Pub Date : 2024-02-29DOI: 10.1017/s0004972724000030
R. J. HIGGS
Let $alpha $ be a complex-valued $2$ -cocycle of a finite group G with $alpha $ chosen so that the $alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $alpha $ -regular are characterised by the values that the irreducible $alpha $ -characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.
{"title":"PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS","authors":"R. J. HIGGS","doi":"10.1017/s0004972724000030","DOIUrl":"https://doi.org/10.1017/s0004972724000030","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline1.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a complex-valued <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline2.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cocycle of a finite group <jats:italic>G</jats:italic> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline3.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> chosen so that the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline4.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-characters of <jats:italic>G</jats:italic> are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of <jats:italic>G</jats:italic> that are also <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline5.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular are characterised by the values that the irreducible <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline6.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-characters of <jats:italic>G</jats:italic> take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline7.png\" /> <jats:tex-math> $G;$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> however, the initial choice of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline8.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> from its cohomology class is not unique in general and it is shown the results can vary for a different choice.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006826","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 : 2024-02-29DOI: 10.1017/s0004972724000042
BENJAMIN STEINBERG
Carlsen [‘ $ast $ -isomorphism of Leavitt path algebras over $Bbb Z$ ’, Adv. Math.324 (2018), 326–335] showed that any $ast $ -homomorphism between Leavitt path algebras over $mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $ast $ -homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.
{"title":"A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS","authors":"BENJAMIN STEINBERG","doi":"10.1017/s0004972724000042","DOIUrl":"https://doi.org/10.1017/s0004972724000042","url":null,"abstract":"Carlsen [‘<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline1.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-isomorphism of Leavitt path algebras over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline2.png\" /> <jats:tex-math> $Bbb Z$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>’, <jats:italic>Adv. Math.</jats:italic>324 (2018), 326–335] showed that any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline3.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-homomorphism between Leavitt path algebras over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline4.png\" /> <jats:tex-math> $mathbb Z$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline5.png\" /> <jats:tex-math> $mathbb C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline6.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006698","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 : 2024-02-29DOI: 10.1017/s0004972724000054
FEIHU LIU
We propose generating functions, $textrm {RGF}_p(x)$ , for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $langle a_1,a_2,a_3rangle $ by p for a small positive integer p, and we characterise the generators of ${langle Arangle }/{p}$ for a general numerical semigroup A and any positive integer p.
我们提出了与西尔维斯特数数相关的数值半群商数的生成函数 $textrm {RGF}_p(x)$ 。利用麦克马洪的分割分析，我们可以通过提取有理函数的常数项得到 $textrm {RGF}_p(x)$ 。我们利用 $textrm {RGF}_p(x)$ 给出了一个小正整数 p 的数值半群 $langle a_1,a_2,a_3rangle $ 的商的生成器系统，并描述了一般数值半群 A 和任意正整数 p 的 ${langle Arangle }/{p}$ 的生成器的特征。
{"title":"GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS","authors":"FEIHU LIU","doi":"10.1017/s0004972724000054","DOIUrl":"https://doi.org/10.1017/s0004972724000054","url":null,"abstract":"We propose generating functions, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000054_inline1.png\" /> <jats:tex-math> $textrm {RGF}_p(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000054_inline2.png\" /> <jats:tex-math> $textrm {RGF}_p(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by extracting the constant term of a rational function. We use <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000054_inline3.png\" /> <jats:tex-math> $textrm {RGF}_p(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to give a system of generators for the quotient of the numerical semigroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000054_inline4.png\" /> <jats:tex-math> $langle a_1,a_2,a_3rangle $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by <jats:italic>p</jats:italic> for a small positive integer <jats:italic>p</jats:italic>, and we characterise the generators of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000054_inline5.png\" /> <jats:tex-math> ${langle Arangle }/{p}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for a general numerical semigroup <jats:italic>A</jats:italic> and any positive integer <jats:italic>p</jats:italic>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007156","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 : 2024-02-28DOI: 10.1017/s0004972724000029
SOURAV DAS, ARIJIT GANGULY
We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3)101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.
{"title":"DIOPHANTINE TRANSFERENCE PRINCIPLE OVER FUNCTION FIELDS","authors":"SOURAV DAS, ARIJIT GANGULY","doi":"10.1017/s0004972724000029","DOIUrl":"https://doi.org/10.1017/s0004972724000029","url":null,"abstract":"We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, <jats:italic>Proc. Lond. Math. Soc. (3)</jats:italic>101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006824","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 : 2024-02-15DOI: 10.1017/s0004972723001466
LIA VAŠ
If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${mathbb {Z}}$ . Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.
如果 E 是一个图，K 是一个域，我们将考虑 E 在 K 上的莱维特路径代数 $L_K(E)$ 的理想 I。我们将描述与包含 I 的最小分级理想相对应的可容许对，其中的分级是 $L_K(E)$ 的自然分级 ${mathbb {Z}}$ 。利用这一描述，我们可以证明 I 的右湮和左湮是相等的（鉴于 I 可能不是自结的）。特别是，我们确定这两个湮没器对应于同一可容许对，并且其描述产生了标题中的特征。然后，我们将讨论任意理想的右湮没器（等同于左湮没器）是直接和这一性质，并回顾具有这一性质的单素环被称为准巴环。我们将展示 E 的一个条件，它等价于具有这一性质的单素 $L_K(E)$。
{"title":"GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS","authors":"LIA VAŠ","doi":"10.1017/s0004972723001466","DOIUrl":"https://doi.org/10.1017/s0004972723001466","url":null,"abstract":"If <jats:italic>E</jats:italic> is a graph and <jats:italic>K</jats:italic> is a field, we consider an ideal <jats:italic>I</jats:italic> of the Leavitt path algebra <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001466_inline1.png\" /> <jats:tex-math> $L_K(E)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>E</jats:italic> over <jats:italic>K</jats:italic>. We describe the admissible pair corresponding to the smallest graded ideal which contains <jats:italic>I</jats:italic> where the grading in question is the natural grading of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001466_inline2.png\" /> <jats:tex-math> $L_K(E)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001466_inline3.png\" /> <jats:tex-math> ${mathbb {Z}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this description, we show that the right and the left annihilators of <jats:italic>I</jats:italic> are <jats:italic>equal</jats:italic> (which may be somewhat surprising given that <jats:italic>I</jats:italic> may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on <jats:italic>E</jats:italic> which is equivalent to unital <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001466_inline4.png\" /> <jats:tex-math> $L_K(E)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having this property.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767114","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 : 2024-02-12DOI: 10.1017/s0004972724000017
JIE CHEN, HONGZHANG CHEN, SHOU-JUN XU
In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by $gamma _{t2}(G)$ , is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if $Gneq N_2$ is a connected claw-free graph of order $ngeq 6$ with minimum degree $delta (G)geq 3$ , then $gamma _{t2}(G)leq frac{4}{11}n$ and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin.33(5) (2017), 1119–1130].
在无孤立图 G 中，如果顶点子集 S 是 G 的支配集，且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内，则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 $gamma _{t2}(G)$ 表示，是 G 中半总支配集的最小心数。利用半总支配集上的边加权函数，我们证明了如果 $Gneq N_2$ 是一个阶数为 $n/geq 6$ 且最小度数为 $delta (G)geq 3$ 的无连接爪图，那么 $gamma _{t2}(G)leq frac{4}{11}n$ 并且这个约束是尖锐的，推翻了 Zhu 等人提出的猜想。['无爪立方图中的半总支配'，Graphs Combin.33(5) (2017), 1119-1130].
{"title":"EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS","authors":"JIE CHEN, HONGZHANG CHEN, SHOU-JUN XU","doi":"10.1017/s0004972724000017","DOIUrl":"https://doi.org/10.1017/s0004972724000017","url":null,"abstract":"In an isolate-free graph <jats:italic>G</jats:italic>, a subset <jats:italic>S</jats:italic> of vertices is a <jats:italic>semitotal dominating set</jats:italic> of <jats:italic>G</jats:italic> if it is a dominating set of <jats:italic>G</jats:italic> and every vertex in <jats:italic>S</jats:italic> is within distance 2 of another vertex of <jats:italic>S</jats:italic>. The <jats:italic>semitotal domination number</jats:italic> of <jats:italic>G</jats:italic>, denoted by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000017_inline1.png\" /> <jats:tex-math> $gamma _{t2}(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the minimum cardinality of a semitotal dominating set in <jats:italic>G</jats:italic>. Using edge weighting functions on semitotal dominating sets, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000017_inline2.png\" /> <jats:tex-math> $Gneq N_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a connected claw-free graph of order <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000017_inline3.png\" /> <jats:tex-math> $ngeq 6$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with minimum degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000017_inline4.png\" /> <jats:tex-math> $delta (G)geq 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000017_inline5.png\" /> <jats:tex-math> $gamma _{t2}(G)leq frac{4}{11}n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and this bound is sharp, disproving the conjecture proposed by Zhu <jats:italic>et al.</jats:italic> [‘Semitotal domination in claw-free cubic graphs’, <jats:italic>Graphs Combin.</jats:italic>33(5) (2017), 1119–1130].","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767182","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 : 2024-02-02DOI: 10.1017/s0004972723001478
MENGQI SHI, JIEYAN WANG
Let $Gamma =langle I_{1}, I_{2}, I_{3}rangle $ be the complex hyperbolic $(4,4,infty )$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $Gamma $ is connected and the closure of a countable union of $mathbb {R}$ -circles.
{"title":"ON THE LIMIT SET OF A COMPLEX HYPERBOLIC TRIANGLE GROUP","authors":"MENGQI SHI, JIEYAN WANG","doi":"10.1017/s0004972723001478","DOIUrl":"https://doi.org/10.1017/s0004972723001478","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001478_inline1.png\" /> <jats:tex-math> $Gamma =langle I_{1}, I_{2}, I_{3}rangle $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the complex hyperbolic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001478_inline2.png\" /> <jats:tex-math> $(4,4,infty )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> triangle group with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001478_inline3.png\" /> <jats:tex-math> $I_1I_3I_2I_3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> being unipotent. We show that the limit set of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001478_inline4.png\" /> <jats:tex-math> $Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is connected and the closure of a countable union of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001478_inline5.png\" /> <jats:tex-math> $mathbb {R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-circles.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139668333","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 : 2024-01-30DOI: 10.1017/s0004972723001430
ADOLFO BALLESTER-BOLINCHES, SESUAI Y. MADANHA, TENDAI M. MUDZIIRI SHUMBA, MARÍA C. PEDRAZA-AGUILERA
A group $G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing $A cap B$ and B permutes with every subgroup of A containing $A cap B$. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra595 (2022), 434–443] who showed that if $G'$ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then $G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}} $, where $ mathfrak {F} $ is a saturated formation containing $ mathfrak {U} $, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning
如果 A 与 B 的每一个包含 $A cap B$ 的子群发生互变，而 B 与 A 的每一个包含 $A cap B$ 的子群发生互变，那么一个群 $G=AB$ 是子群 A 和 B 的弱互变积。第一、第二和第四作者提出了弱互变积['广义互变积与饱和形式'，《代数学报》595 (2022)，434-443]，他们证明了如果 $G'$ 是零幂的，A 与 B 的每个 Sylow 子群互变，B 与 A 的每个 Sylow 子群互变，那么 $G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}}.$, 其中 $ mathfrak {F} $ 是包含 $ mathfrak {U} $ 的饱和形成，即超可溶群类。在这篇文章中，我们证明了关于$ mathfrak {F} $残差、$ mathfrak {F} $投影和$ mathfrak {F}$ 归一的弱互变积的结果。作为我们一些论证的应用，我们统一了关于弱互斥 $sn$ 积的一些结果。
{"title":"GENERALISED MUTUALLY PERMUTABLE PRODUCTS AND SATURATED FORMATIONS, II","authors":"ADOLFO BALLESTER-BOLINCHES, SESUAI Y. MADANHA, TENDAI M. MUDZIIRI SHUMBA, MARÍA C. PEDRAZA-AGUILERA","doi":"10.1017/s0004972723001430","DOIUrl":"https://doi.org/10.1017/s0004972723001430","url":null,"abstract":"<p>A group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G=AB$</span></span></img></span></span> is the weakly mutually permutable product of the subgroups <span>A</span> and <span>B</span>, if <span>A</span> permutes with every subgroup of <span>B</span> containing <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$A cap B$</span></span></img></span></span> and <span>B</span> permutes with every subgroup of <span>A</span> containing <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$A cap B$</span></span></img></span></span>. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, <span>J. Algebra</span> <span>595</span> (2022), 434–443] who showed that if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G'$</span></span></img></span></span> is nilpotent, <span>A</span> permutes with every Sylow subgroup of <span>B</span> and <span>B</span> permutes with every Sylow subgroup of <span>A</span>, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}} $</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$ mathfrak {F} $</span></span></img></span></span> is a saturated formation containing <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$ mathfrak {U} $</span></span></img></span></span>, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139589993","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 : 2024-01-29DOI: 10.1017/s0004972723001387
ROBERT E. GAUNT
We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.
{"title":"ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION","authors":"ROBERT E. GAUNT","doi":"10.1017/s0004972723001387","DOIUrl":"https://doi.org/10.1017/s0004972723001387","url":null,"abstract":"We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586405","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 : 2024-01-29DOI: 10.1017/s0004972723001405
SHIBO LIU
We consider the Dirichlet problem for $p(x)$ -Laplacian equations of the form $$ begin{align*} -Delta_{p(x)}u+b(x)vert uvert ^{p(x)-2}u=f(x,u),quad uin W_{0}^{1,p(x)}(Omega). end{align*} $$ The odd nonlinearity $f(x,u)$ is $p(x)$ -sublinear at $u=0$ but the related limit need not be uniform for $xin Omega $ . Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$ .
{"title":"MULTIPLE SOLUTIONS FOR -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO","authors":"SHIBO LIU","doi":"10.1017/s0004972723001405","DOIUrl":"https://doi.org/10.1017/s0004972723001405","url":null,"abstract":"We consider the Dirichlet problem for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline2.png\" /> <jats:tex-math> $p(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-Laplacian equations of the form <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_eqnu1.png\" /> <jats:tex-math> $$ begin{align*} -Delta_{p(x)}u+b(x)vert uvert ^{p(x)-2}u=f(x,u),quad uin W_{0}^{1,p(x)}(Omega). end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> The odd nonlinearity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline3.png\" /> <jats:tex-math> $f(x,u)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline4.png\" /> <jats:tex-math> $p(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-sublinear at <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline5.png\" /> <jats:tex-math> $u=0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> but the related limit need not be uniform for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline6.png\" /> <jats:tex-math> $xin Omega $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Except being subcritical, no additional assumption is imposed on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline7.png\" /> <jats:tex-math> $f(x,u)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline8.png\" /> <jats:tex-math> $|u|$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001405_inline9.png\" /> <jats:tex-math> $u=0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586220","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}
$G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing 
$A cap B$ and B permutes with every subgroup of A containing 
$A cap B$. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra 595 (2022), 434–443] who showed that if 
$G'$ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then 
$G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}} $, where 
$ mathfrak {F} $ is a saturated formation containing 
$ mathfrak {U} $, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning 
如果 A 与 B 的每一个包含 $A cap B$ 的子群发生互变，而 B 与 A 的每一个包含 $A cap B$ 的子群发生互变，那么一个群 $G=AB$ 是子群 A 和 B 的弱互变积。第一、第二和第四作者提出了弱互变积['广义互变积与饱和形式'，《代数学报》595 (2022)，434-443]，他们证明了如果 $G'$ 是零幂的，A 与 B 的每个 Sylow 子群互变，B 与 A 的每个 Sylow 子群互变，那么 $G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}}.$, 其中 $ mathfrak {F} $ 是包含 $ mathfrak {U} $ 的饱和形成，即超可溶群类。在这篇文章中，我们证明了关于$ mathfrak {F} $残差、$ mathfrak {F} $投影和$ mathfrak {F}$ 归一的弱互变积的结果。作为我们一些论证的应用，我们统一了关于弱互斥 $sn$ 积的一些结果。
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