Pub Date : 2022-11-21DOI: 10.4153/s0008414x22000633
Imin Chen, Aisosa Efemwonkieke, David Sun
{"title":"FERMAT’S LAST THEOREM OVER AND","authors":"Imin Chen, Aisosa Efemwonkieke, David Sun","doi":"10.4153/s0008414x22000633","DOIUrl":"https://doi.org/10.4153/s0008414x22000633","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76572386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-14DOI: 10.4153/s0008414x23000238
L. EL KAOUTIT, Aryan Ghobadi, P. Saracco, J. Vercruysse
We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.
{"title":"Correspondence theorems for Hopf algebroids with applications to affine groupoids","authors":"L. EL KAOUTIT, Aryan Ghobadi, P. Saracco, J. Vercruysse","doi":"10.4153/s0008414x23000238","DOIUrl":"https://doi.org/10.4153/s0008414x23000238","url":null,"abstract":"We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88727652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-05DOI: 10.4153/s0008414x23000354
Shivajee Gupta, A. Vatwani
We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for $L$-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer $G$-function of the type $G^{n 0}_{0 n}$.
{"title":"Riesz-type criteria for L-functions in the Selberg class","authors":"Shivajee Gupta, A. Vatwani","doi":"10.4153/s0008414x23000354","DOIUrl":"https://doi.org/10.4153/s0008414x23000354","url":null,"abstract":"We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for $L$-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer $G$-function of the type $G^{n 0}_{0 n}$.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"66 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78661854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-26DOI: 10.4153/S0008414X22000578
S. Charpentier, A. Mouze
Abstract Given a sequence $varrho =(r_n)_nin [0,1)$ tending to $1$ , we consider the set ${mathcal {U}}_A({mathbb {D}},varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $mathbb {D}$ such that for any compact set K included in the unit circle ${mathbb {T}}$ , different from ${mathbb {T}}$ , the set ${zmapsto f(r_n cdot )vert _K:nin mathbb {N}}$ is dense in the space ${mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${mathcal {U}}_A({mathbb {D}},varrho )$ is residual in $H(mathbb {D})$ . We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${mathcal {C}}(K)$ for any compact set $Ksubset {mathbb {T}}$ different from ${mathbb {T}}$ . Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:fmapsto f(r_n cdot )$ acting on $H(mathbb {D})$ . Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.
{"title":"Abel universal functions","authors":"S. Charpentier, A. Mouze","doi":"10.4153/S0008414X22000578","DOIUrl":"https://doi.org/10.4153/S0008414X22000578","url":null,"abstract":"Abstract Given a sequence \u0000$varrho =(r_n)_nin [0,1)$\u0000 tending to \u0000$1$\u0000 , we consider the set \u0000${mathcal {U}}_A({mathbb {D}},varrho )$\u0000 of Abel universal functions consisting of holomorphic functions f in the open unit disk \u0000$mathbb {D}$\u0000 such that for any compact set K included in the unit circle \u0000${mathbb {T}}$\u0000 , different from \u0000${mathbb {T}}$\u0000 , the set \u0000${zmapsto f(r_n cdot )vert _K:nin mathbb {N}}$\u0000 is dense in the space \u0000${mathcal {C}}(K)$\u0000 of continuous functions on K. It is known that the set \u0000${mathcal {U}}_A({mathbb {D}},varrho )$\u0000 is residual in \u0000$H(mathbb {D})$\u0000 . We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at \u0000$0$\u0000 are dense in \u0000${mathcal {C}}(K)$\u0000 for any compact set \u0000$Ksubset {mathbb {T}}$\u0000 different from \u0000${mathbb {T}}$\u0000 . Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators \u0000$T_n:fmapsto f(r_n cdot )$\u0000 acting on \u0000$H(mathbb {D})$\u0000 . Thus, we study the dynamical properties of \u0000$(T_n)_n$\u0000 such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"6 1","pages":"1957 - 1985"},"PeriodicalIF":0.7,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74729135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-25DOI: 10.4153/s0008414x22000566
Pengtao Li, Zhichun Zhai
Abstract We apply capacities to explore the space–time fractional dissipative equation: (0.1) $$ begin{align} left{begin{aligned} &partial^{beta}_{t}u(t,x)=-nu(-Delta)^{alpha/2}u(t,x)+f(t,x),quad (t,x)inmathbb R^{1+n}_{+}, &u(0,x)=varphi(x), xinmathbb R^{n}, end{aligned}right. end{align} $$ where $alpha>n$ and $beta in (0,1)$ . In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term $R_{alpha ,beta }(varphi )$ and inhomogeneous term $G_{alpha ,beta }(g)$ , respectively. Second, we obtain some space–time estimates for $G_{alpha ,beta }(g).$ Based on these estimates, we prove that the continuity of $R_{alpha ,beta }(varphi )(t,x)$ and the Hölder continuity of $G_{alpha ,beta }(g)(t,x)$ on $mathbb {R}^{1+n}_+,$ which implies a Moser–Trudinger-type estimate for $G_{alpha ,beta }.$ Then, for a newly introduced $L^{q}_{t}L^p_{x}$ -capacity related to the space–time fractional dissipative operator $partial ^{beta }_{t}+(-Delta )^{alpha /2},$ we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in $mathbb {R}^{1+n}_+$ by using the Strichartz estimates and the Moser–Trudinger-type estimate for $G_{alpha ,beta }.$ A strong-type estimate of the $L^{q}_{t}L^p_{x}$ -capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the $L^{q}_{t}L^p_{x}$ -capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).
{"title":"Application of capacities to space–time fractional dissipative equations I: regularity and the blow-up set","authors":"Pengtao Li, Zhichun Zhai","doi":"10.4153/s0008414x22000566","DOIUrl":"https://doi.org/10.4153/s0008414x22000566","url":null,"abstract":"Abstract We apply capacities to explore the space–time fractional dissipative equation: (0.1) \u0000$$ begin{align} left{begin{aligned} &partial^{beta}_{t}u(t,x)=-nu(-Delta)^{alpha/2}u(t,x)+f(t,x),quad (t,x)inmathbb R^{1+n}_{+}, &u(0,x)=varphi(x), xinmathbb R^{n}, end{aligned}right. end{align} $$\u0000 where \u0000$alpha>n$\u0000 and \u0000$beta in (0,1)$\u0000 . In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term \u0000$R_{alpha ,beta }(varphi )$\u0000 and inhomogeneous term \u0000$G_{alpha ,beta }(g)$\u0000 , respectively. Second, we obtain some space–time estimates for \u0000$G_{alpha ,beta }(g).$\u0000 Based on these estimates, we prove that the continuity of \u0000$R_{alpha ,beta }(varphi )(t,x)$\u0000 and the Hölder continuity of \u0000$G_{alpha ,beta }(g)(t,x)$\u0000 on \u0000$mathbb {R}^{1+n}_+,$\u0000 which implies a Moser–Trudinger-type estimate for \u0000$G_{alpha ,beta }.$\u0000 Then, for a newly introduced \u0000$L^{q}_{t}L^p_{x}$\u0000 -capacity related to the space–time fractional dissipative operator \u0000$partial ^{beta }_{t}+(-Delta )^{alpha /2},$\u0000 we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in \u0000$mathbb {R}^{1+n}_+$\u0000 by using the Strichartz estimates and the Moser–Trudinger-type estimate for \u0000$G_{alpha ,beta }.$\u0000 A strong-type estimate of the \u0000$L^{q}_{t}L^p_{x}$\u0000 -capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the \u0000$L^{q}_{t}L^p_{x}$\u0000 -capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"299 8","pages":"1904 - 1956"},"PeriodicalIF":0.7,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72391809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-03DOI: 10.4153/S0008414X22000505
G. Leng, Chang Liu, Dongmeng Xi
Abstract First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.
{"title":"Reconstruction problems of convex bodies from surface area measures and lightness functions","authors":"G. Leng, Chang Liu, Dongmeng Xi","doi":"10.4153/S0008414X22000505","DOIUrl":"https://doi.org/10.4153/S0008414X22000505","url":null,"abstract":"Abstract First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"84 1","pages":"1685 - 1710"},"PeriodicalIF":0.7,"publicationDate":"2022-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84287393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-01DOI: 10.4153/s0008414x2200044x
Ming Xu, Vladimir S. Matveev
{"title":"CJM volume 74 issue 5 Cover and Front matter","authors":"Ming Xu, Vladimir S. Matveev","doi":"10.4153/s0008414x2200044x","DOIUrl":"https://doi.org/10.4153/s0008414x2200044x","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"69 1","pages":"f1 - f3"},"PeriodicalIF":0.7,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87143996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-01DOI: 10.4153/s0008414x22000451
{"title":"CJM volume 74 issue 5 Cover and Back matter","authors":"","doi":"10.4153/s0008414x22000451","DOIUrl":"https://doi.org/10.4153/s0008414x22000451","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"74 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84861241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-16DOI: 10.4153/s0008414x23000172
H. Murakami
We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $expbigl((u+2ppii)/Nbigr)$, where $u$ is a small real number and $p$ is a positive integer. We show that it is asymptotically equivalent to the product of the $p$-dimensional colored Jones polynomial evaluated at $expbigl(4Npi^2/(u+2ppii)bigr)$ and a term that grows exponentially with growth rate determined by the Chern--Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.
{"title":"THE COLORED JONES POLYNOMIAL OF THE FIGURE-EIGHT KNOT AND A QUANTUM MODULARITY","authors":"H. Murakami","doi":"10.4153/s0008414x23000172","DOIUrl":"https://doi.org/10.4153/s0008414x23000172","url":null,"abstract":"We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of the figure-eight knot evaluated at $expbigl((u+2ppii)/Nbigr)$, where $u$ is a small real number and $p$ is a positive integer. We show that it is asymptotically equivalent to the product of the $p$-dimensional colored Jones polynomial evaluated at $expbigl(4Npi^2/(u+2ppii)bigr)$ and a term that grows exponentially with growth rate determined by the Chern--Simons invariant. This indicates a quantum modularity of the colored Jones polynomial.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"111 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80845780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-13DOI: 10.4153/S0008414X22000414
Yujun Liu
Abstract In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data $(mathbf {u}_0, mathbf {w}_0)in H^1(mathbb {R}^3)$ .
{"title":"Global existence of the strong solution to the 3D incompressible micropolar equations with fractional partial dissipation","authors":"Yujun Liu","doi":"10.4153/S0008414X22000414","DOIUrl":"https://doi.org/10.4153/S0008414X22000414","url":null,"abstract":"Abstract In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data \u0000$(mathbf {u}_0, mathbf {w}_0)in H^1(mathbb {R}^3)$\u0000 .","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"31 1","pages":"1516 - 1539"},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90650954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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