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《数学物理学报(A辑)》2020,(3)
假设R∈M_n(Z)为扩张矩阵和N元数字集D={0,a_1,a_2,…,a_(N-1)}u≡{0,1,…,N-1}u (modN),这里u∈Z~n\{0}.该文主要研究由D和R生成的自仿测度μ_(R,D)的谱性,得到了μ_(R,D)为谱测度的一个充分条件.对于一些特殊情况,得到了μ_(R,D)为谱测度的一个充分必要条件,并给出其谱的具体表达式. 相似文献
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《数学年刊A辑(中文版)》2019,(4)
设μ_(M,D)是由仿射迭代函数系{φ_d(x)=M~(-1)-(x+d)}_(d∈D)唯一确定的自仿测度,它的谱与非谱性质与Hilbert空间L~2(μ_(M,D))中正交指数函数系的有限性和无限性有着直接的关系.本文将利用矩阵的初等变换给出μ_(M,D)正交指数函数系有限性的一个充分条件.由于这个条件只与矩阵M的行列式有关,因此,它在μ_(M,D)的非谱性的判断方面便于直接验证. 相似文献
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设p_1,p_2,p_3∈Z\{0,±1},e_1,e_2,e_3是R~3上标准的单位正交基,由扩张矩阵M=diag[p_1,p_2,p_3]和数字集D={0,e_1,e_2,e_3}确定的自仿测度μM,D是支撑在空间Sierpinski垫T(M,D)上,其对应的Hilbert空间L~2(μM,D)上正交指数系的有限性与无限性问题已经解决.在有限的情形下,空间L~2(μM,D)上正交指数系基数的最佳上界为"4"的猜测还未完全解决.本文构造出了此空间上一列五元素正交指数函数系,说明上述最佳上界为"4"的猜测是错误的. 相似文献
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A.题组新编1 .设集合 M ={3,4,5},N ={6 ,7,8,9,1 0 }.( 1 )映射 f:M→ N ,使对任意的 x∈ M都有 x f( x) xf ( x)是奇数 ,这样的映射f的个数是 ( ) .( A) 2 5 ( B) 50 ( C) 75 ( D) 1 2 5( 2 )若映射 f :M→ N,使对任意的 x∈ M都有 x f( x) xf ( x)是偶数 ,这样的映射f的个数是 ( ) .( A) 0 ( B) 50 ( C) 75 ( D) 1 2 5(吴新华供题 )2 .函数 y =x 5- x的值域是;函数 y =x - 5- x的值域是.(向国华供题 )3.已知圆 C:( x - a) 2 ( y - a) 2 =a2 ,直线 l:3x 4 y 3=0 .( 1 )若圆上有… 相似文献
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本文涉及光滑流形 M 上 C~2 映射的局部极大扩张不变集Λ的拓扑与遍历性质.文中讨论了由 Ruelle-Perron-Frobenius 算子的迭代逼近 H(?)lder 函数的平衡态的问题,证明了Λ成为整个流形 M 的若干等价条件,其中包括Λ有非空内部、Λ或其非游荡集有正的 Lebesgue 测度及Λ上存在关于 Lebesgue 测度绝对连续的不变测度等. 相似文献
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《中学数学》2006,(Z1)
考点1集合的概念与运算1.(湖北,文1)集合P={x x2-16<0},Q={x x=2n,n∈Z},则P∩Q=().(A){-2,2}(B){-2,2,-4,4}(C){-2,0,2}(D){-2,2,0,-4,4}2.(安徽,文1)设全集U={1,2,3,4,5,6,7,8},集合S={1,3,5},T={3,6},则CU(S∪T)等于(A)(B){2,4,7,8}(C){1,3,5,6}(D){2,4,6,8}3.(全国,1)设集合M={x x2-x<0},N={x x<2},则().(A)M∩N=(B)M∩N=M(C)M∪N=M(D)M∪N=R4.(重庆,1)已知集合U={1,2,3,4,5,6,7},A={2,4,5,7},B={3,4,5},则(CUA)∪(CUB)=(A){1,6}(B){4,5}(C){2,3,4,5,7}(D){1,2,3,6,7}5.(辽宁,1)设集合A={1,2},则满足A∪B={1,2,3}… 相似文献
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设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立. 相似文献
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Jian-Lin Li 《Journal of Functional Analysis》2011,260(4):1086-1095
In the present paper we will study the spectral property of a class of self-affine measures under the condition of compatible pair. We first answer a question of Dutkay and Jorgensen concerning the relationship between spectral self-affine measure and compatible pair. We then consider the spectra of Bernoulli convolutions and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman. Finally, we provide a structural property for the integer spectrum of a spectral self-affine measure. 相似文献
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Jian- Lin Li 《Monatshefte für Mathematik》2013,169(3-4):397-407
The self-affine measure $\mu _{M,D}$ relating to an expanding matrix $M\in M_{n}(\mathbb Z )$ and a finite digit set $D\subset \mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $\mu _{M,D}$ in the case when $|\det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(\mu _{M,D})$ , then the self-affine measure $\mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure. 相似文献
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On the spectra of a Cantor measure 总被引:1,自引:0,他引:1
We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis. 相似文献
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D. A. Popov 《Functional Analysis and Its Applications》2003,37(3):215-220
We consider the problem of reconstructing a function on the disk
from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk
under the additional condition that the curves that do not meet
coincide with the corresponding straight lines. 相似文献
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给定两个环R,R’.对于满足一定条件的环R,本文证明了若M:R→R’,M*:R’→R为满射且对A,C∈R和B,D∈R’满足M(AM*(B)C+CM*(B)A)=M(A)BM(C)+M(C)BM(A),M*(BM(A)D+DM(A)B)=M*(B)AM*(D)+M*(D)AM*(B)则M和M*是可加的;若R和R’分别包含单位I和I’,M(I),M*(I’)可逆,则存在环同构N使得M(A)=N(A)M(I),M*(B)=N-1(BM(I)).特别地,若R=R’为标准算子代数或Hilbert空间套代数,则M和M*可加且存在有界可逆的线性或共轭线性算子S和T使得M(A)=SAT,M*(B)=TBS或M(A)=TA*S,M*(B)=(SBT)*对任意的A,B∈R成立. 相似文献
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Archiv der Mathematik - This work investigates the spectrality of a self-affine measure $$\mu _{M,D}$$ and the related digit set D in the case when $$|\mathrm{det}(M)|=p^{\alpha }$$ is a prime... 相似文献
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《复变函数与椭圆型方程》2012,57(2):95-110
Let $ k \in {\shadN} $ , $ w(x) = (1+x^2)^{1/2} $ , $ V^{\prime} _k = w^{k+1} {\cal D}^{\prime} _{L^1} = \{{ \,f \in {\cal S}^{\prime}{:}\; w^{-k-1}f \in {\cal D}^{\prime} _{L^1}}\} $ . For $ f \in V^{\prime} _k $ , let $ C_{\eta ,k\,}f = C_0(\xi \,f) + z^k C_0(\eta \,f/t^k)$ where $ \xi \in {\cal D} $ , $ 0 \leq \xi (x) \leq 1 $ $ \xi (x) = 1 $ in a neighborhood of the origin, $ \eta = 1 - \xi $ , and $ C_0g(z) = \langle g, \fraca {1}{(2i \pi (\cdot - z))} \rangle $ for $ g \in V^{\,\prime} _0 $ , z = x + iy , y p 0 . Using a decomposition of C 0 in terms of Poisson operators, we prove that $ C_{\eta ,k,y} {:}\; f \,\mapsto\, C_{\eta ,k\,}f(\cdot + iy) $ , y p 0 , is a continuous mapping from $ V^{\,\prime} _k $ into $ w^{k+2} {\cal D}_{L^1}$ , where $ {\cal D}_{L^1} = \{ \varphi \in C^\infty {:}\; D^\alpha \varphi \in L^1\ \forall \alpha \in {\shadN} \} $ . Also, it is shown that for $ f \in V^{\,\prime} _k $ , $ C_{\eta ,k\,}f $ admits the following boundary values in the topology of $ V^{\,\prime} _{k+1} : C^+_{\eta ,k\,}f = \lim _{y \to 0+} C_{\eta ,k\,}f(\cdot + iy) = (1/2) (\,f + i S_{\eta ,k\,}f\,); C^-_{\eta ,k\,}f = \lim _{y \to 0-} C_{\eta ,k\,} f(\cdot + iy)= (1/2) (-f + i S_{\eta ,k\,}f ) $ , where $ S_{\eta ,k} $ is the Hilbert transform of index k introduced in a previous article by the first named author. Additional results are established for distributions in subspaces $ G^{\,\prime} _{\eta ,k} = \{ \,f \in V^{\,\prime} _k {:}S_{\eta ,k\,}f \in V^{\,\prime} _k \} $ , $ k \in {\shadN} $ . Algebraic properties are given too, for products of operators C + , C m , S , for suitable indices and topologies. 相似文献