共查询到20条相似文献,搜索用时 15 毫秒
1.
Let C be the classical Cantor triadic set. For a,b ? [-1,1]{alpha,betain [-1,1]} , a sufficient and necessary condition for (C×C)?(C×C+(a,b)){(Ctimes C)cap (Ctimes C+(alpha,beta))} to be self-similar is obtained. 相似文献
2.
Let C be the classical Cantor triadic set. For ${\alpha,\beta\in [-1,1]}$ , a sufficient and necessary condition for ${(C\times C)\cap (C\times C+(\alpha,\beta))}$ to be self-similar is obtained. 相似文献
3.
Guo-Tai Deng Xing-Gang He Zhi-Xiong Wen 《Journal of Mathematical Analysis and Applications》2008,337(1):617-631
Let C be the triadic Cantor set. We characterize the all real number α such that the intersection C∩(C+α) is a self-similar set, and investigate the form and structure of the all iterated function systems which generate the self-similar set. 相似文献
4.
Yuru ZouWenxia Li Caiguang Yan 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(14):4660-4670
A scheme is given to compute the Hausdorff dimensions for the intersection of a class of nonhomogeneous Cantor sets with their translations. 相似文献
5.
Mei Feng Dai 《数学学报(英文版)》2008,24(8):1313-1318
We pursue the study on homogeneous Cantor sets with their translations. We get the fractal structure of intersection I(t), and find that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with the coding of t. Concretely, a very brief calculation formula of the measure with the coding of t is given. 相似文献
6.
Science China Mathematics - In this paper, we study three types of Cantor sets. For any integer m ? 4, we show that every real number in [0, k] is the sum of at most k m-th powers of... 相似文献
7.
H.J Ryser 《Journal of Combinatorial Theory, Series A》1973,14(1):79-92
Let X1, X2, …, Xm be finite sets. The present paper is concerned with the m2 ? m intersection numbers |Xi ∩ Xj| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T1, T2, …, Tm be finite sets and let m ? 3. We assume that each of the elements in the set union T1 ∪ T2 ∪ … ∪ Tm occurs in at least two of the subsets T1, T2, …, Tm. We further assume that every pair of sets Ti and Tj (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set Tk (k ≠ i, k ≠ j) such that Tk intersects both Ti and Tj. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}. 相似文献
8.
Γ是齐次对称康托集,对n个实数t_1,…,t_n讨论了交集Γ∩(Γ+t_1)∩…∩(Γ+t_n)≠(?)的条件,以及计算出Γ∩(Γ+t_1)∩…∩(Γ+t_n)的Hausdorff维数的精确表达式. 相似文献
9.
Projections of random Cantor sets 总被引:1,自引:0,他引:1
K. J. Falconer 《Journal of Theoretical Probability》1989,2(1):65-70
Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases. 相似文献
10.
Ignacio Garcia Ursula Molter Roberto Scotto 《Proceedings of the American Mathematical Society》2007,135(10):3151-3161
We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.
11.
12.
S. Astels 《Transactions of the American Mathematical Society》2000,352(1):133-170
For let be a Cantor set constructed from the interval , and let . We derive conditions under which
When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets , where is a set of positive integers and is the set of real numbers such that all partial quotients of , except possibly the first, are members of .
13.
In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate. 相似文献
14.
We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R[Hdim(C)]+1, where [Hdim(C)] denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural sequence of refining coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of Rd is bi-Lipschitz embeddable in Rd+1.We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry. 相似文献
15.
Roger L. Kraft 《Transactions of the American Mathematical Society》2000,352(3):1315-1328
Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .
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17.
Dhurjati Prasad Datta Santanu Raut Anuja Raychoudhuri 《P-Adic Numbers, Ultrametric Analysis, and Applications》2011,3(1):7-22
A class of ultrametric Cantor sets (C, d
u
) introduced recently (S. Raut and D. P. Datta, Fractals 17, 45–52 (2009)) is shown to enjoy some novel properties. The ultrametric d
u
is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrization invariant, is identified
with the Cantor function associated with a Cantor set $
\tilde C
$
\tilde C
, where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically
inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps of $
\tilde C
$
\tilde C
. The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions
are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrization invariance of the valuation it is shown how the scale
factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the
valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of
Cantor sets with Hausdorff dimension log3 2 and thickness 1 are constructed explicitly. 相似文献
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19.
Jürgen Eckhoff 《Israel Journal of Mathematics》1988,62(3):283-301
LetP be a family ofn boxes inR
d
(with edges parallel to the coordinate axes). Fork=0, 1, 2, …, denote byf
k
(P) the number of subfamilies ofP of sizek+1 with non-empty intersection. We show that iff
r
(P)=0 for somer≦n, then
where thef
k
(n, d, r) are ceg196rtain definite numbers defined by (3.4) below. The result is best possible for eachk. Fork=1 it was conjectured by G. Kalai (Israel J. Math.48 (1984), 161–174). As an application, we prove a ‘fractional’ Helly theorem for families of boxes inR
d
. 相似文献
20.