. For a prime p ≡ 3 (mod 4) and m ≥ 2, Romik raised a question about whether the Taylor coefficients around √− 1 of the classical Jacobi theta function θ 3 eventually vanish modulo p m . This question can be extended to a class of modular forms of half-integral weight on Γ 1 (4) and CM points; in this paper, we prove an affirmative answer to it for primes p ≥ 5. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on SL 2 ( Z ).
{"title":"$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $Gamma_1(4)$","authors":"Jigu Kim, Yoonjin Lee","doi":"10.11650/tjm/220802","DOIUrl":"https://doi.org/10.11650/tjm/220802","url":null,"abstract":". For a prime p ≡ 3 (mod 4) and m ≥ 2, Romik raised a question about whether the Taylor coefficients around √− 1 of the classical Jacobi theta function θ 3 eventually vanish modulo p m . This question can be extended to a class of modular forms of half-integral weight on Γ 1 (4) and CM points; in this paper, we prove an affirmative answer to it for primes p ≥ 5. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on SL 2 ( Z ).","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47112368","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 generalize Robinson’s version of an order cancellation law for subsets of vector spaces in which we cancel by unbounded sets. We introduce the notion of weakly narrow sets in normed spaces, study their properties and prove the order cancellation law where the canceled set is weakly narrow. Also we prove the order cancellation law for closed convex subsets of topological vector space where the canceled set has bounded Hausdorff-like distance from its recession cone. We topologically embed the semigroup of closed convex sets sharing a recession cone having bounded Hausdorff-like distance from it into a topological vector space. This result extends Bielawski and Tabor’s generalization of R̊adström theorem. 2010 Mathematics Subject classification. 52A07, 18E20, 46A99.
{"title":"Order Cancellation Law in a Semigroup of Closed Convex Sets","authors":"J. Grzybowski, H. Przybycień","doi":"10.11650/tjm/220603","DOIUrl":"https://doi.org/10.11650/tjm/220603","url":null,"abstract":"In this paper generalize Robinson’s version of an order cancellation law for subsets of vector spaces in which we cancel by unbounded sets. We introduce the notion of weakly narrow sets in normed spaces, study their properties and prove the order cancellation law where the canceled set is weakly narrow. Also we prove the order cancellation law for closed convex subsets of topological vector space where the canceled set has bounded Hausdorff-like distance from its recession cone. We topologically embed the semigroup of closed convex sets sharing a recession cone having bounded Hausdorff-like distance from it into a topological vector space. This result extends Bielawski and Tabor’s generalization of R̊adström theorem. 2010 Mathematics Subject classification. 52A07, 18E20, 46A99.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44716331","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 a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.
{"title":"An Extension of a Depth Inequality of Auslander","authors":"Olgur Celikbas, U. Le, H. Matsui","doi":"10.11650/tjm/220501","DOIUrl":"https://doi.org/10.11650/tjm/220501","url":null,"abstract":"In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49092282","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 a fixed nonnegative integer m, an analytic map φ and an analytic function ψ, the generalized integration operator I (m) φ,ψ is defined by I (m) φ,ψ f(z) = ∫ z
{"title":"Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions","authors":"Jiale Chen, Maofa Wang","doi":"10.11650/tjm/201208","DOIUrl":"https://doi.org/10.11650/tjm/201208","url":null,"abstract":"For a fixed nonnegative integer m, an analytic map φ and an analytic function ψ, the generalized integration operator I (m) φ,ψ is defined by I (m) φ,ψ f(z) = ∫ z","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64993527","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}
An almost self-centered graph is a connected graph of order n with exactly n−2 central vertices, and an almost peripheral graph is a connected graph of order n with exactly n − 1 peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order n; (2) the maximum independence number of an almost self-centered graph of order n and radius r; (3) the minimum order of a k-regular almost self-centered graph and (4) the maximum size of an almost peripheral graph of order n; (5) which numbers are possible for the maximum degree of an almost peripheral graph of order n; (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order n whose maximum degree is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.
{"title":"On Almost Self-centered Graphs and Almost Peripheral Graphs","authors":"Yanan Hu, Xingzhi Zhan","doi":"10.11650/tjm/220401","DOIUrl":"https://doi.org/10.11650/tjm/220401","url":null,"abstract":"An almost self-centered graph is a connected graph of order n with exactly n−2 central vertices, and an almost peripheral graph is a connected graph of order n with exactly n − 1 peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order n; (2) the maximum independence number of an almost self-centered graph of order n and radius r; (3) the minimum order of a k-regular almost self-centered graph and (4) the maximum size of an almost peripheral graph of order n; (5) which numbers are possible for the maximum degree of an almost peripheral graph of order n; (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order n whose maximum degree is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45328457","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 tensor product T (G1, G2, G3) of graphs G1, G2 and G3 is defined by V T (G1, G2, G3) = V (G1)× V (G2)× V (G3) and ET (G1, G2, G3) = {[(u1, u2, u3), (v1, v2, v3)] : |{i : (ui, vi) ∈ E(Gi)}| ≥ 2}. From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of T (G1, G2, G3). So αT (G1, G2, G3) ≥ max{α(G1)α(G2)|G3|, α(G1)α(G3)|G2|, α(G2)α(G3)|G1|}. In this paper, we prove that the equality holds if G1 and G2 are vertex-transitive graphs and G3 is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of T (G1, G2, G3) is determined.
{"title":"Independent Sets in Tensor Products of Three Vertex-transitive\u0000 Graphs","authors":"Hu Mao, Huajun Zhang","doi":"10.11650/TJM/210104","DOIUrl":"https://doi.org/10.11650/TJM/210104","url":null,"abstract":"The tensor product T (G1, G2, G3) of graphs G1, G2 and G3 is defined by V T (G1, G2, G3) = V (G1)× V (G2)× V (G3) and ET (G1, G2, G3) = {[(u1, u2, u3), (v1, v2, v3)] : |{i : (ui, vi) ∈ E(Gi)}| ≥ 2}. From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of T (G1, G2, G3). So αT (G1, G2, G3) ≥ max{α(G1)α(G2)|G3|, α(G1)α(G3)|G2|, α(G2)α(G3)|G1|}. In this paper, we prove that the equality holds if G1 and G2 are vertex-transitive graphs and G3 is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of T (G1, G2, G3) is determined.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47244577","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 well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we show a generalized fundamental identity for arbitrary Dedekind domains. As an application, we show that any Japanese Dedekind domain is excellent.
{"title":"Japanese Dedekind Domains Are Excellent","authors":"Chia-Fu Yu","doi":"10.11650/tjm/220102","DOIUrl":"https://doi.org/10.11650/tjm/220102","url":null,"abstract":". The well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we show a generalized fundamental identity for arbitrary Dedekind domains. As an application, we show that any Japanese Dedekind domain is excellent.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47904347","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}
Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $langle x rangle subset langle y rangle$ or $langle y rangle subset langle x rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.
{"title":"On the Metric Dimension of the Reduced Power Graph of a Finite Group","authors":"Xuanlong Ma, Lan Li","doi":"10.11650/tjm/210905","DOIUrl":"https://doi.org/10.11650/tjm/210905","url":null,"abstract":"Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $langle x rangle subset langle y rangle$ or $langle y rangle subset langle x rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42613033","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 study a predator-prey system with general response function and nonlocal diffusion in high dimensional space and investigate the propagation properties of its solution. More precisely, we study the invasion speed of the predator into habitat of the aborigine prey and obtain the existence of the planar waves by constructing the upper and lower solutions. Finally, we present some numerical simulations to support our results.
{"title":"The Spreading Speed and the Existence of Planar Waves for a Class of Predator-prey System with Nonlocal Diffusion","authors":"Min Zhao, Zhaohai Ma, R. Yuan","doi":"10.11650/tjm/211001","DOIUrl":"https://doi.org/10.11650/tjm/211001","url":null,"abstract":"In this paper, we study a predator-prey system with general response function and nonlocal diffusion in high dimensional space and investigate the propagation properties of its solution. More precisely, we study the invasion speed of the predator into habitat of the aborigine prey and obtain the existence of the planar waves by constructing the upper and lower solutions. Finally, we present some numerical simulations to support our results.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43440465","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 obtain the L p ( R 2 )-boundedness of the maximal functions where p ∈ (1 , ∞ ] and γ is a convex curve satisfying some suitable curvature conditions.
{"title":"Maximal Functions Along Convex Curves with Lacunary Directions","authors":"Haixia Yu","doi":"10.11650/tjm/211208","DOIUrl":"https://doi.org/10.11650/tjm/211208","url":null,"abstract":". In this paper, we obtain the L p ( R 2 )-boundedness of the maximal functions where p ∈ (1 , ∞ ] and γ is a convex curve satisfying some suitable curvature conditions.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43565690","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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