Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups,with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.     

On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding

We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu [K. Knight, W. Fu, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356–1378] and Fan and Li [J. Fan, R. Li, Variable selection via non-concave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (2001) 1348–1360]. We show that the distributions are typically highly non-normal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than n^−1/2 in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators’ distribution function is also provided.     

The mixed Lipschitz space and its dual for tree metrics

This paper develops a theory of harmonic analysis on spaces with tree metrics, extending previous work in this direction by Gavish, Nadler and Coifman (2010) [30] and Gavish and Coifman (2011, 2012) [28], [29]. We show how a natural system of martingales and martingale differences induced by a partition tree leads to simple and effective characterizations of the Lipschitz norm and its dual for functions on a single tree metric space. The restrictions we place on the tree metrics are far more general than those considered in previous work. As the dual norm is equal to the Earth Mover's Distance (EMD) between two probability distributions, we recover a simple formula for EMD with respect to tree distances presented by Charikar (2002) [36].We also consider the situation where an arbitrary metric is approximated by the average of a family of dominating tree metrics. We show that the Lipschitz norm and its dual for the tree metrics can be combined to yield an approximation to the corresponding norms for the underlying metric.The main contributions of this paper, however, are the generalizations of the aforementioned results to the setting of the product of two or more tree metric spaces. For functions on a product space, the notion of regularity we consider is not the Lipschitz condition, but rather the mixed Lipschitz condition that controls the size of a function's mixed difference quotient. This condition is extremely natural for datasets that can be described as a product of metric spaces, such as word-document databases. We develop effective formulas for norms equivalent to the mixed Lipschitz norm and its dual, and extend our results on combining pairs of trees.     

An exact inversion method for the determination of spin–orbit potentials from scattering data

A generalized Newton–Sabatier inversion method which permits extraction from scattering data of central and spin-orbit potentials is presented. The inversion method originally developed by Sabatier and further elaborated by Hooshyar and Richardson, has been reformulated to lead to physically reasonable solutions and to allow for its numerical implementation. Numerical problems due to the occurrence of singularities in the transformation kernel are discussed and a successful application using schematic scattering data is reported. Received: 30 April 1997 / Revised version: 20 October 1997     

Quasi-isometries preserve the geometric decomposition of Haken manifolds

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds. Oblatum 27-III-1996 & 5-IX-1996     

Asymptotic properties of the spectral test, diaphony, and related quantities

This paper presents the limit laws of discrepancies defined via exponential sums, and algorithms (with error bounds) to approximate the corresponding distribution functions. The results cover the weighted and the nonweighted spectral test of Hellekalek and various instances of the general discrepancies of Hickernell and Hoogland and Kleiss for the exponential function system, as well as classical quantities like the spectral test, diaphony, and the Zaremba figure of merit.

     

Plasma-enhanced atomic layer deposition of Co using Co（MeCp）2 precursor

Cobalt（Co） thermal or plasma enhanced atomic layer deposition（PE-ALD） was investigated using a novel metal organic precursor,Co（MeCp）₂, and NH₃ or H₂ or their plasma as a reactant.The growth characteristics,electrical and microstructural properties were investigated.Especially, PE-ALD produced Co thin films at low growth temperature down to 100℃.Interestingly,the low temperature growth of Co films showed the formation of columnar structure at substrate temperature below 300℃.The growth characteristics and films properties of PE-ALD Co using bis（η-methylcyclopentadienyl） Co（Ⅱ）（Co（MeCp）₂） was compared with those of PE-ALD Co using other Cp based metal organic precursors, bis-cyclopentadienyl cobalt（II）（CoCP₂） and cyclopentadienyl isopropyl acetarrudinato-cobalt（Co（CpAMD））.