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1.
For 1/4 < a <√2/4, let S1(x) = ax, S2(x) = 1 - a ax, x ∈ [0,1]. Ca is the attractor of the iterated function system {S1, S2}, then the packing measure of Ca × Ca is Ps(a)(Ca × Ca) = 4.2s(a)(1 - a)s(a),where s(a) = -loga4. 相似文献
2.
Let $D\subset {\mathbb{R}}^3$ be the set of double points of a three-dimensional Brownian motion. We show that, if ξ = ξ3(2,2) is the intersection exponent of two packets of two independent Brownian motions, then almost surely, the ?-packing measure of D is zero if $$ \int_{0^+} r^{-1-\xi} \phi(r)^{\xi} \, dr < \infty,$$ and infinity otherwise. As an important step in the proof we show up-to-constants estimates for the tail at zero of Brownian intersection local times in dimensions two and three. 相似文献
3.
The hausdorff dimension of the sample path of a subordinator 总被引:1,自引:0,他引:1
Joseph Horowitz 《Israel Journal of Mathematics》1968,6(2):176-182
The Hausdorff dimension of the range of an arbitrary subordinator is exactly determined in terms of the rate of linear drift
and the Levy measure of the subordinator. This generalizes the result of Blumenthal and Getoor: that for a stable subordinator
of indexσ, the dimension of the range isσ. 相似文献
4.
Thomas Duquesne 《Stochastic Processes and their Applications》2012,122(3):968-1002
We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant. 相似文献
5.
We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension;
for instance, for almost everyu ∈ [3, 6], the set of all sums ∑
0
8
a
n
4−n
a
n
4−n
with digits witha
n
∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections
of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar
sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates
the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections.
Research of Y. Peres was partially supported by NSF grant #DMS-9803597.
Research of K. Simon was supported in part by the OTKA foundation grant F019099.
Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics
at The Hebrew University of Jerusalem. 相似文献
6.
Following the work of Anily et?al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously. 相似文献
7.
8.
Ignacio Garcia Leandro Zuberman 《Journal of Mathematical Analysis and Applications》2012,386(2):801-812
In this paper we consider a class of symmetric Cantor sets in . Under certain separation condition we determine the exact packing measure of such a Cantor set through the computation of the lower density of the uniform probability measure supported on the set. With an additional restriction on the dimension we give also the exact centered Hausdorff measure by computing the upper density. 相似文献
9.
For the packing measure of the Cartesian product of the middle third Cantor set with itself, the exact value
10.
A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3. 相似文献
11.
Santiago Boza 《Indagationes Mathematicae》2008,19(1):33-51
For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees. 相似文献
12.
Michel L. Lapidus 《Integral Equations and Operator Theory》1989,12(2):163-210
Let u(t) be the operator associated by path integration with the Feynman-Kac functional in which the time integration is performed with respect to an arbitrary Borel measure instead of ordinary Lebesgue measurel. We show that u(t), considered as a function of time t, satisfies a Volterra-Stieltjes integral equation, denoted by (*). We refer to this result as the Feynman-Kac formula with a Lebesgue-Stieltjes measure. Indeed, when n=l, we recover the classical Feynman-Kac formula since (*) then yields the heat (resp., Schrödinger) equation in the diffusion (resp., quantum mechanical) case. We stress that the measure is in general the sum of an absolutely continuous, a singular continuous and a (countably supported) discrete part. We also study various properties of (*) and of its solution. These results extend and use previous work of the author dealing with measures having finitely supported discrete part (Stud. Appl. Math.76 (1987), 93–132); they seem to be new in the diffusion (or imaginary time) as well as in the quantum mechanical (or real time) case.Research partially supported by the National Science Foundation under Grant DMS 8703138. This work was also supported in part by NSF Grant 8120790 at the Mathematical Sciences Research Institute in Berkeley, U.S.A., the CNPq and the Organization of Latin American States at theInstituto de Matemática Pura E Aplicada (IMPA) in Rio de Janeiro, Brazil, as well as theUniversité Pierre et Marie Curie (Paris VI) and the Université Paris Dauphine in Paris, France. 相似文献
13.
H. Joyce 《Proceedings of the American Mathematical Society》1999,127(4):985-991
We construct a separable metric space on which 1-dimensional diameter-type packing measure is not Borel regular.
14.
J. Mijnheer 《Journal of Mathematical Sciences》1995,76(2):2283-2287
Let {X(t): 0≤t<∞) be a stable subordinator with α∈(0,1). For increasing sequences tk we give normalizing constants ak such thatliminf
k→∞ a
k
−1
X(tk) is a.s constant. We also derive a.s. upper bounds.
Proceedings of the XVI Seminar on Stability Problems for Stochastic Models Part I, Eger, Hungary, 1994. 相似文献
15.
16.
Aleksandras Baltrūnas Jonas Šiaulys 《Journal of Mathematical Analysis and Applications》2007,332(1):22-31
Suppose that {a(n)} is a discrete probability distribution on the set N0={0,1,2,…} and {p(n)} is some non-negative sequence defined on the same set. The equation defines a new sequence {b(n)}. Here {a*k(n)} denotes the k-fold convolution of the distribution {a(n)}. In the paper the asymptotic behaviour of the sequence {b(n)} is investigated. It is known that for the large classes of the sequences {a(n)} and {p(n)}, b(n)∼σp([σn]), where 1/σ is the mean of the distribution {a(n)}. The main object of the present work is to estimate the difference b(n)−σp([σn]) for some classes of the sequences {a(n)} and {p(n)}. 相似文献
17.
A clutter (V, E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum
number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. An m×n 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed
as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves
several properties of minimally nonpacking clutters and matrices.
Received: December 1, 1997 / Accepted: April 6, 1999?Published online October 18, 2000 相似文献
18.
Yong Xin Gui 《数学学报(英文版)》2010,26(7):1369-1382
In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite. 相似文献
19.
The exit measure of a supermartingale 总被引:2,自引:0,他引:2
Hans Föllmer 《Probability Theory and Related Fields》1972,21(2):154-166
20.
Let e be a homogeneous subset of ℝ in the sense of Carleson. Let μ be a finite positive measure on ℝ and H
μ(x) its Hilbert transform. We prove that if lim
t→∞
t|e∩{x ‖H
μ(x)| > t}| = 0, then μ
s
(e) = 0, where μs is the singular part of μ. 相似文献