Let X be a Hausdorff continuum (a nondegenerate, compact, connected Hausdorff space). Let C(X) denote the hyperspace of its subcontinua, endowed with the Vietoris topology. We extend some results of the metric case about unicoherence and the existence of selections for C(X). We also introduce two definitions of contractibility of C(X) and discuss their relation with some properties of X. In particular, we show that both definitions are equivalent in the metrizable case, but one of them is more general in the Hausdorff continuum case.
{"title":"ON UNICOHERENCE AND CONTRACTIBILITY OF HYPERSPACES OF NONMETRIZABLE CONTINUA","authors":"Luis Miguel García-Velázquez","doi":"10.1216/rmj.2023.53.435","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.435","url":null,"abstract":"Let X be a Hausdorff continuum (a nondegenerate, compact, connected Hausdorff space). Let C(X) denote the hyperspace of its subcontinua, endowed with the Vietoris topology. We extend some results of the metric case about unicoherence and the existence of selections for C(X). We also introduce two definitions of contractibility of C(X) and discuss their relation with some properties of X. In particular, we show that both definitions are equivalent in the metrizable case, but one of them is more general in the Hausdorff continuum case.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135673148","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}
Jianqiang Xie, M. Ali, H. Budak, Michal Feckan, T. Sitthiwirattham
{"title":"FRACTIONAL HERMITE–HADAMARD INEQUALITY, SIMPSON’S AND OSTROWSKI’S TYPE INEQUALITIES FOR CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF FUNCTIONS","authors":"Jianqiang Xie, M. Ali, H. Budak, Michal Feckan, T. Sitthiwirattham","doi":"10.1216/rmj.2023.53.611","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.611","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41738023","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":"CONTINUITY ON A CLASS OF F-ALGEBRAS","authors":"A. Naziri-Kordkandi","doi":"10.1216/rmj.2023.53.541","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.541","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47799503","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}
In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.
{"title":"DENOMINATORS OF SPECIAL VALUES OF ζ-FUNCTIONS COUNT KU-LOCAL HOMOTOPY GROUPS OF MOD p MOORE SPECTRA","authors":"A. Salch","doi":"10.1216/rmj.2023.53.915","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.915","url":null,"abstract":"In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of $KU$-local stable homotopy groups.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41427820","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}
A BSTRACT . A commutative ring R is called strongly regular associate if, for any a , b ∈ R , Ra = Rb implies that a = rb and sa = b for some regular elements r , s ∈ R . In this paper, we first give a characterization of strongly regular associate rings. A ring R is said to have regular range 1 if, for any a , b ∈ R , Ra + Rb = R implies that a + bx is a regular for some x ∈ R . We show that the ring of continuous functions C ( X ) is strongly regular associate if and only if it has regular range 1. Finally, we generalize a theorem of Anderson and Chun, which states that C ([ a , b ]) is a strongly regular associate ring.
摘要。如果对任意A, b∈R, Ra = Rb意味着对某些正则元素R, s∈R, A = Rb和sa = b，则交换环R称为强正则环R。本文首先给出了强正则环的一个性质。如果对于任意A, b∈R, Ra + Rb = R意味着A + bx对于某个x∈R是正则的，那么我们说环R具有正则范围1。证明了连续函数环C (X)是强正则关联的当且仅当其正则值域为1。最后，我们推广了Anderson和Chun的一个定理，证明C ([a, b])是一个强正则环。
{"title":"ON A THEOREM OF ANDERSON AND CHUN","authors":"A. R. Aliabad, Farimah Farrokhpay, M. Siavoshi","doi":"10.1216/rmj.2023.53.1","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1","url":null,"abstract":"A BSTRACT . A commutative ring R is called strongly regular associate if, for any a , b ∈ R , Ra = Rb implies that a = rb and sa = b for some regular elements r , s ∈ R . In this paper, we first give a characterization of strongly regular associate rings. A ring R is said to have regular range 1 if, for any a , b ∈ R , Ra + Rb = R implies that a + bx is a regular for some x ∈ R . We show that the ring of continuous functions C ( X ) is strongly regular associate if and only if it has regular range 1. Finally, we generalize a theorem of Anderson and Chun, which states that C ([ a , b ]) is a strongly regular associate ring.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43522279","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}
In our previous work, we introduced a series of new derivation Lie algebras Lk(V ) associated to an isolated hypersurface singularity (V, 0). These are new analytic invariants of singularities. In this article, on the one hand, we investigate L2(V ) for fewnomial isolated singularities and obtain the formula of λk(V ) (i.e., the dimension of Lk(V )) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for the L2(V ). This part is a continuous work of our previous work in [HYZ8]. On the other hand, we proposed two new conjectures for the τk(V ) and λk(V ) and we prove these conjectures for a large class of singularities.
{"title":"DERIVATIONS OF LOCAL k-TH HESSIAN ALGEBRAS OF SINGULARITIES","authors":"Naveed Hussain, S. Yau, Huaiqing Zuo","doi":"10.1216/rmj.2023.53.65","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.65","url":null,"abstract":"In our previous work, we introduced a series of new derivation Lie algebras Lk(V ) associated to an isolated hypersurface singularity (V, 0). These are new analytic invariants of singularities. In this article, on the one hand, we investigate L2(V ) for fewnomial isolated singularities and obtain the formula of λk(V ) (i.e., the dimension of Lk(V )) for trinomial singularities. Furthermore, we prove the sharp upper estimate conjecture for the L2(V ). This part is a continuous work of our previous work in [HYZ8]. On the other hand, we proposed two new conjectures for the τk(V ) and λk(V ) and we prove these conjectures for a large class of singularities.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41798187","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}
. The Jensen–Mercer inequality, which is well known in the literature, has an important place in mathematics and related disciplines. In this work, we obtain the Hermite–Jensen–Mercer inequality for post-quantum integrals by utilizing Jensen–Mercer inequalities. Then we investigate the connections between our results and those in earlier works. Moreover, we give some examples to illustrate the main results in this paper. This is the first paper about Hermite–Jensen–Mercer inequalities for post-quantum integrals.
{"title":"POST-QUANTUM HERMITE–JENSEN–MERCER INEQUALITIES","authors":"M. Bohner, H. Budak, Hasan Kara","doi":"10.1216/rmj.2023.53.17","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.17","url":null,"abstract":". The Jensen–Mercer inequality, which is well known in the literature, has an important place in mathematics and related disciplines. In this work, we obtain the Hermite–Jensen–Mercer inequality for post-quantum integrals by utilizing Jensen–Mercer inequalities. Then we investigate the connections between our results and those in earlier works. Moreover, we give some examples to illustrate the main results in this paper. This is the first paper about Hermite–Jensen–Mercer inequalities for post-quantum integrals.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43476562","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}
. In this paper, we consider the real pure quartic number field K = Q ( 4 (cid:112) pd 2 ) , where p is a prime number and d is a square-free positive integer such that d is prime to p . We compute r 2 ( K ) the 2 -rank of the class group of K and as an application we exhibit all possible forms of d for which the 2 -class group of K is trivial (equivalently: the class number of K is odd), cyclic or isomorphic to Z / 2 n 1 Z × Z / 2 n 2 Z , where n i ∈ N ∗ .
{"title":"THE 2-RANK OF THE REAL PURE QUARTIC NUMBER FIELD K=ℚ(pd24)","authors":"Mbarek Haynou, B. Sodaïgui, M. Taous","doi":"10.1216/rmj.2023.53.27","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.27","url":null,"abstract":". In this paper, we consider the real pure quartic number field K = Q ( 4 (cid:112) pd 2 ) , where p is a prime number and d is a square-free positive integer such that d is prime to p . We compute r 2 ( K ) the 2 -rank of the class group of K and as an application we exhibit all possible forms of d for which the 2 -class group of K is trivial (equivalently: the class number of K is odd), cyclic or isomorphic to Z / 2 n 1 Z × Z / 2 n 2 Z , where n i ∈ N ∗ .","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42398445","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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