in{\mathbb Z}^d:\|g\|\mbox{ is odd }\}\) such that \(T\cdot y\cap B\ne\emptyset\) for any \(y\in[x]\).
Theorem. Let \(d\ge 1\). If \(B\subseteq F(2^{{\mathbb Z}^d})\) is a Borel complete section, then there is an \(x\in F(2^{{\mathbb Z}^d})\) and a lattice \(L=k+\{(i_1h_1,\ldots,i_dh_d):i_1,\ldots,i_d\in{\mathbb Z}\}\subseteq{\mathbb Z}^d\) such that \(L\cdot x\subseteq B\).
In \({\mathbb Z}^d\), we put \(e_1=(1,0,\ldots,0),\ldots,e_2=(0,1,\ldots,0),\ldots,e_d=(0,0,\ldots,1)\), and denote \(S=\{\pm e_i:i=1,2,\ldots,d\}\subseteq{\mathbb Z}^d\).
Definition. Let \((T_n)\) be a sequence of subequivalence relation of \(E_{{\mathbb Z}^d}\) on some subsets \(\mathrm{dom}(T_n)\subseteq F(2^{{\mathbb Z}^d})\) with each \(T_n\)-equivalence class finite, and \(\bigcup_n\mathrm{dom}(T_n)=F(2^{{\mathbb Z}^d})\). We say \((T_n)\) is a \textit{(unlayered) toast} if:
\begin{itemize}
\item[(1)] For each \(T_n\)-equivalence class \(C\), and each \(T_m\)-equivalence class \(C'\) whit \(m>n\), if \(C\cap C'\ne\emptyset\), then \(C\subseteq C'\).
\item[(2)] For each \(T_n\)-equivalence class \(C\), there is \(m>n\) and a \(T_m\)-equivalence class \(C'\) such that \(C\cup(C+S)\subseteq C'\).
\end{itemize}
We say \((T_n)\) is a \textit{layered toast} if, instead of \((2)\) above, we have
\begin{itemize}
\item[(2')] For each \(T_n\)-equivalence class \(C\), there is a \(T_{n+1}\)-equivalence class \(C'\) such that \(C\cup(C+S)\subseteq C'\).
\end{itemize}
Theorem. There is no Borel layered toast on \(F(2^{{\mathbb Z}^d})\).
The last result concerning the shape of marker regions, which is a useful tool in studying Borel actions of countable abelian groups.
Theorem. There does not exist a sequence of \({\mathcal R}_n\) of Borel subequivalence relations on \(F(2^{{\mathbb Z}^2})\) satisfying all the following:
\begin{itemize}
\item[(1)] (regular shape) For each \(n\), each equivalence class (marker region) \(R\) of \({\mathcal R}_n\) is a rectangle.
\item[(2)] (bounded size) For each \(n\), there is an upper bound \(w(n)\) on the size of the edge lengths of the marker regions \(R\) in \({\mathcal R}_n\).
\item[(3)] (increasing size) Letting \(v(n)\) denote the smallest edge length of a marker region \(R\) of \({\mathcal R}_n\), we have \(\lim_nv(n)=+\infty\).
\item[(4)] (vanishing boundary) For each \(x\in F(2^{{\mathbb Z}^2})\), we have \(\lim_n\rho(x,\partial{\mathcal R}_n)=+\infty\).
\end{itemize}
Reviewer: Longyun Ding (Tianjin)Subsets of virtually nilpotent groups with the SBM propertyhttps://zbmath.org/1526.200522024-02-15T19:53:11.284213Z"Burkhart, Ryan"https://zbmath.org/authors/?q=ai:burkhart.ryan"Goldbring, Isaac"https://zbmath.org/authors/?q=ai:goldbring.isaac\textit{S. Leth} [Stud. Log. 47, No. 3, 265--278 (1989; Zbl 0675.03042)] introduced a property of subsets of the natural numbers that he called the SIM (short for standard interval measure) property. The property is motivated by ideas from nonstandard analysis and roughly speaking, demands that there be a connection between the internal notion of small gap sizes on hyperfinite intervals and the external notion of the (normalized) standard part of the set having large Lebesgue measure.
In the paper under review, the authors extend Leth's notion of subsets of the integers satisfying SIM to the class of virtually nilpotent groups. In order to do this they define a natural measure on closed balls in asymptotic cones associated with such groups and show that this measure satisfies the Lebesgue density theorem. They then prove analogs of various properties known to hold for SIM sets in this broader context, occasionally assuming extra properties of the group, such as the small spheres property and the small gaps property.
Reviewer: Egle Bettio (Venezia)Polish topologies on groups of non-singular transformationshttps://zbmath.org/1526.220162024-02-15T19:53:11.284213Z"Le Maître, François"https://zbmath.org/authors/?q=ai:le-maitre.francoisSummary: In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel \(\sigma\)-finite measures \(\lambda\) on a standard Borel space for which the group of \(\lambda\)-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.Mixing and double recurrence in probability groupshttps://zbmath.org/1526.370042024-02-15T19:53:11.284213Z"Tserunyan, Anush"https://zbmath.org/authors/?q=ai:tserunyan.anushThe author defines a class probability groups, namely a class of groups equipped with an invariant probability measure. This class also contains the ultraproducts of all locally compact unimodular amenable groups. The basics of the theory of probability/measure-preserving actions for probability groups, including a natural notion of mixing, are developed. A number of connections between mixing and double recurrence in this setting is obtained.
Reviewer: Michael L. Blank (Moskva)The geometry of discrete \(L\)-algebrashttps://zbmath.org/1526.510012024-02-15T19:53:11.284213Z"Rump, Wolfgang"https://zbmath.org/authors/?q=ai:rump.wolfgangSummary: The relationship of discrete \(L\)-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete \(L\)-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete \(L\)-algebra \(X\) is determined and shown to be a complete invariant. It is proved that \(X \setminus \{1\}\) is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality \(n > 3\), a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated \(L\)-algebra is determined.Characterizing time computational complexity classes with polynomial differential equationshttps://zbmath.org/1526.680012024-02-15T19:53:11.284213Z"Gozzi, Riccardo"https://zbmath.org/authors/?q=ai:gozzi.riccardo"Graça, Daniel"https://zbmath.org/authors/?q=ai:graca.daniel-silvaThe paper presents characterisations of the complexity classes FEXPTIME and EXPTIME as well as for the classes defining the Grzegorczyk hierarchy by polynomial ordinary differential equations (ODEs). From previous work [\textit{O. Bournez} et al., LIPIcs -- Leibniz Int. Proc. Inform. 55, Article 109, 15 p. (2016; Zbl 1388.68041); J. ACM 64, No. 6, Article No. 38, 76 p. (2017; Zbl 1426.68088)] a characterisation by ODEs of the complexity class P was known. This led to the question whether the presented approach could be transferred to other (larger) classes. The paper is written in a very good style and explains well many of the approaches, ideas and concepts used in the paper. The definitions and results presented are illustrated with helpful examples and figures so that the reader cannot get lost.
As a first step, the authors provide the reader with the approach to model computational processes, e.g. those of a Turing machine, within their framework of ODEs. Intuitively, one uses limits that approximate the correct answer with decreasing error as the computation time goes to infinity. This is where the class ATS comes in, which abbreviates \textit{analog time space}. Note that -- unlike for Turing machines -- in this class the time and space bounds are completely independent. A crucial point in extending the well-known characterisation of P/FP [\textit{O. Bournez} et al., J. Complexity 36, 106--140 (2016; Zbl 1474.68151)] to higher classes was to understand which properties of ATS need to be preserved.
Continuing their approach, they show how to encode the transition function of a TM into a function in ATSP (the variant of ATS restricted to functions with polynomial time and space bounds). Simulating the computation of a TM over real numbers requires iterating this function over the set of its configurations. This poses some problems and requires a more sophisticated approach, which the authors provide.
In Section 4, the authors begin to focus on the main problem in lifting the known construction to FEXPTIME: the closure under composition of ATSE (the exponential variant of ATS) is not obvious. It is clear that compositions of exponentials do not generally yield exponentials, and therefore this class is not closed under this operation. However, the authors explain how to solve this problem and again present a nice intuition with an example. After some work on the properties of the exponential analog classes, the authors present an analog characterisation of FEXPTIME (Theorem 29).
They then show how to go beyond the exponential case and finally use Theorem 35 to generalise the Grzegorczyk hierarchy with ODEs. They present an analog characterisation of each level of this hierarchy.
Finally, the authors arrive at a characterisation of the class EXPTIME. To do this, they have to define what acceptance of input strings means at this level and present a threshold-like approach (see Fig.~7).
Finally, conclusions are drawn and natural open questions are posed, e.g., do such characterisations exist for the complexity class NP? The previous results are only for deterministic classes, so this question is quite interesting.
Reviewer: Arne Meier (Hannover)On CDCL-based proof systems with the ordered decision strategyhttps://zbmath.org/1526.680112024-02-15T19:53:11.284213Z"Mull, Nathan"https://zbmath.org/authors/?q=ai:mull.nathan"Pang, Shuo"https://zbmath.org/authors/?q=ai:pang.shuo"Razborov, Alexander"https://zbmath.org/authors/?q=ai:razborov.alexander-aSummary: We prove that conflict-driven clause learning (CDCL) SAT-solvers with the ordered decision strategy and the DECISION learning scheme are equivalent to ordered resolution. We also prove that by replacing this learning scheme with its opposite, which stops after the first nonconflict clause when backtracking, they become equivalent to general resolution. To the best of our knowledge, along with [\textit{M. Vinyals}, ``Hard examples for common variable decision heuristics,'' in: Proceedings of the 34th AAAI conference on artificial intelligence 2020, 1652--1659 (2020)], this is the first theoretical study of the interplay between specific decision strategies and clause learning. For both results, we allow nondeterminism in the solver's ability to perform unit propagation, conflict analysis, and restarts, in a way that is similar to previous works in the literature. To aid the presentation of our results, and possibly future research, we define a model and language for discussing CDCL-based proof systems that allow for succinct and precise theorem statements.
For the conference version of this paper see [Lect. Notes Comput. Sci. 12178, 149--165 (2020; Zbl 1523.68166)].Thermodynamically stable phases of asymptotically flat Lovelock black holeshttps://zbmath.org/1526.830132024-02-15T19:53:11.284213Z"Wu, Jerry"https://zbmath.org/authors/?q=ai:wu.jerry-chun-teh"Mann, Robert B."https://zbmath.org/authors/?q=ai:mann.robert-bSummary: We present the first examples of phase transitions in asymptotically flat black hole solutions. We analyse the thermodynamic properties of black holes in order \(N \geqslant 3\) Lovelock gravity, with zero cosmological constant. We find a new type of `inverted' swallowtail indicative of stable temperature regions for an otherwise unstable neutral black hole, and demonstrate multiple such stable phases can exist and coexist at multi-critical points. We also find that for charged black holes, ordinary swallowtails can exist on the stable Gibbs free energy branch, allowing for multiple first order phase transitions as seen for anti-de Sitter (AdS) black holes. A triple point for \(N = 5\) and a quadruple point for \(N = 7\) are presented explicitly. We investigate changes in the Gibbs free energy as the lowest order Lovelock constant is varied, and draw comparisons to pressure changes for AdS black hole systems.A multicriteria group decision-making method based on AIVIFSs, Z-numbers, and trapezium cloudshttps://zbmath.org/1526.900232024-02-15T19:53:11.284213Z"Jia, Qianlei"https://zbmath.org/authors/?q=ai:jia.qianlei"Hu, Jiayue"https://zbmath.org/authors/?q=ai:hu.jiayue"He, Qizhi"https://zbmath.org/authors/?q=ai:he.qizhi"Zhang, Weiguo"https://zbmath.org/authors/?q=ai:zhang.weiguo.1|zhang.weiguo|zhang.weiguo.2"Safwat, Ehab"https://zbmath.org/authors/?q=ai:safwat.ehabSummary: Multicriteria group decision-making (MCGDM), with the strong uncertainty and randomness, has always been a hotspot in the world. The chief purpose of the paper is to address the problem with Atanassov's interval-valued intuitionistic fuzzy sets (AIVIFSs), Z-numbers, and trapezium clouds. First, some related concepts and former operators of AIVIFSs, Z-numbers, and trapezium clouds are reviewed, meanwhile, AIVIFSs and Z-numbers are synthesized to come up with a novel linguistic expression. Then, Z-trapezium-trapezium cloud (ZTTC) is proposed to quantify the linguistic evaluation information to avoid excessive computation caused by traditional methods. Later, a new approach of calculating the objective weight vector is presented based on entropy weight method (EWM). To take the huge advantages of technique for order preference by similarity to ideal solution (TOPSIS) method in ranking, 2-norm in mathematical theory is applied to derive a way of calculating the distance between different ZTTCs. Finally, an example about the grade assessment of coronavirus Disease 2019 (COVID-19) is given. For further confirming the validity and feasibility, sensitivity analysis and comparison with other methods are conducted.Stationary status of discrete and continuous age-structured population modelshttps://zbmath.org/1526.920502024-02-15T19:53:11.284213Z"Srinivasa Rao, Arni S. R."https://zbmath.org/authors/?q=ai:rao.arni-s-r-srinivasa"Carey, James R."https://zbmath.org/authors/?q=ai:carey.james-rSummary: From Leonhard Euler to Alfred Lotka and in recent years understanding the stationary process of the human population has been of central interest to scientists. Population reproductive measure NRR (net reproductive rate) has been widely associated with measuring the status of population stationarity and it is also included as one of the measures in the millennium development goals. This article argues how the partition theorem-based approach provides more up-to-date and timely measures to find the status of the population stationarity of a country better than the NRR-based approach. We question the timeliness of the value of NRR in deciding the stationary process of the country. We prove associated theorems on discrete and continuous age distributions and derive measurable functional properties. The partitioning metric captures the underlying age structure dynamic of populations at or near stationarity. As the population growth rates for an ever-increasing number of countries trend towards replacement levels and below, new demographic concepts and metrics are needed to better characterize this emerging global demography.