The convex hull property and topology of hypersurfaces with nonnegative curvature

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.     

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.     

Likelihood and conditional likelihood inference for generalized additive mixed models for clustered data

Lin and Zhang (J. Roy. Statist. Soc. Ser. B 61 (1999) 381) proposed the generalized additive mixed model (GAMM) as a framework for analysis of correlated data, where normally distributed random effects are used to account for correlation in the data, and proposed to use double penalized quasi-likelihood (DPQL) to estimate the nonparametric functions in the model and marginal likelihood to estimate the smoothing parameters and variance components simultaneously. However, the normal distributional assumption for the random effects may not be realistic in many applications, and it is unclear how violation of this assumption affects ensuing inferences for GAMMs. For a particular class of GAMMs, we propose a conditional estimation procedure built on a conditional likelihood for the response given a sufficient statistic for the random effect, treating the random effect as a nuisance parameter, which thus should be robust to its distribution. In extensive simulation studies, we assess performance of this estimator under a range of conditions and use it as a basis for comparison to DPQL to evaluate the impact of violation of the normality assumption. The procedure is illustrated with application to data from the Multicenter AIDS Cohort Study (MACS).     

聚芳香杂环甲烯的三阶非线性光学效应与时间分辨光克尔效应

合成了一系列聚芳香杂环甲烯,包括聚吡咯甲烯和聚噻吩甲烯.采用四波混频法研究材料的共振三阶非线性光学效应(λ＝532nm),其共振二阶超分子极化率γ三阶非线性光学系数分别达到10^-30esu和10^-8esu.选择具有良好溶解性、成膜性的聚吡咯对二甲氨基苯甲烯(PPDMAB),采用飞秒时间分辨光克尔效应方法研究材料的非共振三阶非线性光学效应(λ=790nm).实验表明,翠绿亚胺碱溶液的光克尔信号仅表现一超快响应的成分,归功于π电子云扭转产生的非共振激发.PPDMAB的非共振二阶超分子极化率γ三阶非线性光学系数分别达到γ=5.78×10^-32esu和χ⁽³⁾=1.26×10^-10esu.     

Mild and efficient synthesis of propargylamines by copper-catalyzed Mannich reaction

Terminal alkynes undergo mild and efficient aminomethylation with aqueous formaldehyde and secondary amines under CuI catalysis. In most cases high to nearly quantitative yields of tertiary propargylamines are obtained in DMSO solution at room temperature. Aromatic, aliphatic and silylated acetylenes as well as alkynols can be used. Primary amines are less reactive and satisfactory yields of secondary propargylamines are obtained only with phenylacetylene.     

Orbital formal normal forms for general Bogdanov-Takens singularity

We solve completely the problem of classification of germs of complex planar vector fields with nilpotent singularity with respect to formal orbital equivalence.     

π1-classification of real arrangements with up to eight lines

One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan (Michigan Math. J. 44(2) (1997) 283), and it turns out that the incidence lattice of such arrangements determines the projective fundamental group. We use actions on the set of wiring diagrams, introduced in (Garber et al. (J. Knot Theory Ramf.), to classify real arrangements of up to 8 lines. In particular, we show that the incidence lattice of such arrangements determines both the affine and the projective fundamental groups.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.