Let $R$ be a ring. An $R$-module $M$ is a weak $w$-projective module if ${rm Ext}_R^1(M,N)=0$ for all $N$ in the class of $GV$-torsion-free $R$-modules with the property that ${rm Ext}^k_R(T,N)=0$ for all $w$-projective $R$-modules $T$ and all integers $kgeq1$. In this paper, we introduce and study some properties of weak $w$-projective modules. We use these modules to characterise some classical rings. For example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective; $R$ is a von Neumann regular ring if and only if every FP-projective module is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projective; and $R$ is $w$-semi-hereditary if and only if every finite type submodule of a free module is weak $w$-projective if and only if every finitely generated ideal of $R$ is weak $w$-projective.
{"title":"note on weak w-projective modules","authors":"Refat Abdelmawla Khaled Assaad","doi":"10.53733/336","DOIUrl":"https://doi.org/10.53733/336","url":null,"abstract":"Let $R$ be a ring. An $R$-module $M$ is a weak $w$-projective module if ${rm Ext}_R^1(M,N)=0$ for all $N$ in the class of $GV$-torsion-free $R$-modules with the property that ${rm Ext}^k_R(T,N)=0$ for all $w$-projective $R$-modules $T$ and all integers $kgeq1$. In this paper, we introduce and study some properties of weak $w$-projective modules. We use these modules to characterise some classical rings. For example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective; $R$ is a von Neumann regular ring if and only if every FP-projective module is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projective; and $R$ is $w$-semi-hereditary if and only if every finite type submodule of a free module is weak $w$-projective if and only if every finitely generated ideal of $R$ is weak $w$-projective.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"69 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141360286","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}
Let $varphi(n)$ be the Euler function, $sigma(n)=sum_{dmid n}d$ the sum of divisors function and $gamma=0.577ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $sigma(n)/n < e^gamma loglog n$ holds for $n > 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/varphi(n)$.
{"title":"Robin inequality for n/phi(n)","authors":"Jean-Louis Nicolas","doi":"10.53733/324","DOIUrl":"https://doi.org/10.53733/324","url":null,"abstract":"Let $varphi(n)$ be the Euler function, $sigma(n)=sum_{dmid n}d$ the sum of divisors function and $gamma=0.577ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $sigma(n)/n < e^gamma loglog n$ holds for $n > 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/varphi(n)$.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"58 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141008860","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}
For $kin Bbb N$, we introduce the notion of $k$-rational homotopy fixed points and we prove, under a certain assumption, that if $X$ is a rational elliptic space of formal dimension $n$, then $X$ admits an $(n -1)$-rational homotopy fixed point.
{"title":"$k$-rational homotopy fixed points, $kin Bbb N$","authors":"Mahmoud Benkhalifa","doi":"10.53733/367","DOIUrl":"https://doi.org/10.53733/367","url":null,"abstract":"For $kin Bbb N$, we introduce the notion of $k$-rational homotopy fixed points and we prove, under a certain assumption, that if $X$ is a rational elliptic space of formal dimension $n$, then $X$ admits an $(n -1)$-rational homotopy fixed point.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"82 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141011268","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}
The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${mathcal R}$-boundedness of the solution operator families of the Lam'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {it N}-dimensional Euclidean space ($N geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.
{"title":"Bent-half space model problem for Lame equation with surface tension","authors":"S. Maryani, Ari Wardayani, R. Renny","doi":"10.53733/321","DOIUrl":"https://doi.org/10.53733/321","url":null,"abstract":"The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${mathcal R}$-boundedness of the solution operator families of the Lam'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {it N}-dimensional Euclidean space ($N geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"36 16","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141010758","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}
It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a $K$-contact manifold $M^{2n+1}$ if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field $V$, (ii) $V$ is an eigenvector of the Ricci operator with eigenvalue $2n$, (iii) $V$ is gradient.
{"title":"Yamabe solitons in contact geometry","authors":"Rahul Poddar, S. Balasubramanian, Ramesh Sharma","doi":"10.53733/286","DOIUrl":"https://doi.org/10.53733/286","url":null,"abstract":"It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a $K$-contact manifold $M^{2n+1}$ if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field $V$, (ii) $V$ is an eigenvector of the Ricci operator with eigenvalue $2n$, (iii) $V$ is gradient.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"28 39","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138955233","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}
This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution $(u, omega)$ is regular if the velocity $u$ satisfies $$int_{0}^{T}| nabla_{h}u_{h}|_{dot{B}_{p,frac{2p}{3}}^{0}}^{q} d t
本文进一步研究了贝索夫空间中三维微极方程的正则性准则。更确切地说，如果速度 $u$ 满足$$int_{0}^{T}| nabla_{h}u_{h}|_{dot{B}_{p,frac{2p}{3}}^{0}^{q} d t
{"title":"note on the regularity criterion for the micropolar fluid equations in homogeneous Besov spaces","authors":"Qiang Li, Mianlu Zou","doi":"10.53733/315","DOIUrl":"https://doi.org/10.53733/315","url":null,"abstract":"This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution $(u, omega)$ is regular if the velocity $u$ satisfies\u0000$$int_{0}^{T}| nabla_{h}u_{h}|_{dot{B}_{p,frac{2p}{3}}^{0}}^{q} d t<infty, with frac{3}{p}+frac{2}{q}=2, frac{3}{2}<pleqinfty,$$or $$int_{0}^{T}| nabla_{h}u|_{dot{B}_{infty ,infty}^{-1}}^{frac{8}{3}} d t<infty,$$or $$int_{0}^{T}|nabla_{h} u_{h}|_{dot{B}_{infty,infty}^{-alpha}}^{frac{2}{2-alpha}} d t<infty, with 0< alpha< 1. $$","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138954762","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}
We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.
{"title":"Group Actions on Product Systems","authors":"Valentin Deaconu, Leonard Huang","doi":"10.53733/311","DOIUrl":"https://doi.org/10.53733/311","url":null,"abstract":"We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666875","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}
This brief corrigendum corrects some minor errors in the paper `"Two new proofs of the fact that triangle groups are distinguished by their finite quotients", published in the New Zealand Journal of Mathematics 52 (2022), 827--844.
{"title":"Corrigendum to: Two new proofs of the fact that triangle groups are distinguished by their finite quotients","authors":"Marston Conder","doi":"10.53733/361","DOIUrl":"https://doi.org/10.53733/361","url":null,"abstract":"This brief corrigendum corrects some minor errors in the paper `\"Two new proofs of the fact that triangle groups are distinguished by their finite quotients\", published in the New Zealand Journal of Mathematics 52 (2022), 827--844.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689761","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}
We give a counterexample in this amendment to show that there is an error in consideration of the statement "{it if $f : X to Y$ and ${bf J}$ is an ideal on $Y$, then $f^{-1}({bf J}) = {f^{-1}(J) : J in {bf J}}$ is an ideal on $X$}" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein.
在此修正中，我们给出了一个反例，说明在考虑Hamlett在他的论文“Lindelöf关于理想”中的命题“{it if $f: X to Y$ and ${bf J}$是$Y$上的理想，那么$f^{-1}({bf J}) = {f^{-1}(J): J in {bf J}}$是$X$}上的理想”时存在错误[新西兰数学学报，42,115-120,2012]。我们在这里也对它作了新的修改，从此正确地提出了基于其中所导出的上述陈述的所有结果。
{"title":"Amendment to \"Lindelöf with respect to an ideal\" [New Zealand J. Math. 42, 115-120, 2012]","authors":"Jiarul Hoque, S. Modak","doi":"10.53733/218","DOIUrl":"https://doi.org/10.53733/218","url":null,"abstract":"We give a counterexample in this amendment to show that there is an error in consideration of the statement \"{it if $f : X to Y$ and ${bf J}$ is an ideal on $Y$, then $f^{-1}({bf J}) = {f^{-1}(J) : J in {bf J}}$ is an ideal on $X$}\" by Hamlett in his paper \"Lindelöf with respect to an ideal\" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75998991","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}
We investigate integer partitions $lambda$ of $n$ that are nearly self-conjugate in the sense that there are $n - 1$ overlapping cells among the Ferrers diagram of $lambda$ and its transpose, by establishing a correspondence, through the method of combinatorial telescoping, to partitions of $n$ in which (i).~there exists at least one even part; (ii).~any even part is of size $2$; (iii).~the odd parts are distinct; and (iv).~no odd part is of size $1$. In particular, this correspondence confirms a conjecture that had been given in the OEIS.
{"title":"Nearly self-conjugate integer partitions","authors":"John M. Campbell, Shane Chern","doi":"10.53733/217","DOIUrl":"https://doi.org/10.53733/217","url":null,"abstract":"We investigate integer partitions $lambda$ of $n$ that are nearly self-conjugate in the sense that there are $n - 1$ overlapping cells among the Ferrers diagram of $lambda$ and its transpose, by establishing a correspondence, through the method of combinatorial telescoping, to partitions of $n$ in which (i).~there exists at least one even part; (ii).~any even part is of size $2$; (iii).~the odd parts are distinct; and (iv).~no odd part is of size $1$. In particular, this correspondence confirms a conjecture that had been given in the OEIS.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73901412","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}
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