SPECTRAL SEQUENCES OF COHOMOLOGIES OF SINGULARITIES OF C∞ MAPPINGS

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Dealing with singularities in nonlinear unconstrained optimization

We propose a new trust region based optimization algorithm for solving unconstrained nonlinear problems whose second derivatives matrix is singular at a local solution. We give a theoretical characterization of the singularity in this context and we propose an iterative procedure which allows to identify a singularity in the objective function during the course of the optimization algorithm, and artificially adds curvature to the objective function. Numerical tests are performed on a set of unconstrained nonlinear problems, both singular and non-singular. Results illustrate the significant performance improvement compared to classical trust region and filter algorithms proposed in the literature. The approach is also shown to be competitive with tensor methods in terms of efficiency while reaching a higher level of robustness.     

不连续三阶两点边值问题的可解性

证明了非线性三阶两点边值问题u′″（t）-q（u″（t））f（t，u（t）），u（O）=a，u（1）=b，u″（0）=c解的一个存在定理．在这个问题中，f（t，u）是一个Carathéodory函数而边界条件是非齐次的．我们的结论表明该问题能够有一个解，只要在R。的某个有界集合上q（υ）的“本性高度”与f（t，u）的“最大高度”积分的乘积是适当的．     

A Remark on the Existence of Solutions to Nonlinear Equations with Degenerate Mappings

The paper studies nonlinear mappings using methods of p-regularity theory and some concepts and technics of set-valued analysis. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case where a linear part may be singular at the considered initial point.      

离轴矢量光束的紧聚焦特性

基于理查德-沃尔夫矢量衍射积分理论,数值研究了离轴矢量光束经过高数值孔径透镜后的紧聚焦特性.通过对比分析偏振奇点离轴的径向、角向矢量光束紧聚焦后,焦点附近的强度和相位分布规律,提出了一种由广义柱矢量光束设计紧聚焦场的方法.此外,通过分析矢量光束自旋分量的聚焦行为和自旋轨道耦合所诱导的自旋角动量分布变化,讨论了偏振奇点离轴对称破缺对高阶矢量光束紧聚焦场空间结构,特别是其自旋角动量空间分布的影响.该研究结果不仅对矢量光场的紧聚焦特性进行了补充,也为改进和完善焦场分布提供了理论参考.     

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.     

Enumeration and limit laws of dissections on a cylinder

We compute the generating function for triangulations on a cylinder, with the restriction that all vertices belong to its boundary and that the intersection of a pair of different faces is either empty, a vertex or an edge. We generalize these results to maps with either constant ({k}-dissections) or unrestricted (unrestricted dissections) face degree. We apply singularity analysis to the resulting generating functions to obtain asymptotic estimates for their coefficients, as well as limit distributions for natural parameters.     

Integrability of a coupled KdV system: Painlevé property, Lax pair and Bäcklund transformation

The integrability of a coupled KdV system is studied by prolongation technique and singularity analysis. As a result, Bäcklund transformation and linear spectral problem associated with this system are obtained. Some special solutions of the system are also proposed.     

Reversibility and quasi-homogeneous normal forms of vector fields

This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.