In this paper, we prove a support theorem of Stroock–Varadhan type for pinned diffusion processes. To this end, we use two powerful results from stochastic analysis. One is quasi-sure analysis for Brownian rough path. The other is Aida–Kusuoka–Stroock’s positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals.
{"title":"SUPPORT THEOREM FOR PINNED DIFFUSION PROCESSES","authors":"Y. Inahama","doi":"10.1017/nmj.2023.25","DOIUrl":"https://doi.org/10.1017/nmj.2023.25","url":null,"abstract":"\u0000 In this paper, we prove a support theorem of Stroock–Varadhan type for pinned diffusion processes. To this end, we use two powerful results from stochastic analysis. One is quasi-sure analysis for Brownian rough path. The other is Aida–Kusuoka–Stroock’s positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47326768","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}
Abstract In this note, we correct an oversight regarding the modules from Definition 4.2 and proof of Lemma 5.12 in Baur et al. (Nayoga Math. J., 2020, 240, 322–354). In particular, we give a correct construction of an indecomposable rank $2$ module $operatorname {mathbb {L}}nolimits (I,J)$ , with the rank 1 layers I and J tightly $3$ -interlacing, and we give a correct proof of Lemma 5.12.
在本文中，我们纠正了Baur et al. (Nayoga Math)中关于定义4.2和引理5.12证明的模块的疏忽。[J] .生物医学工程学报，2020,24(2):322-354。特别地，我们给出了秩为$2$的不可分解模块$operatorname {mathbb {L}}nolimits (I,J)$的正确构造，其中秩为1的层I和J紧密交错，并给出了引理5.12的正确证明。
{"title":"CORRIGENDUM TO “CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS”","authors":"K. Baur, Dusko Bogdanic, Ana GARCIA ELSENER","doi":"10.1017/nmj.2022.7","DOIUrl":"https://doi.org/10.1017/nmj.2022.7","url":null,"abstract":"Abstract In this note, we correct an oversight regarding the modules from Definition 4.2 and proof of Lemma 5.12 in Baur et al. (Nayoga Math. J., 2020, 240, 322–354). In particular, we give a correct construction of an indecomposable rank \u0000$2$\u0000 module \u0000$operatorname {mathbb {L}}nolimits (I,J)$\u0000 , with the rank 1 layers I and J tightly \u0000$3$\u0000 -interlacing, and we give a correct proof of Lemma 5.12.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"249 1","pages":"269 - 273"},"PeriodicalIF":0.8,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41648792","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}
Abstract A class of exotic $_3F_2(1)$ -series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these $_3F_2(1)$ -series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.
{"title":"EVALUATION OF CERTAIN EXOTIC \u0000$_3F_2$\u0000 (1)-SERIES","authors":"Marta Na Chen, W. Chu","doi":"10.1017/nmj.2022.23","DOIUrl":"https://doi.org/10.1017/nmj.2022.23","url":null,"abstract":"Abstract A class of exotic \u0000$_3F_2(1)$\u0000 -series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these \u0000$_3F_2(1)$\u0000 -series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"249 1","pages":"107 - 118"},"PeriodicalIF":0.8,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48019451","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}
C'edric Dion, Antonio Lei, Anwesh Ray, Daniel Vallières
Let $ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $ell $ -towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $ell $ and the abelian $ell $ -tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $mu $ and $lambda $ invariants.
{"title":"ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS","authors":"C'edric Dion, Antonio Lei, Anwesh Ray, Daniel Vallières","doi":"10.1017/nmj.2023.18","DOIUrl":"https://doi.org/10.1017/nmj.2023.18","url":null,"abstract":"\u0000\t <jats:p>Let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$ell $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$ell $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$ell $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and the abelian <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$ell $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$mu $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0027763023000181_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> invariants.</jats:p>","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45398484","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}
Abstract Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $mathcal {B}(R)$ over a local domain R essentially of finite type over $mathbb {C}$ . We show that if R is normal and $Delta $ is an effective $mathbb {Q}$ -Weil divisor on $operatorname {Spec} R$ such that $K_R+Delta $ is $mathbb {Q}$ -Cartier, then the BCM test ideal $tau _{widehat {mathcal {B}(R)}}(widehat {R},widehat {Delta })$ of $(widehat {R},widehat {Delta })$ with respect to $widehat {mathcal {B}(R)}$ coincides with the multiplier ideal $mathcal {J}(widehat {R},widehat {Delta })$ of $(widehat {R},widehat {Delta })$ , where $widehat {R}$ and $widehat {mathcal {B}(R)}$ are the $mathfrak {m}$ -adic completions of R and $mathcal {B}(R)$ , respectively, and $widehat {Delta }$ is the flat pullback of $Delta $ by the canonical morphism $operatorname {Spec} widehat {R}to operatorname {Spec} R$ . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.
{"title":"BIG COHEN–MACAULAY TEST IDEALS IN EQUAL CHARACTERISTIC ZERO VIA ULTRAPRODUCTS","authors":"T. Yamaguchi","doi":"10.1017/nmj.2022.41","DOIUrl":"https://doi.org/10.1017/nmj.2022.41","url":null,"abstract":"Abstract Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra \u0000$mathcal {B}(R)$\u0000 over a local domain R essentially of finite type over \u0000$mathbb {C}$\u0000 . We show that if R is normal and \u0000$Delta $\u0000 is an effective \u0000$mathbb {Q}$\u0000 -Weil divisor on \u0000$operatorname {Spec} R$\u0000 such that \u0000$K_R+Delta $\u0000 is \u0000$mathbb {Q}$\u0000 -Cartier, then the BCM test ideal \u0000$tau _{widehat {mathcal {B}(R)}}(widehat {R},widehat {Delta })$\u0000 of \u0000$(widehat {R},widehat {Delta })$\u0000 with respect to \u0000$widehat {mathcal {B}(R)}$\u0000 coincides with the multiplier ideal \u0000$mathcal {J}(widehat {R},widehat {Delta })$\u0000 of \u0000$(widehat {R},widehat {Delta })$\u0000 , where \u0000$widehat {R}$\u0000 and \u0000$widehat {mathcal {B}(R)}$\u0000 are the \u0000$mathfrak {m}$\u0000 -adic completions of R and \u0000$mathcal {B}(R)$\u0000 , respectively, and \u0000$widehat {Delta }$\u0000 is the flat pullback of \u0000$Delta $\u0000 by the canonical morphism \u0000$operatorname {Spec} widehat {R}to operatorname {Spec} R$\u0000 . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"251 1","pages":"549 - 575"},"PeriodicalIF":0.8,"publicationDate":"2022-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44361830","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}
Abstract In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.
{"title":"SEMIAMPLENESS FOR CALABI–YAU SURFACES IN POSITIVE AND MIXED CHARACTERISTIC","authors":"F. Bernasconi, L. Stigant","doi":"10.1017/nmj.2022.32","DOIUrl":"https://doi.org/10.1017/nmj.2022.32","url":null,"abstract":"Abstract In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"250 1","pages":"365 - 384"},"PeriodicalIF":0.8,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48103663","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}
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