Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system [ mbox{(P)} qquad cases{ partial_t u=D_1Delta u+v^p, & $xin{bf R}^N,,,,t>0,$ partial_t v=D_2Delta v+u^q, & $xin{bf R}^N,,,,t>0,$ (u(cdot,0),v(cdot,0))=(mu,nu), & $xin{bf R}^N,$ } ] where $D_1$, $D_2>0$, $0 1$ and $(mu,nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
{"title":"Optimal singularities of initial functions for solvability of a semilinear parabolic system","authors":"Y. Fujishima, Kazuhiro Ishige","doi":"10.2969/jmsj/86058605","DOIUrl":"https://doi.org/10.2969/jmsj/86058605","url":null,"abstract":"Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system [ mbox{(P)} qquad cases{ partial_t u=D_1Delta u+v^p, & $xin{bf R}^N,,,,t>0,$ partial_t v=D_2Delta v+u^q, & $xin{bf R}^N,,,,t>0,$ (u(cdot,0),v(cdot,0))=(mu,nu), & $xin{bf R}^N,$ } ] where $D_1$, $D_2>0$, $0 1$ and $(mu,nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps. Default functions were introduced in [7] and it is known that vanishing of the default function of a local martingale implies that it is a true martingale. Positivity of the default function then indicates the singularity of local martingale. Such local martingales are called strictly local martingales. Recently strictly local martingales are playing important roles in the theory of financial bubbles (cf. [19]), so the notion of default function becomes important in mathematical finance area. We consider them in a different context. In mathematical analysis of subharmonic functions it is classical and natural to consider the functions along Brownian motions. A stochastic process derived from a subharmonic function composed with Brownian motion is a local submartingale. Then if we know that the process is a true submartingale, which follows from vanishing of the default function of the submartingale, it effects simpleness and clearness in analysis of subharmonic functions. In this paper we intend to show that this probabilistic notion plays effective roles in some analysis such as L-Liouville type theorems of subharmonic functions and Liouville type theorems for functions satisfying some nonlinear differential inequalities. It covers and extends the precedent results about L-Liouville theorem for subharmonic functions 2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.
{"title":"Default functions and Liouville type theorems based on symmetric diffusions","authors":"A. Atsuji","doi":"10.2969/jmsj/82398239","DOIUrl":"https://doi.org/10.2969/jmsj/82398239","url":null,"abstract":"Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps. Default functions were introduced in [7] and it is known that vanishing of the default function of a local martingale implies that it is a true martingale. Positivity of the default function then indicates the singularity of local martingale. Such local martingales are called strictly local martingales. Recently strictly local martingales are playing important roles in the theory of financial bubbles (cf. [19]), so the notion of default function becomes important in mathematical finance area. We consider them in a different context. In mathematical analysis of subharmonic functions it is classical and natural to consider the functions along Brownian motions. A stochastic process derived from a subharmonic function composed with Brownian motion is a local submartingale. Then if we know that the process is a true submartingale, which follows from vanishing of the default function of the submartingale, it effects simpleness and clearness in analysis of subharmonic functions. In this paper we intend to show that this probabilistic notion plays effective roles in some analysis such as L-Liouville type theorems of subharmonic functions and Liouville type theorems for functions satisfying some nonlinear differential inequalities. It covers and extends the precedent results about L-Liouville theorem for subharmonic functions 2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43576089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Beata Osińska-Ulrych, Grzegorz Skalski, S. Spodzieja
{"title":"Effective Łojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs","authors":"Beata Osińska-Ulrych, Grzegorz Skalski, S. Spodzieja","doi":"10.2969/jmsj/83378337","DOIUrl":"https://doi.org/10.2969/jmsj/83378337","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41454374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $Nin ]-infty,1]cup [n,+infty]$ with $varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $Nin [n,+infty[$ and $varepsilon=1$, and for $Nin ]-infty,1]$ and $varepsilon=0$ by the second named author.
{"title":"Comparison geometry of manifolds with boundary under lower $N$-weighted Ricci curvature bounds with $varepsilon$-range","authors":"K. Kuwae, Y. Sakurai","doi":"10.2969/jmsj/87278727","DOIUrl":"https://doi.org/10.2969/jmsj/87278727","url":null,"abstract":"We study comparison geometry of manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $Nin ]-infty,1]cup [n,+infty]$ with $varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting theorems, and also comparison geometric results for inscribed radius, volume around the boundary, and smallest Dirichlet eigenvalue of the weighted $p$-Laplacian. Our results interpolate those for $Nin [n,+infty[$ and $varepsilon=1$, and for $Nin ]-infty,1]$ and $varepsilon=0$ by the second named author.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41582718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A mass transportation proof of the sharp one-dimensional Gagliardo–Nirenberg inequalities","authors":"V. H. Nguyen","doi":"10.2969/jmsj/82258225","DOIUrl":"https://doi.org/10.2969/jmsj/82258225","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49035489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extremal trigonal fibrations on rational surfaces","authors":"C. Gong, S. Kitagawa, Jun Lu","doi":"10.2969/jmsj/82438243","DOIUrl":"https://doi.org/10.2969/jmsj/82438243","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41402088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any compact riemannian surface of genus three $(Sigma,ds^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian $lambda_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24pi$. In this paper we improve the result and we show that $lambda_1(ds^2)Area(ds^2)leq16(4-sqrt{7})pi approx 21.668,pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $approx 21.414,pi$.
{"title":"On the first eigenvalue of the Laplacian on compact surfaces of genus three","authors":"A. Ros","doi":"10.2969/jmsj/85898589","DOIUrl":"https://doi.org/10.2969/jmsj/85898589","url":null,"abstract":"For any compact riemannian surface of genus three $(Sigma,ds^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian $lambda_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24pi$. In this paper we improve the result and we show that $lambda_1(ds^2)Area(ds^2)leq16(4-sqrt{7})pi approx 21.668,pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $approx 21.414,pi$.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45167362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove categorical systolic inequalities for the derived categories of 2-Calabi-Yau Ginzburg dg algebras associated to ADE quivers and explore their symplecto-geometric aspects.
{"title":"Systolic inequalities, Ginzburg dg algebras and Milnor fibers","authors":"Jongmyeong Kim","doi":"10.2969/jmsj/85878587","DOIUrl":"https://doi.org/10.2969/jmsj/85878587","url":null,"abstract":"We prove categorical systolic inequalities for the derived categories of 2-Calabi-Yau Ginzburg dg algebras associated to ADE quivers and explore their symplecto-geometric aspects.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47069698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $mathcal{C}$ is given by an element $O(T)in H^2(G,operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $operatorname{Hom}(G,operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}))$, where $operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed in arXiv:0909.3140, allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided crossed category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $mathbb{Z}/2$-crossed structures over Tambara-Yamagami fusion categories and, consequently, a conceptual interpretation of the results in arXiv:math/0011037 about the classification of braidings over Tambara-Yamagami categories.
对于阿贝尔群$ A $，我们研究了$ A $-作用和$ A $-分级编织张量范畴之间的紧密联系。此外，我们证明了一元范畴$mathcal{C}$上的范畴群作用$T$的平凡化存在的障碍是由H^2(G，operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}) $中的元素$O(T)给出的。在$O(T)=0$的情况下，障碍物集合在$operatorname{hm}(G，operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}))$上形成一个torsor，其中$operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}})$是单位元的张量自然自同构的阿贝尔群。平凡化的上同解释，以及arXiv:0909.3140中提出的(忠实分级)编织A交叉张量范畴的同局部分类，使我们能够提供一种构造忠实A分级编织张量范畴的方法。我们算出两个例子。首先，我们计算了与点半简单张量范畴相关的编织交叉范畴的琐屑化存在的障碍。在第二个例子中，我们计算了Tambara-Yamagami融合范畴上编织的$mathbb{Z}/2$-交叉结构的显式公式，从而对arXiv:math/0011037中关于Tambara-Yamagami范畴上编织分类的结果进行了概念解释。
{"title":"Trivializing group actions on braided crossed tensor categories and graded braided tensor categories","authors":"César Galindo","doi":"10.2969/jmsj/85768576","DOIUrl":"https://doi.org/10.2969/jmsj/85768576","url":null,"abstract":"For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $mathcal{C}$ is given by an element $O(T)in H^2(G,operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $operatorname{Hom}(G,operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}}))$, where $operatorname{Aut}_otimes(operatorname{Id}_{mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. \u0000The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed in arXiv:0909.3140, allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided crossed category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $mathbb{Z}/2$-crossed structures over Tambara-Yamagami fusion categories and, consequently, a conceptual interpretation of the results in arXiv:math/0011037 about the classification of braidings over Tambara-Yamagami categories.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43538658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Poincaré inequalities with exact missing terms on homogeneous groups","authors":"T. Ozawa, D. Suragan","doi":"10.2969/jmsj/83738373","DOIUrl":"https://doi.org/10.2969/jmsj/83738373","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47752995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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