Local Geometry of Fractals Given by Tangent Measure Distributions

 Let μ be a self-similar-measure and ν an ergodic shift-invariant measure on a self-similar set A. We show that under weak conditions ν-almost all points x in A show the same local structure, that is, the same tangent measure distribution of μ. (Received 10 October 2000, in revised form 8 March 2001)     

The Tangent Measure Distribution of Self-Conformal Fractals

We show that the tangent measure distribution of a self-conformal measure exists at almost all points of the support of the measure. Moreover, we prove, that it is the same for almost all points.     

The Tangent Measure Distribution of Self-Conformal Fractals

We show that the tangent measure distribution of a self-conformal measure exists at almost all points of the support of the measure. Moreover, we prove, that it is the same for almost all points.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

一类带有给定切线多边形的样条曲线

利用带有形状参数的基函数,构造与给定切线多边形相切的样条曲线,所构造的曲线是C2和C3连续的,且对切线多边形是保形的.曲线上的所有控制点可由多边形顶点直接计算产生,曲线具有局部修改性.最后,以实例说明算法是有效的.     

与给定多边形相切的可调广义Ball闭曲线

讨论了与给定控制多边形相切的分段三次、五次和六次可调广义Ball曲线的构造方法,所构造的曲线分别是C1,C2和C3连续的,而且对切线多边形是保形的.曲线上的所有广义Ball曲线段的控制点由切线多边形的顶点直接计算产生.给出了在保持公共连接点处相应连续的情况下,内控制点的活动范围.曲线可以在一定范围内做局部修改.计算实例...     

Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M,g) be an n-dimensional Riemannian manifold and T2M be its secondorder tangent bundle equipped with a lift metric (g).In this paper,first,the authors construct some Riemannian almost product structures on (T2M,(g)) and present some results concerning these structures.Then,they investigate the curvature properties of (T2M,(g)).Finally,they study the properties of two metric connections with nonvanishing torsion on (T2 M,(g)):The H-lift of the Levi-Civita connection of g to T2 M,and the product conjugate connection defined by the Levi-Civita connection of (g) and an almost product structure.     

带有给定凸切线多边形的保形五次样条逼近

本讨论带有给定切线多边形的保形逼近问题．给出了一条与给定切线多边形相切的保形五次参数祥条曲线。     

Parametric Families of Multivariate Distributions with Given Margins

Parametric families of continuous bivariate distributions with given margins that include independence and perfect positive dependence are compared on the basis on some important properties. Since many such families exist, the comparisons are helpful for deciding on suitable models for multivariate data. The study of the properties has motivation from applications in extreme value inference. One property considered for bivariate families is whether they extend to multivariate families, and extensions are given when possible. Several new bivariate and multivariate families are included and some open research problems in the area of multivariate families are mentioned.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

Eigenvalue Distributions and Geometry of Banach Spaces

We consider the sums Z⁽ⁿ⁾ of i.i.d. random vectors taking values in a d-dimensional Euclidean space. It is assumed that the so-called CRAMER condition holds in a neighbourhood of the origin. We establish lower bounds for the large deviation probabilities P(Z⁽ⁿ⁾ ? A) with A belonging to a large class of sets.     

与给定多边形相切的一类含参数三角样条曲线

基于一类与给定多边形相切的三角样条曲线,通过在基函数中引入形状参数λ,在保持原曲线的光滑性及其他基本性质不变的条件下,构造出一类能自由调控曲线形态的含参数三角样条曲线,并结合图例讨论了其相关性质.     

Optimal Joint Distributions of Several Random Variables with Given Marginals

Solutions of a singularly perturbed vector boundary-value problem are studied under the principle assumption that the trivial solution of the unperturbed equation is stable in certain senses. This is accomplished by constructing special invariant regions in which solutions display the kind of nonuniformity known as boundary-layer behavior.     

Harmonic Calculus on Fractals - A Measure Geometric Approach I

Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linear Cantor-type measures with arbitrary weights.     

具有给定西洛子群正规化子性质的局部群系

本文研究了满足条件N^F?F的局部群系集合的代数性质,同时对于这类群类F,给出了极小非F-群的结构.     

多个离散分布的信息差度量的渐近分布

本文中对于多信息源的信息差异性度量B,我们研究了B(u1,u2,…,us)的渐近分布,其中u1,u2,…us是某个概率分布的样本估计．在某些较弱的条件下,我们证明了nB(u1,u2,…,us)的极限分布是自由度为1的X2分布的加权和,其中n是样本容量,u1,u2,…,us不必相互独立．     

On the Lagrangian Geometry of the Tangent Bundle of a Toric Variety

Mathematical Notes -     

Tangent Fields and the Local Structure of Random Fields

A tangent field of a random field X on ^N at a point z is defined to be the limit of a sequence of scaled enlargements of X about z. This paper develops general properties of tangent fields, emphasising their rich structure and strong invariance properties which place considerable constraints on their form. The theory is illustrated by a variety of examples, both of a smooth and fractal nature.     

Geometry of Local Adaptive Galerkin Bases

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as ɛ ² have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ ⁴ give the different curvatures of the manifold, the ones that scale as ɛ ⁶ give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E _m , where E _m is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ ^m . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold. Accepted 14 October 1998