Multilinear Operators, Discrete Decompositions, and Spectral Resolution of Nanostructures

纳米结构的多线性算子、离散分解和光谱分辨率

• 批准号: 0070514
• 负责人: Rodolfo Torres
• 金额: $ 10.1万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070514&HistoricalAwards=false
• 关键词: Multilinear; Operators; Discrete; Decompositions; Spectral

ABSTRACT:The research to be conducted includes the analysis of operators associated withmultilinear singular integrals and the study of discrete functions spaces. Theinvestigations will be based on techniques related to Littlewood-Paley theory,molecular decompositions, and time-frequency analysis tools. Specific aspectsproposed in the analysis of operators are to continue collaborations in thedevelopment of a multilinear counterpart of the linear Calderon-Zygmundtheory, and to study various classes of multilinear pseudodifferentialoperators. Problems in the analysis of discrete function spaces include issuesabout the sampling of functions with controlled mean oscillations and theapproximation of band limited signals. The proposal also contains aninterdisciplinary component. In particular, theoretical problems arising in theanalytic formulation of the scattering of light by structurally colored tissuesof living organisms will be considered. The spectral resolution andmathematical properties of quasi-ordered geometries will be investigated. Thelast part of the research will be assisted by numerical computation and datavisualization.Operators associated with singular integrals arise as technical tools inanalysis and also as transformations encountered in the mathematical modeling ofcertain physical phenomena. Such transformations can be used to describe thechanges in the properties of a function or the transition from the input to theoutput of a system. Properties of functions or signals often need to beunderstood from the information encoded in samples of the data. Suchinformation can be quantified by function spaces and decoded by Fourier analysisand related time-frequency techniques. Fourier analysis is the mathematicalversion of a diffracting physical prism. It resolves a signal into a spectrumof waves of different amplitudes and oscillations in a similar way that a prismdiffracts a ray of light into a rainbow of colors of different wavelengths.Modern decomposition techniques in analysis provide a universal language for theprocessing of complicated information. Progress in this area of analysis alwaysproduces important advances in scientific problems where large and complicatedsets of data need to be analyzed to search for ordered patterns, reduceunnecessary information, or visualize intricate structures.

摘要：本论文的研究内容包括多线性奇异积分算子的分析和离散函数空间的研究。 这些调查将基于Littlewood-Paley理论，分子分解和时频分析工具相关的技术。 具体方面提出的分析运营商是继续合作的发展中的一个多线性对应的线性Calderon-Zygmund理论，并研究各类多线性pseudodifferentialoperators。 离散函数空间分析中的问题包括受控平均振荡函数的采样问题和带限信号的逼近问题。 该提案还包含一个跨学科的组成部分。 特别是，理论问题所产生的分析制定的光散射的结构彩色tissuesof活的有机体将被考虑。 研究了准有序几何的光谱分辨率和数学性质。 最后一部分的研究将通过数值计算和数据可视化来辅助，与奇异积分相关的算子作为分析中的技术工具出现，也作为某些物理现象的数学建模中遇到的变换. 这种变换可以用来描述函数性质的变化或系统从输入到输出的过渡。 函数或信号的性质通常需要从数据样本中编码的信息中理解。 这些信息可以通过函数空间量化，并通过傅立叶分析和相关的时频技术解码。 傅里叶分析是衍射物理棱镜的几何变换。 它把信号分解成不同振幅和振荡的波谱，就像棱镜把一束光衍射成不同波长的彩虹一样。现代分析中的分解技术为处理复杂信息提供了一种通用语言。 这一分析领域的进步总是在科学问题上产生重要的进步，这些问题需要分析大量复杂的数据集以寻找有序的模式，减少不必要的信息，或可视化复杂的结构。