Multifractal structure and product of matrices

There is a well established multifractal theory for self-similar measures generated by non-overtapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict our attentionto the important classes of self-similar measures that have matrix representations. The dimension spectra andthe L-spectra are analyzed through the product of matrices. There are abnormal behaviors on the multifrac-tal structure and they will be discussed in detail.     

ENDPOINT ESTIMATE FOR MAXIMAL COMMUTATORS WITH NON-DOUBLING MEASUES

The authors establish the weak type endpoint estimate for the maximal commutators generated by Calderon-Zygmund singular integrals and Orlicz type functions with non-doubling measures.     

CARLESON MEASURES FOR BESOV-SOBOLEV SPACES WITH APPLICATIONS IN THE UNIT BALL OF C^n

This paper is devoted to give the connections between Carleson measures for Besov-Sobolev spaces Bσp (B) and p-Carleson measure in the unit ball of Cn. As appli-cations, we characterize the Riemann-Sti...     

Generalized area operators on L^p（R^n）

We introduce the generalized area operators by using nonnegative measures defined on upper half-spaces R＋^n＋1. The characterization of the boundedness and compactness of the generalized area operator from LP（]Rn） to Lq（IRn） is investigated in terms of s-Carleson measures with 1 〈 p, q 〈 ＋∞. In the case of p = q = 1, the weak type estimate is also obtained.     

Measures of Asymmetry Dual to Mean Minkowski Measures of Asymmetry for Convex Bo dies

We introduce a family of measures (functions) of asymmetry for convex bodies and discuss their properties. It turns out that this family of measures shares many nice properties with the mean Minkowski measures. As the mean Minkowski measures describe the symmetry of lower dimensional sections of a convex body, these new measures describe the symmetry of lower dimensional orthogonal projections.     

支于自仿射Tile上的细分分布

In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.     

GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS Ⅱ

We present a definition of general Sobolev spaces with respect to arbitrary measures ,W^k,p(Ω,μ) for 1≤p≤∞,In[RARP] we proved that these spaces are complete under very light conditions.Now we prove that if we consider certain general types of measures,then Cc^∞(R) is dense in these spaces,As an application to Sobolev orthogonal polynomials,we study the boundedness of the multiplication poerator,THis gives an estimation of the zeroes of Sobolev orthogonal polynomials.     

THE SET OF THE INVARIANT MEASURES OF BOUNDED SPIN-FLIP PROCESSES WITH POTENTIAL

In this paper, some properties of the invariant measures of bounded spin-flip processes are discussed. It is proved that all invariant measures of a bounded spinflip process with potential are reversible measures if its speed functions have locally finite range interaction. Therefore, all invariant measures of the process are the Gibbs states of the potential. It is also proved that for each given potential, there exists a spin-flip process suth that all of its invariant measures are the Gibbs states of the potential.     

NORMALLY DISTRIBUTED PROBABILITY MEASURE ON THE METRIC SPACE OF NORMS

In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of ...     

Lacunary Measures and Self-similar Probability Measures in Function Spaces

The paper deals with multifractal quantities for some types of Radon measures,especiallyself-similar probability measures,and their relations to Besov spaces.     

Surface Measures Generated by Differentiable Measures

We study surface measures on level sets of functions on general probability spaces with measures differentiable along vector fields and suggest a new simple construction. Our construction applies also to level sets of mappings with values in finite-dimensional spaces. The standard surface measures arising for Gaussian measures in the Malliavin calculus can be obtained in this way. A positive answer is given to a question raised by M. Röckner concerning continuity of surface measures with respect to a parameter.     

Hessian Measures of Convex Functions and Applications to Area Measures

The Hessian measures of a (semi-)convex function can be introducedas coefficients of a local Steiner formula. The investigationof Hessian measures is continued by the provision of a geometriccharacterization of the support of these measures. Then theRadon–Nikodym derivative and the absolute continuity ofHessian measures with respect to Lebesgue measure are explored.As special cases of the results, known results for surface areameasures of convex bodies are recovered.     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.     

Vague Convergence of Semimartingale Random Measures

Abstract

In this paper, we introduce and research the vague convergence of semimartingale random measures in distribution. The conditions are provided for the vague convergence of semimartingale random measures and the convergence of stochastic integrals with respect to semimartingale random measures in distribution.     

Measures on finite concrete logics

We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets.

     

Generalized Random Spectral Measures

In an attempt to examine the random version of the spectral theorem, the notion of random spectral measures and generalized random spectral measures are introduced and investigated. It is shown that each generalized random spectral measure on $(\mathbb C ,\mathcal{B}(\mathbb C ))$ admits a modification which is a random spectral measure.     

Boundary Measures for Geometric Inference

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.     

Systematic Measures of Biological Networks I: Invariant Measures and Entropy

This paper is part I of a two‐part series devoted to the study of systematic measures in a complex biological network modeled by a system of ordinary differential equations. As the mathematical complement to our previous work with collaborators, the series aims at establishing a mathematical foundation for characterizing three important systematic measures: degeneracy, complexity, and robustness, in such a biological network and studying connections among them. To do so, we consider in part I stationary measures of a Fokker‐Planck equation generated from small white noise perturbations of a dissipative system of ordinary differential equations. Some estimations of concentration of stationary measures of the Fokker‐Planck equation in the vicinity of the global attractor are presented. The relationship between the differential entropy of stationary measures and the dimension of the global attractor is also given.© 2016 Wiley Periodicals, Inc.     

Curvature Measures and Random Sets,I

The paper deals with signed curvature measures as introduced by Federer for sets with positive reach. An integral representation and a local Steiner formula for these measures are given. The main result is the additive extension of the curvature measures to locally finite unions of compatible sets with positive reach. Within this comprehensive class of subsets of R^d a generalized Steiner polynomial (local version) and section theorems (principal kinematic formula, Crofton formula) for the curvature measures are derived.