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We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.
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On the geometry of random Cantor sets and fractal percolation 总被引:1,自引:0,他引:1
Random Cantor sets are constructions which generalize the classical Cantor set, middle third deletion being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1]
e
. This answers a question of Dekking and Meester,(9) (b) The special case of fractal percolation arises when the substitution is as follows: The cube [0,1]
d
is divided intoM
d subcubes of side-lengthM
–, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep
c(M, d) ofp, marking the existence of crossings of [0,1]
d
contained in the limit set, satisfiesp
c(M, d)p
c(d) asM, wherep
c(d) is the critical probability of site percolation on a latticeL
d
obtained by adding certain edges to the hypercubic lattice
d
. This result generalizes in an unexpected way a finding of Chayes and Chayes,(4) who studied the special case whend=2. 相似文献
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本文考虑一类广义Cantor集Γ_(β,の)={∞∑n=1dnβn:dn∈Dn,n≥1}的自相似性,其中0β1且对任意的n≥1,D_n为整数集Z的非空有限子集;并且给出Γ_(β,の)为齐次生成自相似集的充分必要条件.作为应用,本文考虑一类广义Cantor集交的自相似性,部分推广了Li,Yao和Zhang(2011)关于自相似性的结果. 相似文献
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In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 相似文献
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Robert Samuel Simon 《International Journal of Game Theory》1999,28(3):367-384
Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically,
as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many
agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held
in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of
formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition
where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition
is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and
combinatorial structure. 相似文献
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Aldo Portela 《Bulletin of the Brazilian Mathematical Society》2009,40(1):53-75
It is known that not every Cantor set of S
1 is C
1-minimal. In this work we prove that every member of a subfamily of what we here call regular interval Cantor set is not C
1-minimal. We also prove that no member of a class of Cantor sets that includes this subfamily is C
1+∈-minimal, for any ∈ > 0.
Partially supported by CNPq-Brasil and PEDECIBA-Uruguay. 相似文献
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S. V. Kolesnikov 《Mathematical Notes》1998,63(1):50-54
This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: |z|<1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE
1,E
2, andE
3 of the unit circle Γ: |z|=1,
= Γ, are the setsI(ƒ) of all Plessner points,F(ƒ) of all Fatou points, andE(ƒ) of all exceptional boundary points, respectively, for a function ƒ holomorphic inD if and only ifE
1 is aG
δ-set andE
3 is a
-set of linear measure zero. In the second part of the paper it is shown that for any
-subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ.
Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 56–61, January, 1998. 相似文献
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给定Rd 中的Moran集类 ,本文证明了对介于该集类中元素的上盒维数的最大值和最小值之间的任何一个数值s,总存在该集类中的一个元素 ,其上盒维数等于s,对下盒维数、修正的下盒维数也有类似的性质成立 ,从而给文 [1 ]中的猜想 1一个肯定的回答 .此外 ,还讨论了齐次Cantor集和偏次Cantor集盒维数存在性之间的关系 . 相似文献
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Alexander Blokh A. M. Bruckner P. D. Humke J. Smí tal 《Transactions of the American Mathematical Society》1996,348(4):1357-1372
We first give a geometric characterization of -limit sets. We then use this characterization to prove that the family of -limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.
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In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value. 相似文献
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A. I. Pavlov 《Mathematical Notes》1999,66(4):442-450
The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function
G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R0, the following inequality holds:
Further, suppose that for some fixed α>0 the deviation DN of the sequence xn={αn}, n=1, 2, ..., as N→∞ has the estimate DN=0(lnB N/N). Then if the function G(z) is not an identical constant and the inequality B+1<A holds, then the power series
converging in the disk |z|<1 cannot be analytically continued to the region |z|>1 across any arc of the circle |z|=1.
Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 540–550, October, 1999. 相似文献
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Let A denote the class of functions f(z) with
f(0)=f′(0)−1=0, 相似文献
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Analytic queueing network models often assume infinite capacity queues due to the difficulty of grasping the between-queue correlation. This correlation can help to explain the propagation of congestion. We present an analytic queueing network model which preserves the finite capacity of the queues and uses structural parameters to grasp the between-queue correlation. Unlike pre-existing models it maintains the network topology and the queue capacities exogenous. Additionally, congestion is directly modeled via a novel formulation of the state space of the queues which explicitly captures the blocking phase. The model can therefore describe the sources and effects of congestion. 相似文献
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陈东立 《纯粹数学与应用数学》2001,17(2):133-137
讨论了乘积一致结构的单子的刻画 .利用一致结构的单子给出了 Cauchy网及含小集集族的非标准特征 .作为应用给出一致空间几个重要定理的离散化证明 . 相似文献
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This paper concerns the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heat-conduction and vacuum as far field density. In particular, the initial density can have compact support. We prove that the Cauchy problem admits a local strong solution provided both the initial density and the initial magnetic field decay not too slow at infinity. 相似文献
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Making use of the familiar convolution structure of analytic functions, in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here. 相似文献