Annals of Pure and Applied Logic最新文献

{"title":"Classification of ℵ0-categorical C-minimal pure C-sets","authors":"Françoise Delon ,&nbsp;Marie-Hélène Mourgues","doi":"10.1016/j.apal.2023.103375","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103375","url":null,"abstract":"<div><p>We classify all <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal <em>C</em>-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical <em>C</em>-minimal sets as a first step. We first define <em>solvable</em> good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without endpoints and the same number of branches at each node. The class of <em>colored</em> good trees is the elementary class of solvable good trees. We show that a pure <em>C</em>-set <em>M</em> is indiscernible, finite or <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal iff its canonical tree <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is a colored good tree. The classification of general <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal <em>C</em>-sets is done via finite trees with labeled vertices and edges, where labels are natural numbers, or infinity and complete theories of indiscernible, <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical or finite, and <em>C</em>-minimal <em>C</em>-sets.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Positive modal logic beyond distributivity","authors":"Nick Bezhanishvili ,&nbsp;Anna Dmitrieva ,&nbsp;Jim de Groot ,&nbsp;Tommaso Moraschini","doi":"10.1016/j.apal.2023.103374","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103374","url":null,"abstract":"<div><p>We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-persistence and show that every weak positive modal logic is <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.<span>¹</span></p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49737990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak saturation properties and side conditions","authors":"Monroe Eskew","doi":"10.1016/j.apal.2023.103356","DOIUrl":"10.1016/j.apal.2023.103356","url":null,"abstract":"<div><p>Towards combining “compactness” and “hugeness” properties at <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42596957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers","authors":"Ur Ya'ar","doi":"10.1016/j.apal.2023.103358","DOIUrl":"10.1016/j.apal.2023.103358","url":null,"abstract":"<div><p>We prove that the theory of the models constructible using finitely many cofinality quantifiers – <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mo>&lt;</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mo>&lt;</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> for <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to &lt;<em>μ</em>-closed sets.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45405170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized fusible numbers and their ordinals","authors":"Alexander I. Bufetov ,&nbsp;Gabriel Nivasch ,&nbsp;Fedor Pakhomov","doi":"10.1016/j.apal.2023.103355","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103355","url":null,"abstract":"<div><p>Erickson defined the <em>fusible numbers</em> as a set <span><math><mi>F</mi></math></span> of reals generated by repeated application of the function <span><math><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Erickson, Nivasch, and Xu showed that <span><math><mi>F</mi></math></span> is well ordered, with order type <span><math><msub><mrow><mi>ε</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. They also investigated a recursively defined function <span><math><mi>M</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span>. They showed that the set of points of discontinuity of <em>M</em> is a subset of <span><math><mi>F</mi></math></span> of order type <span><math><msub><mrow><mi>ε</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. They also showed that, although <em>M</em> is a total function on <span><math><mi>R</mi></math></span>, the fact that the restriction of <em>M</em> to <span><math><mi>Q</mi></math></span> is total is not provable in first-order Peano arithmetic <span><math><mi>PA</mi></math></span>.</p><p>In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets <span><math><mi>F</mi></math></span> of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>R</mi></math></span>.</p><p>The most straightforward generalization of <span><math><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> to an <em>n</em>-ary function is the function <span><math><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>. We show that this function generates a set <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whose order type is just <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo></math></span>. For this, we develop recursively defined functions <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> naturally generalizing the function <em>M</em>.</p><p>Furthermore, we prove that for any <em>linear</em> function <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>R</mi></math></span>, the order type of the resulting <span><math><mi>F</mi></math></span> is at most <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>0</mn><","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49724785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extensions of Solovay's system S without independent sets of axioms","authors":"Igor Gorbunov ,&nbsp;Dmitry Shkatov","doi":"10.1016/j.apal.2023.103360","DOIUrl":"10.1016/j.apal.2023.103360","url":null,"abstract":"<div><p>Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system <strong>S</strong>, which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of <strong>S</strong> without deductively independent sets of axioms.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45257728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Indestructibility of some compactness principles over models of PFA","authors":"Radek Honzik ,&nbsp;Chris Lambie-Hanson ,&nbsp;Šárka Stejskalová","doi":"10.1016/j.apal.2023.103359","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103359","url":null,"abstract":"<div><p>We show that <span><math><mi>PFA</mi></math></span> (Proper Forcing Axiom) implies that adding any number of Cohen subsets of <em>ω</em> will not add an <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Aronszajn tree or a weak <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-Kurepa tree, and moreover no <em>σ</em>-centered forcing can add a weak <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-Kurepa tree (a tree of height and size <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> with at least <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> cofinal branches). This partially answers an open problem whether ccc forcings can add <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-Aronszajn trees or <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-Kurepa trees (with <span><math><mo>¬</mo><msub><mrow><mo>□</mo></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math></span> in the latter case).</p><p>We actually prove more: We show that a consequence of <span><math><mi>PFA</mi></math></span>, namely the <em>guessing model principle</em>, <span><math><mi>GMP</mi></math></span>, which is equivalent to the <em>ineffable slender tree property</em>, <span><math><mi>ISP</mi></math></span>, is preserved by adding any number of Cohen subsets of <em>ω</em>. And moreover, <span><math><mi>GMP</mi></math></span> implies that no <em>σ</em>-centered forcing can add a weak <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-Kurepa tree (see Section <span>2.1</span> for definitions).</p><p>For more generality, we study variations of the principle <span><math><mi>GMP</mi></math></span> at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-Kurepa trees and no <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>ω</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>-Aronszajn trees.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49724782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On countably perfectly meager and countably perfectly null sets","authors":"Tomasz Weiss ,&nbsp;Piotr Zakrzewski","doi":"10.1016/j.apal.2023.103357","DOIUrl":"10.1016/j.apal.2023.103357","url":null,"abstract":"<div><p>We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set.</p><p>We say that a subset <em>A</em> of a perfect Polish space <em>X</em> is countably perfectly meager (respectively, countably perfectly null) in <em>X</em>, if for every perfect Polish topology <em>τ</em> on <em>X</em>, giving the original Borel structure of <em>X</em>, <em>A</em> is covered by an <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>-set <em>F</em> in <em>X</em> with the original Polish topology such that <em>F</em> is meager with respect to <em>τ</em> (respectively, for every finite, non-atomic, Borel measure <em>μ</em> on <em>X</em>, <em>A</em> is covered by an <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>σ</mi></mrow></msub></math></span>-set <em>F</em> in <em>X</em> with <span><math><mi>μ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>).</p><p>We prove that if <span><math><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>≤</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then there exists a universally meager set in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span> which is not countably perfectly meager in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span> (respectively, a universally null set in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span> which is not countably perfectly null in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span>).</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45641003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Primitive recursive reverse mathematics","authors":"Nikolay Bazhenov ,&nbsp;Marta Fiori-Carones ,&nbsp;Lu Liu ,&nbsp;Alexander Melnikov","doi":"10.1016/j.apal.2023.103354","DOIUrl":"10.1016/j.apal.2023.103354","url":null,"abstract":"<div><p>We use a second-order analogy <span><math><msup><mrow><mi>PRA</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of <span><math><mi>PRA</mi></math></span> to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that <span><math><msup><mrow><mi>PRA</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is sufficiently robust to serve as an alternative base system below <span><math><msub><mrow><mi>RCA</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> to study the proof-theoretic content of theorems in ordinary mathematics. (The most popular alternative is perhaps <span><math><msubsup><mrow><mi>RCA</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>.) We discover that many theorems that are known to be true in <span><math><msub><mrow><mi>RCA</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> either hold in <span><math><msup><mrow><mi>PRA</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> or are equivalent to <span><math><msub><mrow><mi>RCA</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> or its weaker (but natural) analogy <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup></math></span>-<span><math><msub><mrow><mi>RCA</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> over <span><math><msup><mrow><mi>PRA</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. However, we also discover that some standard mathematical and combinatorial facts are incomparable with these natural subsystems.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42904058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large cardinals at the brink","authors":"W. Hugh Woodin","doi":"10.1016/j.apal.2023.103328","DOIUrl":"10.1016/j.apal.2023.103328","url":null,"abstract":"<div><p>Kunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44436569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}