Estimates for the Norm of Green's Matrix Based on the Integral Performance Criterion for Dichotomy and Hausdorff Set Bounds

For a system of ordinary differential equations of the form , , we obtain new upper bounds for the second norm of Green's matrix using the integral performance criterion for dichotomy and bounds of the Hausdorff set of the matrix A. These estimates are considerably better for many applications than some well-known ones.     

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to metrize the space of rooted compact real trees. Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields. SNE supported in part by NSF grants DMS-0071468 and DMS-0405778, and a Miller Institute for Basic Research in Science research professorship JP supported in part by NSF grants DMS-0071448 and DMS-0405779 AW supported by a DFG Forchungsstipendium     

N指标d维广义Wiener过程像集的一致维数

研究了N指标d维广义Wiener过程像集的一致维数和测度，得到了其像集的致Hausdorff维数和一致Packing维数。     

Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

Removable singularities of weak solutions to linear partial differential equations

Suppose that P(x, D) is a linear differential operator of order m > 0 with smooth coefficients whose derivatives up to order m are continuous functions in the domain G ⁿ (n 1), 1 < p > , s > 0, and q=p/(p – 1). In this paper, we show that if n, m, p, and s satisfy the two-sided bound 0 n – q(m – s)< n, then for a weak solution of the equation P(x, D)u=0 from the Sharpley-DeVore class C _p ^s (G)_loc, any closed set in G is removable if its Hausdorff measure of order n – q(m – s) is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation P(x, D)_u=0 from the Sobolev classes and extends it to the case of noninteger orders of smoothness.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 584–591.Original Russian Text Copyright © 2005 by A. V. Pokrovskii.This revised version was published online in April 2005 with a corrected issue number.     

An approximation method to estimate the Hausdorff measure of the Sierpinski gasket

In this paper, we firstly define a decreasing sequence {P^n(S)} by the generation of the Sierpinski gasket where each P^n(S) can be obtained in finite steps. Then we prove that the Hausdorff measure H^8(S) of the Sierpinski gasket S can be approximated by {P^n(S)} with P^n(S)/(1 1/2^n-3)s ≤ H^8(S)≤ Pb(S).An algorithm is presented to get P^n(S) for n≤ 5. As an application, we obtain the best lower bound of H^8(S) till now: H^8(S) ≥ 0.5631.     

Beyond Besov Spaces,Part 2: Oscillation Spaces

We study several extensions of Besov spaces; these extensions include the oscillation spaces O_p^s,swhich take into account correlations between the positions of large wavelet coefficients through the scales and, more generally, spaces defined through the distributions of suprema of wavelet coefficients on wavelet subtrees. These spaces are independent of the particular wavelet basis chosen. Examples of applications will be taken from image processing and multifractal analysis.     

The multifractal spectrum of some moran measures

The multifractal formalism is shown to hold for a class of Moran measures supported on the Moran fractals associated with the sequences of which the frequency of the letter exists.     

Hausdorff Dimensions of Self-Similar and Self-Affine Fractals in the Heisenberg Group

We study the Hausdorff dimensions of invariant sets for self-similarand self-affine iterated function systems in the Heisenberggroup. In our principal result we obtain almost sure formulaefor the dimensions of self-affine invariant sets, extendingto the Heisenberg setting some results of Falconer and Solomyakin Euclidean space. As an application, we complete the proofof the comparison theorem for Euclidean and Heisenberg Hausdorffdimension initiated by Balogh, Rickly and Serra-Cassano. 2000Mathematics Subject Classification 22E30, 28A78 (primary), 26A18,28A78 (secondary).