Pub Date : 2023-11-06DOI: 10.1017/s0004972723001077
SAUNAK BHATTACHARJEE, ANUP B. DIXIT, DISHANT SAIKIA
Abstract For $kgeq 2$ and a nonzero integer n , a generalised Diophantine m -tuple with property $D_k(n)$ is a set of m positive integers $S = {a_1,a_2,ldots , a_m}$ such that $a_ia_j + n$ is a k th power for $1leq i< jleq m$ . Define $M_k(n):= text {sup}{|S| : S$ having property $D_k(n)}$ . Dixit et al . [‘Generalised Diophantine m -tuples’, Proc. Amer. Math. Soc. 150 (4) (2022), 1455–1465] proved that $M_k(n)=O(log n)$ , for a fixed k , as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$ . In particular, we show that for $kgeq 2$ , $M_k(n) leq 3,phi (k) log n$ if n is sufficiently large compared to k .
{"title":"AN EFFECTIVE BOUND FOR GENERALISED DIOPHANTINE <i>m</i>-TUPLES","authors":"SAUNAK BHATTACHARJEE, ANUP B. DIXIT, DISHANT SAIKIA","doi":"10.1017/s0004972723001077","DOIUrl":"https://doi.org/10.1017/s0004972723001077","url":null,"abstract":"Abstract For $kgeq 2$ and a nonzero integer n , a generalised Diophantine m -tuple with property $D_k(n)$ is a set of m positive integers $S = {a_1,a_2,ldots , a_m}$ such that $a_ia_j + n$ is a k th power for $1leq i< jleq m$ . Define $M_k(n):= text {sup}{|S| : S$ having property $D_k(n)}$ . Dixit et al . [‘Generalised Diophantine m -tuples’, Proc. Amer. Math. Soc. 150 (4) (2022), 1455–1465] proved that $M_k(n)=O(log n)$ , for a fixed k , as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$ . In particular, we show that for $kgeq 2$ , $M_k(n) leq 3,phi (k) log n$ if n is sufficiently large compared to k .","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135634080","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 : 2023-11-06DOI: 10.1017/s0004972723001089
MINGLIANG FANG, HUI LI, XIAO YAO
Abstract We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=exp (h_1(z)+C_1 z)$ , where $h_1$ is an entire function of period c and $exp (C_1 c)neq 1$ , or $f(z)=exp (h_2(z)+C_2 z)$ , where $h_2$ is an entire function of period $2c$ and $C_2$ satisfies $$ begin{align*} bigg(frac{1+exp(C_2c)}{1-exp(C_2 c)}bigg)^{2n}=1. end{align*} $$
{"title":"THE DIFFERENCE ANALOGUE OF THE TUMURA–HAYMAN–CLUNIE THEOREM","authors":"MINGLIANG FANG, HUI LI, XIAO YAO","doi":"10.1017/s0004972723001089","DOIUrl":"https://doi.org/10.1017/s0004972723001089","url":null,"abstract":"Abstract We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=exp (h_1(z)+C_1 z)$ , where $h_1$ is an entire function of period c and $exp (C_1 c)neq 1$ , or $f(z)=exp (h_2(z)+C_2 z)$ , where $h_2$ is an entire function of period $2c$ and $C_2$ satisfies $$ begin{align*} bigg(frac{1+exp(C_2c)}{1-exp(C_2 c)}bigg)^{2n}=1. end{align*} $$","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135634229","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 : 2023-11-06DOI: 10.1017/s0004972723001107
KEVIN LIMANTA
Abstract We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $mathbb {F}$ of characteristic zero as a normalised $mathbb {F}$ -linear functional on $mathbb {F}[alpha _1, alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $mathrm {SO}(2,mathbb {F})$ -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$ begin{align*} S(m,n) = frac{(2m)!(2n)!}{m!n!(m+n)!} end{align*} $$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $alpha _1^{2m}alpha _2^{2n}$ .
{"title":"AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS","authors":"KEVIN LIMANTA","doi":"10.1017/s0004972723001107","DOIUrl":"https://doi.org/10.1017/s0004972723001107","url":null,"abstract":"Abstract We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $mathbb {F}$ of characteristic zero as a normalised $mathbb {F}$ -linear functional on $mathbb {F}[alpha _1, alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $mathrm {SO}(2,mathbb {F})$ -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$ begin{align*} S(m,n) = frac{(2m)!(2n)!}{m!n!(m+n)!} end{align*} $$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $alpha _1^{2m}alpha _2^{2n}$ .","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135634245","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 : 2023-10-27DOI: 10.1017/s0004972723001053
SHUANG-SHUANG LI, YU-QING SHAN, XIAO-HUI YAN
Abstract For any positive integers $k_1,k_2$ and any set $Asubseteq mathbb {N}$ , let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2in A$ . Let g be a fixed integer. We prove that if $k_1$ and $k_2$ are two integers with $2le k_1
摘要对于任意正整数$k_1,k_2$和任意集合$Asubseteq mathbb {N}$，设$R_{k_1,k_2}(A, N)$是方程$ N =k_1a_1+k_2a_2$与$a_1,a_2在A$中的解的个数。设g为一个固定整数。证明了如果$k_1$和$k_2$是两个整数，且$2le k_1<k_2$和$(k_1,k_2)=1$，则不存在任何集$Asubseteq mathbb {N}$使得$R_{k_1,k_2}(A, N)-R_{k_1,k_2}( math_1 <k_2$)=g$，如果$1=k_1<k_2$，则存在一个集A使得$R_{k_1,k_2}(A, N)-R_{k_1,k_2}( math_1 <k_2}(mathbb {N} set- A, N)对所有正整数N =1$。
{"title":"PARTITIONS OF NATURAL NUMBERS AND THEIR WEIGHTED REPRESENTATION FUNCTIONS","authors":"SHUANG-SHUANG LI, YU-QING SHAN, XIAO-HUI YAN","doi":"10.1017/s0004972723001053","DOIUrl":"https://doi.org/10.1017/s0004972723001053","url":null,"abstract":"Abstract For any positive integers $k_1,k_2$ and any set $Asubseteq mathbb {N}$ , let $R_{k_1,k_2}(A,n)$ be the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2in A$ . Let g be a fixed integer. We prove that if $k_1$ and $k_2$ are two integers with $2le k_1<k_2$ and $(k_1,k_2)=1$ , then there does not exist any set $Asubseteq mathbb {N}$ such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(mathbb {N}setminus A,n)=g$ for all sufficiently large integers n , and if $1=k_1<k_2$ , then there exists a set A such that $R_{k_1,k_2}(A,n)-R_{k_1,k_2}(mathbb {N}setminus A,n)=1$ for all positive integers n .","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"276 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136261755","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 : 2023-10-26DOI: 10.1017/s0004972723001065
JONATHAN MUI
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":"EVENTUAL POSITIVITY AND ASYMPTOTIC BEHAVIOUR FOR HIGHER-ORDER EVOLUTION EQUATIONS","authors":"JONATHAN MUI","doi":"10.1017/s0004972723001065","DOIUrl":"https://doi.org/10.1017/s0004972723001065","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":"164 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136381371","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 : 2023-10-23DOI: 10.1017/s0004972723001028
ANUP B. DIXIT, VEEKESH KUMAR, SIDDHI S. PATHAK
Abstract We establish the linear independence of values of the q -analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q -series.
{"title":"LINEAR INDEPENDENCE OF VALUES OF THE <i>q</i>-EXPONENTIAL AND RELATED FUNCTIONS","authors":"ANUP B. DIXIT, VEEKESH KUMAR, SIDDHI S. PATHAK","doi":"10.1017/s0004972723001028","DOIUrl":"https://doi.org/10.1017/s0004972723001028","url":null,"abstract":"Abstract We establish the linear independence of values of the q -analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q -series.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"46 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368326","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 : 2023-10-20DOI: 10.1017/s000497272300103x
JUNYANG ZHANG, HANG ZHOU
Abstract It is proved that Tutte’s $3$ -flow conjecture is true for Cayley graphs on groups of order $8p$ where p is an odd prime.
摘要证明了在p为奇素数的$8p$群上的Cayley图上Tutte的$3$流猜想的成立。
{"title":"NOWHERE-ZERO -FLOWS IN CAYLEY GRAPHS OF ORDER","authors":"JUNYANG ZHANG, HANG ZHOU","doi":"10.1017/s000497272300103x","DOIUrl":"https://doi.org/10.1017/s000497272300103x","url":null,"abstract":"Abstract It is proved that Tutte’s $3$ -flow conjecture is true for Cayley graphs on groups of order $8p$ where p is an odd prime.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135570151","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 : 2023-10-20DOI: 10.1017/s0004972723001041
DIEGO ALVES, JEAN LELIS, DIEGO MARQUES, PAVEL TROJOVSKÝ
Abstract We prove that any subset of $overline {mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [ Lectures on Transcendental Numbers , Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].
{"title":"ON THE EXCEPTIONAL SET OF TRANSCENDENTAL ENTIRE FUNCTIONS IN SEVERAL VARIABLES","authors":"DIEGO ALVES, JEAN LELIS, DIEGO MARQUES, PAVEL TROJOVSKÝ","doi":"10.1017/s0004972723001041","DOIUrl":"https://doi.org/10.1017/s0004972723001041","url":null,"abstract":"Abstract We prove that any subset of $overline {mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [ Lectures on Transcendental Numbers , Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135570154","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/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":"3 1","pages":"0"},"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":"119 1","pages":"0"},"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}
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