Pub Date : 2022-11-11DOI: 10.1017/s1446788721000367
{"title":"JAZ volume 113 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s1446788721000367","DOIUrl":"https://doi.org/10.1017/s1446788721000367","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"2 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78475879","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}
Pub Date : 2022-11-11DOI: 10.1017/s1446788721000355
{"title":"JAZ volume 113 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s1446788721000355","DOIUrl":"https://doi.org/10.1017/s1446788721000355","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"83 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91192987","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}
Pub Date : 2022-09-07DOI: 10.1017/S1446788721000331
{"title":"JAZ volume 113 issue 2 Cover and Front matter","authors":"","doi":"10.1017/S1446788721000331","DOIUrl":"https://doi.org/10.1017/S1446788721000331","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"41 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87050710","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}
Pub Date : 2022-09-07DOI: 10.1017/S1446788721000343
{"title":"JAZ volume 113 issue 2 Cover and Back matter","authors":"","doi":"10.1017/S1446788721000343","DOIUrl":"https://doi.org/10.1017/S1446788721000343","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"18 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81681192","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}
Pub Date : 2022-08-23DOI: 10.1017/S1446788722000131
Cuicui Li, Fang Liu, P. Zhao
Abstract In this paper, we study the boundary blow-up problem related to the infinity Laplacian $$ begin{align*}begin{cases} Delta_{infty}^h u=u^q &mathrm{in}; Omega, u=infty &mathrm{on} ;partialOmega, end{cases} end{align*} $$ where $Delta _{infty }^h u=|Du|^{h-3} langle D^2uDu,Du rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q leq h.$
{"title":"BOUNDARY BLOW-UP SOLUTIONS TO EQUATIONS INVOLVING THE INFINITY LAPLACIAN","authors":"Cuicui Li, Fang Liu, P. Zhao","doi":"10.1017/S1446788722000131","DOIUrl":"https://doi.org/10.1017/S1446788722000131","url":null,"abstract":"Abstract In this paper, we study the boundary blow-up problem related to the infinity Laplacian \u0000$$ begin{align*}begin{cases} Delta_{infty}^h u=u^q &mathrm{in}; Omega, u=infty &mathrm{on} ;partialOmega, end{cases} end{align*} $$\u0000 where \u0000$Delta _{infty }^h u=|Du|^{h-3} langle D^2uDu,Du rangle $\u0000 is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When \u0000$q>h>1$\u0000 , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case \u0000$q leq h.$","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"337 - 358"},"PeriodicalIF":0.7,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91267317","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}
Pub Date : 2022-08-06DOI: 10.1017/s1446788723000046
M. Gran, J. Scherer
We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.
{"title":"CONDITIONAL FLATNESS, FIBERWISE LOCALIZATIONS, AND ADMISSIBLE REFLECTIONS","authors":"M. Gran, J. Scherer","doi":"10.1017/s1446788723000046","DOIUrl":"https://doi.org/10.1017/s1446788723000046","url":null,"abstract":"\u0000 We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"48 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84761063","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}
Pub Date : 2022-07-12DOI: 10.1017/s144678872100032x
{"title":"JAZ volume 113 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s144678872100032x","DOIUrl":"https://doi.org/10.1017/s144678872100032x","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"117 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79528322","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}
Pub Date : 2022-07-12DOI: 10.1017/s1446788721000318
{"title":"JAZ volume 113 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s1446788721000318","DOIUrl":"https://doi.org/10.1017/s1446788721000318","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"16 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72793738","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}
Pub Date : 2022-06-17DOI: 10.1017/S1446788722000374
S. V. Gusev, Mikhail Volkov
For every group G, the set $mathcal {P}(G)$ of its subsets forms a semiring under set-theoretical union $cup $ and element-wise multiplication $cdot $ , and forms an involution semigroup under $cdot $ and element-wise inversion ${}^{-1}$ . We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring $(mathcal {P}(G),cup ,cdot )$ nor the involution semigroup $(mathcal {P}(G),cdot ,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.
{"title":"SEMIRING AND INVOLUTION IDENTITIES OF POWER GROUPS","authors":"S. V. Gusev, Mikhail Volkov","doi":"10.1017/S1446788722000374","DOIUrl":"https://doi.org/10.1017/S1446788722000374","url":null,"abstract":"\u0000\t <jats:p>For every group <jats:italic>G</jats:italic>, the set <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=\"S1446788722000374_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$mathcal {P}(G)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> of its subsets forms a semiring under set-theoretical union <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=\"S1446788722000374_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$cup $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and element-wise multiplication <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$cdot $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, and forms an involution semigroup under <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$cdot $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and element-wise inversion <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000${}^{-1}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. We show that if the group <jats:italic>G</jats:italic> is finite, non-Dedekind, and solvable, neither the semiring <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$(mathcal {P}(G),cup ,cdot )$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> nor the involution semigroup <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$(mathcal {P}(G),cdot ,{}^{-1})$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.</jats:p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"148 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84842984","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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