Lorenz Curve for Truncated and Censored Data

In this paper, we consider a nonparametric estimator of the Lorenz curve and Gini index when the data are subjected to random left truncation and right censorship. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. A law of the iterated logarithm for the Lorenz curve is also derived. Lastly, we obtain a central limit theorem for the corresponding Gini index.     

The Heighway Dragon Revisited

We prove that the Heighway dragon is a countable union of closed geometrically similar disk-like planar sets which intersect each other in a linear order: any two of them intersect at no more than one cut point and for any three disks there exist at least two with an empty intersection. Consequently, the interior of the Heighway dragon is a countable union of disjoint open disk-like planar sets. We determine all the cut points of the dragon and show that each disk-like subset between two cut points is a graph self-similar set defined by a graph-directed iterated function system consisting of four seed sets. Our results describe a fairly complete picture of the topological and geometric structure of the Heighway dragon.     

Extremely narrow-dispersed high molecular weight polyethylene from living polymerization at elevated temperatures with o-F substituted Ti enolatoimines

With a new titanium imineenolato complex, ethylene is polymerized in an unprecedented living fashion affording a polymer with a high molecular weight (Mn > 10(5) g mol-1) and an extremely narrow distribution (Mw/Mn 1.01) at the same time; the living character is also retained even at an elevated temperature of 75 degrees C, and blockcopolymers are accessible.     

Topology of connected self-similar tiles in the plane with disconnected interiors

We study the topological structure of connected self-similar tiles in defined by injective contractions satisfying the open set condition. We emphasize on tiles each of whose interior consists of either finitely or infinitely many components. In the former case, we show in particular that the closure of some component is a topological disk. In the latter case we show that the closure of each component is a locally connected continuum. We introduce the finite tail and infinite replication properties and show that under these assumptions the closure of each component is a disk. As an application we prove that the closure of each component of the interior of the Lévy dragon is a disk.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of \mathbbR^d{{\mathbb{R}}^d} . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.     

Synthesis, polymerisation and degradation of poly(lactide-co-propylene glycol) dimethacrylate adhesives

Fluid ABA triblock poly(lactide-co-propylene glycol-co-lactide)s with 7, 17 or 34 propylene glycol and 2, 4, or 8 lactic acid units in each B and A block, respectively, end capped with methacrylate groups were prepared and polymerisation rates upon blue light exposure for 60, 120 or 240 s using 0.5, 1 or 2 wt.% initiators determined. The shortest monomer formulation, with 0.5 wt.% initiators, was most promising as an injectable degradable adhesive for biomedical applications, achieving 96% polymerisation by 120 s after the start of 60 s illumination, forming then a semi-rigid polymer that in water released linearly with time poly(methacrylic-co-lactic) acid degradation products. Over 14 weeks, 2 mm thick specimens largely maintained their dimensions as water sorption balanced the 19 wt.% material loss. Raising initiator concentration or monomer length reduced polymerisation rate on the lower surface of samples. Increasing the number of lactic acid units in each A block from 2 to 8 also enhanced photocrosslinked polymer water sorption and increased average total mass loss to 60 wt.% over 14 weeks but acid product release rate decreased with time. Monomers produced with longer polypropylene glycol B blocks required longer periods of light exposure for full cure and the final polymers exhibited slower non-linear hydrolytic degradation.     

A dimension result arising from the -spectrum of a measure

We give a rigorous proof of the following heuristic result: Let be a Borel probability measure and let be the -spectrum of . If is differentiable at , then the Hausdorff dimension and the entropy dimension of equal . Our result improves significantly some recent results of a similar nature; it is also of particular interest for computing the Hausdorff and entropy dimensions of the class of self-similar measures defined by maps which do not satisfy the open set condition.

     

The Heighway Dragon Revisited

We prove that the Heighway dragon is a countable union of closed geometrically similar disk-like planar sets which intersect each other in a linear order: any two of them intersect at no more than one cut point and for any three disks there exist at least two with an empty intersection. Consequently, the interior of the Heighway dragon is a countable union of disjoint open disk-like planar sets. We determine all the cut points of the dragon and show that each disk-like subset between two cut points is a graph self-similar set defined by a graph-directed iterated function system consisting of four seed sets. Our results describe a fairly complete picture of the topological and geometric structure of the Heighway dragon.     

Separation conditions for conformal iterated function systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. The authors are supported in part by an HKRGC grant.     

色散扩散控制的单分子和双分子过程

根据最近提出的低频响应的等待时间分布函数,考虑了色散扩散控制的过程的时间变化,预言和所得到的实验数据符合很好。 关键词：