Mathematical Proceedings of the Royal Irish Academy最新文献

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AN UNCERTAINTY PRINCIPLE RELATED TO THE EUCLIDEAN MOTION GROUP 与欧几里得运动群有关的一种测不准原理

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 2004-01-30 DOI: 10.3318/PRIA.2004.104.2.249

J. Christensen, H. Schlichtkrull

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/", g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), ".-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.o

我们证明了一个众所周知的圆上函数的测不准原理可以由欧几里得运动群的测不准原理导出。1. 设G是李代数G的李群，设(n, H)是G的酉表示，则每个元素Xe G在H × n(exptX) X - X n(X) X = hm '-o t上生成一个封闭的斜伴随算子n(X)，其域D(n(X))由存在极限的所有Xe H组成。与n相关的不确定性原理表明，对于由X,Y和[X,Y]生成的算子，下面的式子成立“n(X) X ''''n(Y) X”>^'(n([X,Y]) X, X) '对于所有xeD(n(X))nD(n(n(Y))nD(n([X, Y]))。我们主张这是实现不确定性原理的一种自然方式。Kraus[4]首先提出了三维或更少维度的李群，最近Christensen在[2]中去掉了维度约束。例如，Un上函数的经典海森堡测不准原理很容易以这种方式从海森堡群的薛定谔表示中推导出来(见[3，第212页])。本文的目的是指出，由Breitenberger[1]提出并在[5]中进一步讨论的圆的不确定性原理;6;7;8;9;10]由U2的欧几里得运动群的主级数表示得到。*通讯作者;e-mail: vepjan@math.ku.dk皇家爱尔兰学院数学学报，104A(2)， 249-252(2004)©皇家爱尔兰学院此内容下载自157.55.39.58星期二，2016年11月15日04:05:14 UTC所有使用受http://about.jstor.org/terms 250皇家爱尔兰学院数学学报2。设G为欧几里得运动群G=j(r,z)=(^or ^|rER,zecj)它的李代数为fl={(j j)| rER,zecj}设H为圆T ={se C´'s' = 1}上的平方可积函数的Hubert空间H = L2(j)，内积(/"，G)= JT f(T) G (T)dt。如在第五章[11]中所述，下面定义了G在^i上的一个幺正表示:7Tfl(r, z)/(?) = ^Re(z^/(^-^)， (j e T)，其中a e c。为了简单起见，我们在下面假设a = l，这对我们的目的来说已经足够了。表示形式n'记为n。我们现在从李代数g的元素生成三个算子。设X,YU r2egbe *-G)，”。- (s i) ^ (o ?)然后exp (ijr) = ^ J), exParl) = ^我)和QxV (tY2) =(^ ^然后我们得到7 t f (s) (X) =我lim - + 0 / - > 0 ?其中/'(a) = jtf{eus)。此运算符的域为{/eL2(T)|?h-»/V0绝对连续与/7 e L2(T)}(3.1)也是rv^^ v hm 1*0, T) f(s) f(s) hm e^s)fis (fis)。, _。7t(rv^^ Fj)/^) = v hm i-o t t-+o t和/VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis…_。……当s=e10时，i /VUM Y2)fis = hm r t^o i /->o t。这两个算子都是在h上定义的。该内容从157.55.39.58下载于2016年11月15日星期二04:05:14 UTC。所有的使用都符合http://about.jstor.org/terms Christensen和Schlichtkrull欧氏运动组251。不确定性原理由于[X,YX] = Y21 [X,Y2]=-Yl，不确定性原理给出1(7*7，)/，/>| = '(n([X, Y2])fJ)'，第一作者要感谢NUI Galway的人们在撰写论文时给予的帮助和友好。[1] E. Breitenberger，角度观测的不确定性测度和不确定性关系，物理基础15(1985)，353-64。[2] J.G. Christensen，李群算子的不确定性原理，出现在傅里叶分析与应用10(2004)，541- 541。[3]王志强，李志强，不确定性原理的数学研究，傅里叶分析与应用3(1997)，207-38。[4]王晓明，王晓明。不确定性关系的研究进展。物理学报，2016(1)，34-41。[5]王志强，周志强，周志强。基于小波变换的周期基函数分析，计算机工程学报(4)，2003,31 - 34。[6]王志强，周志强，周志强，周期不确定性原理与多分辨率分析的最优函数，数学学报，42(1999)，425 - 425。[7]王晓明，王晓明，王晓明，关于圆上函数与实线上函数的不确定原理，傅里叶分析与应用9(2003)，387-409。[8]胡晓明，周晓明，周晓明，周晓明，周晓明，周晓明，周晓明，周晓明。可在线获取:http://www.ma.tum.de/gkaam/personen/rauhut/此内容于2016年11月15日星期二从157.55.39.58下载，04:05:14 UTC所有使用均须遵守http://about.jstor.org/terms 252爱尔兰皇家科学院数学学报[9]M. Rosier和M. Voit，超球膨胀的不确定性原理，数学分析与应用学报209(1997)，624-34。[10]株式会社

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引用次数: 0

THE HOMOLOGY OF HEISENBERG LIE ALGEBRAS OVER FIELDS OF CHARACTERISTIC TWO 特征二域上Heisenberg李代数的同调

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 2003-12-05 DOI: 10.3318/PRIA.2005.105.2.47

Emil Sköldberg

The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory.

利用离散莫尔斯理论计算了海森堡李代数在特征为2的场上的贝蒂数的生成函数。

引用次数: 6

COTYPE AND NONLINEAR ABSOLUTELY SUMMING MAPPINGS 共型和非线性绝对和映射

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 2003-07-23 DOI: 10.3318/PRIA.2005.105.1.75

D. Pellegrino

In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. We give new examples of absolutely summing analytic mappings and polynomial/multilinear versions of linear coincidence theorems.

本文通过Banach空间的定义域和值域的共型，研究了Banach空间上的绝对和映射。我们给出了绝对求和解析映射和线性重合定理的多项式/多线性版本的新例子。

引用次数: 15

ON THE JACOBI EQUATION AND MANIFOLDS WITH MULTIPLE CONJUGATE POINTS 雅可比方程与多共轭点流形

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/PRIA.2013.113.03

J. Burns, Eoghan J. Staunton, D. Wraith

We investigate the phenomenon of multiple conjugate points along a geodesic. In the first instance, we investigate conjugate points in the context of the Jacobi equation, a second order ordinary differential equation, which captures precisely the geometry of conjugate points on surfaces. We then construct geometric examples which exhibit similar properties in higher dimensions.

研究了沿测地线的多个共轭点的现象。在第一个例子中，我们在雅可比方程的背景下研究共轭点，雅可比方程是一个二阶常微分方程，它精确地捕获了曲面上共轭点的几何形状。然后我们构造几何例子，在高维中表现出类似的性质。

引用次数: 1

Geometry of vector arrays and operator ampliations in Hilbert spaces 希尔伯特空间中矢量阵列和算子放大的几何

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/pria.2019.119.09

Klímek

引用次数: 0

A CONTRIBUTION TO THE INVERSE EIGENVALUE PROBLEM FOR NON-NEGATIVE MATRICES 对非负矩阵的特征值反问题的贡献

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/PRIA.2013.113.09

F. Holland

Sometime in the early 1970s Trevor introduced me to the spectral theory of positive linear operators which owes its origins to the celebrated Perron-Frobenius theorem according to which the spectral radius of a non-negative matrix is one of its eigenvalues, and possesses a corresponding eigenvector whose components are nonnegative real numbers. This was a subject dear to his heart, and a recurring theme to which he often returned in later years. In this article we make a tangential contribution to the converse of the PerronFrobenius theorem, the so-called inverse eigenvalue problem for non-negative matrices, namely, under what circumstances are the components of a vector of complex numbers the eigenvalues of such a matrix? To this end, we associate with each vector of unit norm an analytic self map of the unit open disc of the complex plane, which is also a rational function, and develop its power series expansion about the origin. Sufficient conditions are presented that ensure that the resulting coefficients which encode information about the chosen vector are non-negative. Conversely, if these are all non-negative, it turns out that the vector satisfies conditions that are necessary ones for it to solve the inverse problem.

在20世纪70年代早期的某个时候，Trevor向我介绍了正线性算子的谱理论，该理论起源于著名的Perron-Frobenius定理，根据该定理，非负矩阵的谱半径是其特征值之一，并且具有相应的特征向量，其分量是非负实数。这是他非常关心的一个问题，也是他晚年经常反复谈到的一个主题。在这篇文章中，我们对PerronFrobenius定理的逆，即所谓的非负矩阵的逆特征值问题做出了切向贡献，即，在什么情况下复数向量的分量是这样一个矩阵的特征值?为此，我们将复平面上的单位开盘的解析自映射(也是有理函数)与每一个单位范数向量联系起来，并展开其在原点上的幂级数展开式。给出了保证编码所选矢量信息的所得系数不为负的充分条件。相反，如果这些都是非负的，那么这个向量就满足解反问题的必要条件。

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引用次数: 4

A GENERAL WOLFF THEOREM FOR ARBITRARY BANACH SPACES 任意巴拿赫空间的一般wolff定理

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.1353/mpr.2004.0009

P. Mellon

引用次数: 3

Dominating Sets for Analytic and Harmonic Functions and Completeness of Weighted Bergman Spaces 解析函数和调和函数的支配集与加权Bergman空间的完备性

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/PRIA.2002.102.2.175

N. Arcozzi, Anders Björn

A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.

一组E c问是问如果一口holomorphically支配,cE (z) I = SUpzcn如果(z)对所有全纯函数f l Q如下从流浪的结果,这个属性的未定性相当于Aleksandrov紧化点* (Q)从Q x E .此外,它相当于大量其他语句(新旧)的全纯,谐波和拓扑性质,包括一定的加权伯格曼空间p = oo是巴拿赫空间。我们将此推广到R'中的调和函数和cV中的全纯函数。我们也给出了当加权Bergman空间是(拟)-Banach空间时的一些结果，p = oo的情况由上述结果表征。

引用次数: 8

EDITORIAL: AN HOMAGE TO TREVOR WEST 社论:向特雷弗·韦斯特致敬

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/pria.2013.113.06

M. Mathieu

引用次数: 0

SOME VARIANTS OF WEBER'S THEOREM 韦伯定理的一些变体

Mathematical Proceedings of the Royal Irish Academy

Pub Date : 1900-01-01 DOI: 10.3318/PRIA.2004.104.1.67

S. Mécheri

Weber’s theorem says that if A : H 0 H is bounded and linear on a separable Hilbert space, then any operator that is compact, commutes with A and lies in the weak closure of the range of the inner derivation induced by A must also be quasinilpotent. In this note we consider related problems for generalised inner derivations associated with operators A and B on H.

韦伯定理指出，如果A: H 0 H在可分离的希尔伯特空间上是有界线性的，那么任何紧的、与A交换的、位于由A引出的内导数范围的弱闭包内的算子也一定是拟无效的。本文考虑了H上与A和B算子相关的广义内导的相关问题。

引用次数: 4

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