Canadian Journal of Mathematics最新文献

{"title":"Higher Chow cycles on a family of Kummer surfaces","authors":"Ken Sato","doi":"10.4153/s0008414x24000415","DOIUrl":"https://doi.org/10.4153/s0008414x24000415","url":null,"abstract":"<p>We construct a collection of families of higher Chow cycles of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918000838399-0173:S0008414X24000415:S0008414X24000415_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(2,1)$</span></span></img></span></span> on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918000838399-0173:S0008414X24000415:S0008414X24000415_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ge 18$</span></span></img></span></span> in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random analytic functions with a prescribed growth rate in the unit disk","authors":"Xiang Fang, Pham Trong Tien","doi":"10.4153/s0008414x24000403","DOIUrl":"https://doi.org/10.4153/s0008414x24000403","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {R}f$</span></span></img></span></span> be the randomization of an analytic function over the unit disk in the complex plane <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*}mathcal{R} f(z) =sum_{n=0}^infty a_n X_n z^n in H({mathbb D}), end{align*} $$</span></span></img></span>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f(z)=sum _{n=0}^infty a_n z^n in H({mathbb D})$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(X_n)_{n geq 0}$</span></span></img></span></span> is a standard sequence of independent Bernoulli, Steinhaus, or complex Gaussian random variables. In this paper, we demonstrate that prescribing a polynomial growth rate for random analytic functions over the unit disk leads to rather satisfactory characterizations of those <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$f in H({mathbb D})$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240516082445098-0753:S0008414X24000403:S0008414X24000403_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal R} f$</span></span></img></span></span> admits a given rate almost surely. In particular, we show that the growth rate of the random functions, the growth rate of their Taylor coefficients, and the asymptotic distribution of their zero sets can mutually, completely determine each other. Although the problem is purely complex analytic, the key strategy in the proofs is to introduce a class of auxiliary Banach spaces, which facilitate quantitative estimates.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Largest exact structures and almost split sequences on hearts of twin cotorsion pairs","authors":"Yu Liu, Wuzhong Yang, Panyue Zhou","doi":"10.4153/s0008414x2400035x","DOIUrl":"https://doi.org/10.4153/s0008414x2400035x","url":null,"abstract":"<p>Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull–Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"9 45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140838302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tensorially absorbing inclusions of C*-algebras","authors":"Pawel Sarkowicz","doi":"10.4153/s0008414x24000324","DOIUrl":"https://doi.org/10.4153/s0008414x24000324","url":null,"abstract":"<p>When <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span> is strongly self-absorbing, we say an inclusion <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$B subseteq A$</span></span></img></span></span> of C*-algebras is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span>-stable if it is isomorphic to the inclusion <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$B otimes mathcal {D} subseteq A otimes mathcal {D}$</span></span></img></span></span>. We give ultrapower characterizations and show that if a unital inclusion is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span>-stable, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span>-stability can be exhibited for countably many intermediate C*-algebras concurrently. We show that such unital embeddings between unital <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span>-stable C*-algebras are point-norm dense in the set of all unital embeddings, and that every unital embedding between <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430053820262-0734:S0008414X24000324:S0008414X24000324_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {D}$</span></span></img></span></span>-stable C*-algebras is approximately unitarily equivalent to a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140838846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Range inclusion and diagonalization of complex symmetric operators","authors":"Cun Wang, Jiayi Zhao, Sen Zhu","doi":"10.4153/s0008414x24000294","DOIUrl":"https://doi.org/10.4153/s0008414x24000294","url":null,"abstract":"<p>We consider the range inclusion and the diagonalization in the Jordan algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {S}_C$</span></span></img></span></span> of <span>C</span>-symmetric operators, that are, bounded linear operators <span>T</span> satisfying <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$CTC =T^{*}$</span></span></img></span></span>, where <span>C</span> is a conjugation on a separable complex Hilbert space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal H$</span></span></img></span></span>. For <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Tin mathcal {S}_C$</span></span></img></span></span>, we aim to describe the set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$C_{mathcal {R}(T)}$</span></span></img></span></span> of those operators <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Ain mathcal {S}_C$</span></span></img></span></span> satisfying the range inclusion <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {R}(A)subset mathcal {R}(T)$</span></span></img></span></span>. It is proved that (i) <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$C_{mathcal {R}(T)}=Tmathcal {S}_C T$</span></span></img></span></span> if and only if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417112820850-0588:S0008414X24000294:S0008414X24000294_inline9.png\"><span d","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"266 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140610047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Catching a robber on a random k-uniform hypergraph","authors":"Joshua Erde, Mihyun Kang, Florian Lehner, Bojan Mohar, Dominik Schmid","doi":"10.4153/s0008414x24000270","DOIUrl":"https://doi.org/10.4153/s0008414x24000270","url":null,"abstract":"<p>The game of <span>Cops and Robber</span> is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The <span>cop number</span> of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an <span>n</span>-vertex connected graph is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$O(sqrt {n})$</span></span></img></span></span>. In 2016, Prałat and Wormald showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreover, Łuczak and Prałat showed that on a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$log $</span></span></img></span></span>-scale the cop number demonstrates a surprising <span>zigzag</span> behavior in dense regimes of the binomial random graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$G(n,p)$</span></span></img></span></span>. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the <span>k</span>-uniform binomial random hypergraph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G^k(n,p)$</span></span></img></span></span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Oleft (sqrt {frac {n}{k}}, log n right )$</span></span></img></span></span> for a broad range of parameters <span>p</span> and <span>k</span> and that on a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240425084516565-0876:S0008414X24000270:S0008414X24000270_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$log $</span></span></img></span></span>-scale our upper bound on the cop number arises as the minimum of <span>two</span> complementary zigzag curves, as opposed to the case of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/bin","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Calibrated geometry in hyperkähler cones, 3-Sasakian manifolds, and twistor spaces","authors":"Benjamin Aslan, Spiro Karigiannis, Jesse Madnick","doi":"10.4153/s0008414x24000282","DOIUrl":"https://doi.org/10.4153/s0008414x24000282","url":null,"abstract":"<p>We systematically study calibrated geometry in hyperkähler cones <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$C^{4n+4}$</span></span></img></span></span>, their 3-Sasakian links <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$M^{4n+3}$</span></span></img></span></span>, and the corresponding twistor spaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Z^{4n+2}$</span></span></img></span></span>, emphasizing the relationships between submanifold geometries in various spaces. Our analysis highlights the role played by a canonical <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {Sp}(n)mathrm {U}(1)$</span></span></img></span></span>-structure <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$gamma $</span></span></img></span></span> on the twistor space <span>Z</span>. We observe that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {Re}(e^{- i theta } gamma )$</span></span></img></span></span> is an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240413094014785-0335:S0008414X24000282:S0008414X24000282_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$S^1$</span></span></img></span></span>-family of semi-calibrations and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri–Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler–Einstein and nearly Kähler structures.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140584792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SOBOLEV TRACE-TYPE INEQUALITIES VIA TIME-SPACE FRACTIONAL HEAT EQUATIONS","authors":"Yongrui Tang, Pengtao Li, Rui Hu, Zhichun Zhai","doi":"10.4153/s0008414x24000269","DOIUrl":"https://doi.org/10.4153/s0008414x24000269","url":null,"abstract":"","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"115 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140381459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence rate of entropy-regularized multi-marginal optimal transport costs","authors":"Luca Nenna, Paul Pegon","doi":"10.4153/s0008414x24000257","DOIUrl":"https://doi.org/10.4153/s0008414x24000257","url":null,"abstract":"<p>We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240404055322598-0693:S0008414X24000257:S0008414X24000257_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$varepsilon $</span></span></img></span></span> tends to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240404055322598-0693:S0008414X24000257:S0008414X24000257_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span>. We establish lower and upper bounds on the difference with the unregularized cost of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240404055322598-0693:S0008414X24000257:S0008414X24000257_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Cvarepsilon log (1/varepsilon )+O(varepsilon )$</span></span></img></span></span> for some explicit dimensional constants <span>C</span> depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semiconcave costs for a finer estimate, and lower bounds for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240404055322598-0693:S0008414X24000257:S0008414X24000257_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {C}^2$</span></span></img></span></span> costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric structures in pseudo-random graphs","authors":"Thang Pham, Steven Senger, Michael Tait, Vu Thi Huong Thu","doi":"10.4153/s0008414x24000245","DOIUrl":"https://doi.org/10.4153/s0008414x24000245","url":null,"abstract":"<p>In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140585317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}