Pub Date : 2023-10-09DOI: 10.1017/s0004972723001004
MUHAMMAD AFIFURRAHMAN
Abstract Given a set X of $ntimes n$ matrices and a positive integer m , we consider the problem of estimating the cardinalities of the product sets $A_1 cdots A_m$ , where $A_iin X$ . When $X={mathcal M}_n(mathbb {Z};H)$ , the set of $ntimes n$ matrices with integer elements of size at most H , we give several bounds on the cardinalities of the product sets and solution sets of related equations such as $A_1 cdots A_m=C$ and $A_1 cdots A_m=B_1 cdots B_m$ . We also consider the case where X is the subset of matrices in ${mathcal M}_n(mathbb {F})$ , where $mathbb {F}$ is a field with bounded rank $kleq n$ . In this case, we completely classify the related product set.
{"title":"SOME COUNTING QUESTIONS FOR MATRIX PRODUCTS","authors":"MUHAMMAD AFIFURRAHMAN","doi":"10.1017/s0004972723001004","DOIUrl":"https://doi.org/10.1017/s0004972723001004","url":null,"abstract":"Abstract Given a set X of $ntimes n$ matrices and a positive integer m , we consider the problem of estimating the cardinalities of the product sets $A_1 cdots A_m$ , where $A_iin X$ . When $X={mathcal M}_n(mathbb {Z};H)$ , the set of $ntimes n$ matrices with integer elements of size at most H , we give several bounds on the cardinalities of the product sets and solution sets of related equations such as $A_1 cdots A_m=C$ and $A_1 cdots A_m=B_1 cdots B_m$ . We also consider the case where X is the subset of matrices in ${mathcal M}_n(mathbb {F})$ , where $mathbb {F}$ is a field with bounded rank $kleq n$ . In this case, we completely classify the related product set.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135095359","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 : 2023-10-09DOI: 10.1017/s0004972723000989
BIN BIN HAN, WEN TING ZHANG, YAN FENG LUO
Abstract An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.
摘要最近发现了一个五阶非有限基对合单阵的例子。我们证实了这个例子在同构上是所有对合模中唯一最小的。
{"title":"FINITE BASIS PROBLEM FOR INVOLUTION MONOIDS OF ORDER FIVE","authors":"BIN BIN HAN, WEN TING ZHANG, YAN FENG LUO","doi":"10.1017/s0004972723000989","DOIUrl":"https://doi.org/10.1017/s0004972723000989","url":null,"abstract":"Abstract An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135095668","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 : 2023-10-09DOI: 10.1017/s0004972723000977
DEKE LI, QINGXUAN WANG
Abstract We consider the two-dimensional minimisation problem for $inf { E_a(varphi ):varphi in H^1(mathbb {R}^2) text {and} |varphi |_2^2=1}$ , where the energy functional $ E_a(varphi )$ is a cubic-quintic Schrödinger functional defined by $E_a(varphi ):=tfrac 12int _{mathbb {R}^2}|nabla varphi |^2,dx-tfrac 14aint _{mathbb {R}^2}|varphi |^4,dx+tfrac 16a^2int _{mathbb {R}^2}|varphi |^6,dx$ . We study the existence and asymptotic behaviour of the ground state. The ground state $varphi _{a}$ exists if and only if the $L^2$ mass a satisfies $a>a_*={lVert QrVert }^2_2$ , where Q is the unique positive radial solution of $-Delta u+ u-u^3=0$ in $mathbb {R}^2$ . We show the optimal vanishing rate $int _{mathbb {R}^2}|nabla varphi _{a}|^2,dxsim (a-a_*)$ as $asearrow a_*$ and obtain the limit profile.
{"title":"A NOTE ON NORMALISED GROUND STATES FOR THE TWO-DIMENSIONAL CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION","authors":"DEKE LI, QINGXUAN WANG","doi":"10.1017/s0004972723000977","DOIUrl":"https://doi.org/10.1017/s0004972723000977","url":null,"abstract":"Abstract We consider the two-dimensional minimisation problem for $inf { E_a(varphi ):varphi in H^1(mathbb {R}^2) text {and} |varphi |_2^2=1}$ , where the energy functional $ E_a(varphi )$ is a cubic-quintic Schrödinger functional defined by $E_a(varphi ):=tfrac 12int _{mathbb {R}^2}|nabla varphi |^2,dx-tfrac 14aint _{mathbb {R}^2}|varphi |^4,dx+tfrac 16a^2int _{mathbb {R}^2}|varphi |^6,dx$ . We study the existence and asymptotic behaviour of the ground state. The ground state $varphi _{a}$ exists if and only if the $L^2$ mass a satisfies $a>a_*={lVert QrVert }^2_2$ , where Q is the unique positive radial solution of $-Delta u+ u-u^3=0$ in $mathbb {R}^2$ . We show the optimal vanishing rate $int _{mathbb {R}^2}|nabla varphi _{a}|^2,dxsim (a-a_*)$ as $asearrow a_*$ and obtain the limit profile.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135044550","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 : 2023-10-06DOI: 10.1017/s0004972723000990
ARUNMARAN MAHENTHIRAM
An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.
{"title":"HARMONIC-MEASURE DISTRIBUTION FUNCTIONS AND RELATED FUNCTIONS FOR SIMPLY CONNECTED AND MULTIPLY CONNECTED TWO-DIMENSIONAL REGIONS","authors":"ARUNMARAN MAHENTHIRAM","doi":"10.1017/s0004972723000990","DOIUrl":"https://doi.org/10.1017/s0004972723000990","url":null,"abstract":"An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350335","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 : 2023-10-06DOI: 10.1017/s0004972723000953
DANIEL GIL-MUÑOZ, MAGDALÉNA TINKOVÁ
Abstract The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $mathbb {Z}$ -form) that is universal over K . We prove the nonexistence of universal $mathbb {Z}$ -forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.
{"title":"THE LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS OVER SIMPLEST CUBIC FIELDS","authors":"DANIEL GIL-MUÑOZ, MAGDALÉNA TINKOVÁ","doi":"10.1017/s0004972723000953","DOIUrl":"https://doi.org/10.1017/s0004972723000953","url":null,"abstract":"Abstract The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $mathbb {Z}$ -form) that is universal over K . We prove the nonexistence of universal $mathbb {Z}$ -forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351411","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 : 2023-10-06DOI: 10.1017/s0004972723000965
HONGSHENG HU
Abstract In 1968, Steinberg [ Endomorphisms of Linear Algebraic Groups , Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.
1968年，Steinberg[线性代数群的自同态，Memoirs of the American Mathematical Society, 1980]证明了欧氏反射群的不可约反射表示的外幂也是不可约且对非同构的定理。我们将这一结果推广到群作用下的内积不变量不一定存在的更一般的情况。
{"title":"ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS","authors":"HONGSHENG HU","doi":"10.1017/s0004972723000965","DOIUrl":"https://doi.org/10.1017/s0004972723000965","url":null,"abstract":"Abstract In 1968, Steinberg [ Endomorphisms of Linear Algebraic Groups , Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351407","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 : 2023-10-06DOI: 10.1017/s0004972723000941
PRECIOUS U. AGIGOR-MIKE, SARAH B. HART, MARTIN C. OBI
Abstract In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc notin S$ for all $a, b, c in S$ . If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.
摘要本文研究了三积集，它类似于广泛研究的无积集的概念。如果群G的非空子集S对于S$中的所有$ A, b, c $是无三积的。如果S是无三积集合并且不是任何其他无三积集合的真子集，我们说S是局部极大的。我们对包含大小为1的局部极大无三积集的所有群进行分类。然后，我们得到了群的一个子集局部极大三积无的充要条件，并与标准三积集的情况作了一些观察和比较。
{"title":"TRIPLE-PRODUCT-FREE SETS","authors":"PRECIOUS U. AGIGOR-MIKE, SARAH B. HART, MARTIN C. OBI","doi":"10.1017/s0004972723000941","DOIUrl":"https://doi.org/10.1017/s0004972723000941","url":null,"abstract":"Abstract In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc notin S$ for all $a, b, c in S$ . If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350331","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 : 2023-09-18DOI: 10.1017/s0004972723000904
MIN CHEN, MIN TANG
Abstract Let $S={s_{1}, s_{2}, ldots }$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $alpha $ as $nrightarrow infty $ and let $beta>max (alpha , 2)$ . We show that for all sufficiently large positive integers l , if $Asubset [0, l]$ with $lin A$ , $gcd A=1$ and $|A|geq (2-{k}/{lambda beta })l/(lambda +1)$ , where $lambda =lceil {k}/{beta }rceil $ , then $kAcap Sneq emptyset $ for $23$ and $kgeq 3$ .
{"title":"SUMSETS CONTAINING A TERM OF A SEQUENCE","authors":"MIN CHEN, MIN TANG","doi":"10.1017/s0004972723000904","DOIUrl":"https://doi.org/10.1017/s0004972723000904","url":null,"abstract":"Abstract Let $S={s_{1}, s_{2}, ldots }$ be an unbounded sequence of positive integers with $s_{n+1}/s_{n}$ approaching $alpha $ as $nrightarrow infty $ and let $beta>max (alpha , 2)$ . We show that for all sufficiently large positive integers l , if $Asubset [0, l]$ with $lin A$ , $gcd A=1$ and $|A|geq (2-{k}/{lambda beta })l/(lambda +1)$ , where $lambda =lceil {k}/{beta }rceil $ , then $kAcap Sneq emptyset $ for $2<beta leq 3$ and $kgeq {2beta }/{(beta -2)}$ or for $beta>3$ and $kgeq 3$ .","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135203016","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 : 2023-09-18DOI: 10.1017/s0004972723000849
SARIKA DEVHARE, VINAYAK JOSHI, JOHN LAGRANGE
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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{"title":"CORRECTION TO ‘ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET’","authors":"SARIKA DEVHARE, VINAYAK JOSHI, JOHN LAGRANGE","doi":"10.1017/s0004972723000849","DOIUrl":"https://doi.org/10.1017/s0004972723000849","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135110028","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 : 2023-09-15DOI: 10.1017/s0004972723000916
YOSSI BOKOR BLEILE
{"title":"GEOMETRIC AND TOPOLOGICAL SHAPE ANALYSIS: INVESTIGATING AND SUMMARISING THE SHAPE OF DATA","authors":"YOSSI BOKOR BLEILE","doi":"10.1017/s0004972723000916","DOIUrl":"https://doi.org/10.1017/s0004972723000916","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135396565","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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