Bulletin of the Australian Mathematical Society最新文献

{"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":"42 1","pages":""},"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}

{"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":"33 1","pages":""},"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}

{"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":"93 1","pages":""},"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}

{"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":"14 1","pages":""},"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}

{"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":"273 ","pages":""},"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}

{"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":"&lt;p&gt;A group &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$G=AB$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is the weakly mutually permutable product of the subgroups &lt;span&gt;A&lt;/span&gt; and &lt;span&gt;B&lt;/span&gt;, if &lt;span&gt;A&lt;/span&gt; permutes with every subgroup of &lt;span&gt;B&lt;/span&gt; containing &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$A cap B$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;B&lt;/span&gt; permutes with every subgroup of &lt;span&gt;A&lt;/span&gt; containing &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$A cap B$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, &lt;span&gt;J. Algebra&lt;/span&gt; &lt;span&gt;595&lt;/span&gt; (2022), 434–443] who showed that if &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$G'$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is nilpotent, &lt;span&gt;A&lt;/span&gt; permutes with every Sylow subgroup of &lt;span&gt;B&lt;/span&gt; and &lt;span&gt;B&lt;/span&gt; permutes with every Sylow subgroup of &lt;span&gt;A&lt;/span&gt;, then &lt;span&gt;&lt;span&gt;&lt;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\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}} $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$ mathfrak {F} $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; is a saturated formation containing &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$ mathfrak {U} $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning &lt;span&gt;&lt;span&gt;&lt;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":"6 1","pages":""},"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}

{"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":"173 1","pages":""},"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}

{"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":"44 1","pages":""},"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}

{"title":"CONGRUENCES FOR RANKS OF PARTITIONS","authors":"RENRONG MAO","doi":"10.1017/s0004972723001454","DOIUrl":"https://doi.org/10.1017/s0004972723001454","url":null,"abstract":"Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001454_inline1.png\" /> <jats:tex-math> $5$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for ranks of partitions.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586215","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":"HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS","authors":"NADIA TAGHIPOUR, SHAMILA BAYATI, FARHAD RAHMATI","doi":"10.1017/s0004972723001363","DOIUrl":"https://doi.org/10.1017/s0004972723001363","url":null,"abstract":"It is well known that the edge ideal &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline1.png\" /&gt; &lt;jats:tex-math&gt; $I(G)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of a simple graph &lt;jats:italic&gt;G&lt;/jats:italic&gt; has linear quotients if and only if &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline2.png\" /&gt; &lt;jats:tex-math&gt; $G^c$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline3.png\" /&gt; &lt;jats:tex-math&gt; $G^c$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; when &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline4.png\" /&gt; &lt;jats:tex-math&gt; $I(G)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has homological linear quotients results in a graph with the same property. In particular, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline5.png\" /&gt; &lt;jats:tex-math&gt; $I(G)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has homological linear quotients when &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline6.png\" /&gt; &lt;jats:tex-math&gt; $G^c$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline7.png\" /&gt; &lt;jats:tex-math&gt; $I(G)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has homological linear quotients for every graph &lt;jats:italic&gt;G&lt;/jats:italic&gt; such that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline8.png\" /&gt; &lt;jats:tex-math&gt; $G^c$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline9.png\" /&gt; &lt;jats:tex-math&gt; $lambda $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-min","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"38 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586223","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}