Pub Date : 2012-11-29DOI: 10.3318/PRIA.2014.114.03
S. Bell, T. Ferguson, Erik Lundberg
For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.
对于一个次正则算子，C. R. Putnam不等式给出了它的自对易子的模的上界。在域的Smirnov空间中具有解析符号的Toeplitz算子的特殊情况下，还存在D. Khavinson(1985)所示的几何下界，该下界与Putnam不等式结合时隐含经典等周期不等式。对于非平凡域，我们将这些估计与精确结果进行比较。然后我们考虑这些算子作用于一个域的Bergman空间上，我们得到了反映该域几何形状的下界。当与Putnam不等式结合在一起时，它们产生了域基频的Faber-Krahn不等式和扭转刚度的Saint-Venant不等式(但具有非尖锐常数)。我们在这个受限的环境中推测出普特南不等式的改进版本。
{"title":"Self-commutators of Toeplitz operators and isoperimetric inequalities","authors":"S. Bell, T. Ferguson, Erik Lundberg","doi":"10.3318/PRIA.2014.114.03","DOIUrl":"https://doi.org/10.3318/PRIA.2014.114.03","url":null,"abstract":"For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134282231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-03-31DOI: 10.3318/PRIA.2011.111.1.8
A. Das, R. K. Nath
The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 .
{"title":"A CHARACTERISATION OF CERTAIN FINITE GROUPS OF ODD ORDER","authors":"A. Das, R. K. Nath","doi":"10.3318/PRIA.2011.111.1.8","DOIUrl":"https://doi.org/10.3318/PRIA.2011.111.1.8","url":null,"abstract":"The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11 .","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127391003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-01-07DOI: 10.3318/PRIA.2011.111.1.3
Matthew Daws
A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $pi:mc Atomc B(E)$ such that $pi(mc A)$ equals its own bicommutant.
对偶巴拿赫代数是一个巴拿赫代数，它是一个对偶空间，其乘法分别是弱$^*$连续的。我们证明了给定一个一元对偶巴拿赫代数$mc a $，我们可以找到一个自反巴拿赫空间$E$，以及一个等距的弱$^*$-弱$^*$-连续同态$pi:mc a $到mc B(E)$使得$pi(mc a)$等于它自己的双元突变体$。
{"title":"A BICOMMUTANT THEOREM FOR DUAL BANACH ALGEBRAS","authors":"Matthew Daws","doi":"10.3318/PRIA.2011.111.1.3","DOIUrl":"https://doi.org/10.3318/PRIA.2011.111.1.3","url":null,"abstract":"A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $pi:mc Atomc B(E)$ such that $pi(mc A)$ equals its own bicommutant.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115395151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-12-11DOI: 10.3318/PRIA.2008.109.1.49
M. Carley
A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) + c(x,y,t)log[(x-t)^2+y^2]^{1/2} + d(x,y,t), without having to explicitly analyze the singularities of $f(x,y,t)$ or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when $yequiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.
{"title":"NUMERICAL QUADRATURES FOR NEAR-SINGULAR AND NEAR-HYPERSINGULAR INTEGRALS IN BOUNDARY ELEMENT METHODS","authors":"M. Carley","doi":"10.3318/PRIA.2008.109.1.49","DOIUrl":"https://doi.org/10.3318/PRIA.2008.109.1.49","url":null,"abstract":"A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) + c(x,y,t)log[(x-t)^2+y^2]^{1/2} + d(x,y,t), without having to explicitly analyze the singularities of $f(x,y,t)$ or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when $yequiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"260 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133840460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-10-23DOI: 10.3318/PRIA.2008.109.1.79
P. Hegarty
We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.
{"title":"FINITE GROUPS WITH AN AUTOMORPHISM CUBING A LARGE FRACTION OF ELEMENTS","authors":"P. Hegarty","doi":"10.3318/PRIA.2008.109.1.79","DOIUrl":"https://doi.org/10.3318/PRIA.2008.109.1.79","url":null,"abstract":"We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. \u0000 \u0000Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117338695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-09-11DOI: 10.3318/PRIA.2009.109.2.187
S. Verpoort
The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.
{"title":"ON THE AREA FUNCTIONAL OF THE SECOND FUNDAMENTAL FORM OF OVALOIDS","authors":"S. Verpoort","doi":"10.3318/PRIA.2009.109.2.187","DOIUrl":"https://doi.org/10.3318/PRIA.2009.109.2.187","url":null,"abstract":"The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called \"mean curvature of the second fundamental form\". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125926220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-03-13DOI: 10.3318/PRIA.2010.110.2.131
S. Pumpluen
A simple sufficient condition for certain cyclic algebras of odd degree d to be split is presented. It employs certain binary forms of degree d and the values they represent. A similar sufficient condition for certain Albert algebras not to be division algebras is found as well.
{"title":"A CONDITION FOR CYCLIC ALGEBRAS TO BE SPLIT","authors":"S. Pumpluen","doi":"10.3318/PRIA.2010.110.2.131","DOIUrl":"https://doi.org/10.3318/PRIA.2010.110.2.131","url":null,"abstract":"A simple sufficient condition for certain cyclic algebras of odd degree d to be split is presented. It employs certain binary forms of degree d and the values they represent. A similar sufficient condition for certain Albert algebras not to be division algebras is found as well.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128383862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-09-24DOI: 10.3318/PRIA.2007.107.1.1
A. Zimmermann
For any algebraically closed field $k$ of positive characteristic $p$ and any non negative integer $n$ K"ulshammer defined ideals $T_nA^perp$ of the centre of a symmetric $k$-algebra $A$. We show that for derived equivalent algebras $A$ and $B$ there is an isomorphism of the centres of $A$ and $B$ mapping $T_nA^perp$ to $T_nB^perp$ for all $n$. Recently H'ethelyi, Horv'ath, K"ulshammer and Murray showed that this holds for Morita equivalent algebras.
对于任意具有正特征的代数闭域$k$和任意非负整数$n$ k$， ulshammer定义了对称$k$-代数$ a $中心的理想$T_nA^perp$。我们证明了对于派生的等价代数$A$和$B$，对于所有$n$， $A$和$B$的中心映射$T_nA^perp$到$T_nB^perp$是同构的。最近，H ethelyi, Horv ath, K ulshammer和Murray证明了这对于Morita等价代数是成立的。
{"title":"INVARIANCE OF GENERALISED REYNOLDS IDEALS UNDER DERIVED EQUIVALENCES","authors":"A. Zimmermann","doi":"10.3318/PRIA.2007.107.1.1","DOIUrl":"https://doi.org/10.3318/PRIA.2007.107.1.1","url":null,"abstract":"For any algebraically closed field $k$ of positive characteristic $p$ and any non negative integer $n$ K\"ulshammer defined ideals $T_nA^perp$ of the centre of a symmetric $k$-algebra $A$. We show that for derived equivalent algebras $A$ and $B$ there is an isomorphism of the centres of $A$ and $B$ mapping $T_nA^perp$ to $T_nB^perp$ for all $n$. Recently H'ethelyi, Horv'ath, K\"ulshammer and Murray showed that this holds for Morita equivalent algebras.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126711704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-04-07DOI: 10.3318/PRIA.2008.108.1.89
B. Solel
We study completely contractive representations of product systems $X$ of correspondences over the semigroup $mathbb{Z}_+^k$. We present a necessary and sufficient condition for such a representation to have a regular isometric dilation. We discuss representations that doubly commute and show that these representations induce completely contractive representations of the norm closed algebra generated by the image of the Fock representation of $X$.
{"title":"REGULAR DILATIONS OF REPRESENTATIONS OF PRODUCT SYSTEMS","authors":"B. Solel","doi":"10.3318/PRIA.2008.108.1.89","DOIUrl":"https://doi.org/10.3318/PRIA.2008.108.1.89","url":null,"abstract":"We study completely contractive representations of product systems $X$ of correspondences over the semigroup $mathbb{Z}_+^k$. We present a necessary and sufficient condition for such a representation to have a regular isometric dilation. We discuss representations that doubly commute and show that these representations induce completely contractive representations of the norm closed algebra generated by the image of the Fock representation of $X$.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127695484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We revisit a simply stated problem of Knuth. Previous approaches rely on the Bernoulli nature of the underlying stochastic process to recover the systems mean behaviour. We show that limiting results hold for a wide range of stochastic processes. A Large Deviation Principle (LDP) is proved, allowing estimates to be made for the probability of rare-events. From the LDP, a weak law of large numbers is deduced.
{"title":"ON KNUTH'S GENERALISATION OF BANACH'S MATCHBOX PROBLEM","authors":"W. Dukes, K. Duffy","doi":"10.1353/mpr.2004.0023","DOIUrl":"https://doi.org/10.1353/mpr.2004.0023","url":null,"abstract":"We revisit a simply stated problem of Knuth. Previous approaches rely on the Bernoulli nature of the underlying stochastic process to recover the systems mean behaviour. We show that limiting results hold for a wide range of stochastic processes. A Large Deviation Principle (LDP) is proved, allowing estimates to be made for the probability of rare-events. From the LDP, a weak law of large numbers is deduced.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130184136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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