Symmetry Parameter Analysis of Singular Integrals

奇异积分的对称参数分析

基本信息

  • 批准号:
    2054863
  • 负责人:
  • 金额:
    $ 19.76万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2021
  • 资助国家:
    美国
  • 起止时间:
    2021-07-01 至 2024-06-30
  • 项目状态:
    已结题

项目摘要

Harmonic Analysis is the branch of mathematics concerned with the rigorous description of signals (functions) and of their processing (operators). Examples of signals are sound, images, time series and weather data. Such signals are analyzed via the overlaying (superposition) of basic harmonics of well-specified duration, intensity and frequency. These basic harmonics are functions called wavelets. Image or audio denoising, compression, or pattern recognition are accomplished by filter processing, which refers to a suitable superposition of each wavelet after the action of the filter on it. This is also known as the time-frequency method. A particular concrete example of a re-construction process is used in tomographic imaging, where the shape of a solid body is re-composed from samples of the body along one or two-dimensional rays of penetrating waves, which can be mathematically described as lines or planes in three dimensional space. One component of this mathematics research project focuses on a new family of methods for the wavelet description of the class of singular integral operators, arising for instance in the time-frequency analysis of highly oscillatory signals. Another component of this research project is concerned with the mathematical properties of sampling solid objects along lines or planes. The integrated broader impact activities focus on strengthening the pool of socioeconomically disadvantaged, ethnical minority students (underrepresented groups) in graduate degrees in mathematics and improving retention. Activities connected to training and mentoring of graduate students in Analysis and topical dissemination of knowledge will also be carried out. The broad aim of the first circle of questions is to produce representation formulas for classes of singular integrals in terms of so-called model operators conserving the same invariance structure. This paradigm applies to Zygmund-type operators and modulation invariant operators akin to the bilinear Hilbert transform, both of which are out of reach for dyadic-probabilistic methods. Concrete applications come from elliptic and dispersive PDE, operator theory and quasi-conformal mappings. The second related family of questions is motivated by pointwise convergence of bilinear ergodic averages for Banach-valued functions, a celebrated theorem by Bourgain in the scalar case. The approach is based on Banach-valued variational estimates for the truncated bilinear Hilbert transform. The central item in a further set of questions concerning directional singular integrals is a version of the Kakeya maximal estimate where tubular averages are replaced with averages over singular line segments, and more generally, n-dimensional subspaces. One source of motivation is the connection with Fourier restriction in higher codimensions. The methods involve algebra-geometric techniques such as polynomial partitioning on manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
调和分析是数学的一个分支,涉及对信号(函数)及其处理(运算符)的严格描述。信号的例子有声音、图像、时间序列和天气数据。这些信号通过特定持续时间、强度和频率的基波叠加(叠加)来分析。这些基本的谐波是被称为小波的函数。图像或音频的去噪、压缩或模式识别是通过滤波处理来完成的,滤波处理指的是在滤波器作用于每个小波之后每个小波的适当叠加。这也被称为时频方法。重建过程的特定具体示例用于断层成像,其中实体的形状由沿穿透波的一维或二维射线的实体样本重新组成,其可以在数学上描述为三维空间中的线或平面。这个数学研究项目的一个组成部分集中于一类奇异积分算子的小波描述的新方法,例如在高振荡信号的时频分析中出现的方法。这项研究项目的另一个组成部分是关于沿线或面采样实体对象的数学特性。综合的、更广泛的影响活动侧重于加强在社会经济上处于不利地位的少数族裔学生(代表人数不足的群体)在数学研究生学位中的地位,并提高保留率。还将开展与研究生在分析和专题知识传播方面的培训和辅导有关的活动。第一个问题圈的主要目的是在保持相同不变结构的情况下,用所谓的模型算子给出奇异积分类的表示公式。这一范例适用于Zygmund型算子和类似于双线性Hilbert变换的调制不变算子,这两者都不是并矢概率方法所能达到的。具体的应用来自于椭圆型和色散型偏微分方程组、算子理论和拟共形映射。第二类相关问题的动机是Banach值函数的双线性遍历平均的逐点收敛,这是Bourain在标量情况下的著名定理。该方法基于截断双线性Hilbert变换的Banach值变分估计。在另一组关于方向奇异积分的问题中,中心项是Kakeya最大估计的一个版本,其中管状平均被奇异线段上的平均替换为更一般地,n维子空间上的平均。动机的一个来源是与高余维中的傅立叶限制的联系。这些方法涉及代数几何技术,如流形上的多项式划分。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(4)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Evolution of superoscillations for spinning particles
旋转粒子超振荡的演化
  • DOI:
    10.1090/bproc/159
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Colombo, Fabrizio;Pozzi, Elodie;Sabadini, Irene;Wick, Brett
  • 通讯作者:
    Wick, Brett
Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights
用于平滑 Calderón-Zygmund 算子和加倍权重的两个权重 Sobolev 范数不等式
  • DOI:
    10.1007/s00209-023-03220-x
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0.8
  • 作者:
    Sawyer, Eric T.;Wick, Brett D.
  • 通讯作者:
    Wick, Brett D.
Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications
  • DOI:
    10.1007/s11785-023-01346-8
  • 发表时间:
    2022-04
  • 期刊:
  • 影响因子:
    0.8
  • 作者:
    Mishko Mitkovski;Cody B. Stockdale;Nathan A. Wagner;B. Wick
  • 通讯作者:
    Mishko Mitkovski;Cody B. Stockdale;Nathan A. Wagner;B. Wick
Maximal subspace averages
最大子空间平均值
  • DOI:
    10.1016/j.aim.2022.108749
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Di Plinio, Francesco;Parissis, Ioannis
  • 通讯作者:
    Parissis, Ioannis
{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Brett Wick其他文献

Steven George Krantz (1951 -) Celebrates his 70th Birthday
  • DOI:
    10.1007/s11785-023-01480-3
  • 发表时间:
    2024-02-08
  • 期刊:
  • 影响因子:
    0.800
  • 作者:
    Arni S. R. Srinivasa Rao;Siqi Fu;Gregory Knese;Kaushal Verma;Brett Wick
  • 通讯作者:
    Brett Wick

Brett Wick的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Brett Wick', 18)}}的其他基金

Testing Theorems in Analytic Function Theory, Harmonic Analysis and Operator Theory
解析函数论、调和分析和算子理论中的检验定理
  • 批准号:
    2349868
  • 财政年份:
    2024
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
Conference: Geometric Measure Theory, Harmonic Analysis, and Partial Differential Equations: Recent Advances
会议:几何测度理论、调和分析和偏微分方程:最新进展
  • 批准号:
    2402028
  • 财政年份:
    2024
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
Conference: Recent Advances and Past Accomplishments in Harmonic Analysis
会议:谐波分析的最新进展和过去的成就
  • 批准号:
    2230844
  • 财政年份:
    2022
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
Singular Integrals with Modulation or Rotational Symmetry
具有调制或旋转对称性的奇异积分
  • 批准号:
    2000510
  • 财政年份:
    2019
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
International Conference on Interpolation in Spaces of Analytic Functions at CIRM
CIRM 解析函数空间插值国际会议
  • 批准号:
    1936503
  • 财政年份:
    2019
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
Applications of Harmonic Analysis to Riesz Transforms and Commutators beyond the Classical Settings
谐波分析在经典设置之外的 Riesz 变换和换向器中的应用
  • 批准号:
    1800057
  • 财政年份:
    2018
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant
Applications of Harmonic Analysis to Function Theory and Operator Theory
调和分析在函数论和算子理论中的应用
  • 批准号:
    1500509
  • 财政年份:
    2015
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Continuing Grant
CAREER: An Integrated Proposal Based on The Corona Problem
职业:基于新冠问题的综合提案
  • 批准号:
    1603246
  • 财政年份:
    2015
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Continuing Grant
Applications of Harmonic Analysis to Function Theory and Operator Theory
调和分析在函数论和算子理论中的应用
  • 批准号:
    1560955
  • 财政年份:
    2015
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Continuing Grant
The Corona Problem: Connections between Operator Theory, Function Theory and Geometry
电晕问题:算子理论、函数论和几何之间的联系
  • 批准号:
    1200994
  • 财政年份:
    2012
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Standard Grant

相似海外基金

Gravitational Wave Data Analysis: Parameter Inference and Black Hole Ringdown
引力波数据分析:参数推断和黑洞衰荡
  • 批准号:
    2308833
  • 财政年份:
    2023
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Continuing Grant
Development of rainfall-runoff and flood inundation integrated analysis model and parameter optimization method for basinwide comprehensive flood disaster prevention
全流域综合防洪雨径洪涝综合分析模型及参数优化方法开发
  • 批准号:
    21K04277
  • 财政年份:
    2021
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Integrated, photoactive-liquid crystal modulators characterised by optical, multi-parameter analysis
以光学多参数分析为特征的集成光敏液晶调制器
  • 批准号:
    2897721
  • 财政年份:
    2020
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Studentship
Pathological elucidation of autism by using morphological multi-parameter analysis in neuron and microglia co-culture system
利用神经元和小胶质细胞共培养系统的形态学多参数分析对自闭症进行病理学阐明
  • 批准号:
    20H03597
  • 财政年份:
    2020
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
CAREER: The relative importance of remote oceanic forcing on estuarine exchange flow across a broad parameter space - Numerical analysis integrated with visualization & educati
职业:远程海洋强迫对跨广泛参数空间的河口交换流的相对重要性 - 数值分析与可视化相结合
  • 批准号:
    1944735
  • 财政年份:
    2020
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Continuing Grant
Analysis of muscle contraction mechanism by multichannel surface electromyogram based on propagation wave parameter distribution
基于传播波参数分布的多通道表面肌电分析肌肉收缩机制
  • 批准号:
    19K11546
  • 财政年份:
    2019
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Comprehensive analysis of pancreatic cancer-associated antigen specific immune responses using high parameter flow cytometry and its clinical significance
高参数流式细胞术综合分析胰腺癌相关抗原特异性免疫反应及其临床意义
  • 批准号:
    19K18114
  • 财政年份:
    2019
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Early-Career Scientists
Mathematical analysis for Navier-Stokes equations with approximate parameter
具有近似参数的纳维-斯托克斯方程的数学分析
  • 批准号:
    19K03577
  • 财政年份:
    2019
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Vehicle Parameter Optimization Based on Bifurcation Analysis for Avoiding Hunting Motion
基于分叉分析的车辆参数优化避免狩猎运动
  • 批准号:
    18K04029
  • 财政年份:
    2018
  • 资助金额:
    $ 19.76万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
SCH:Smartphone Wound Image Parameter Analysis and Decision Support in Mobile Env
SCH:移动环境中的智能手机伤口图像参数分析和决策支持
  • 批准号:
    9496652
  • 财政年份:
    2018
  • 资助金额:
    $ 19.76万
  • 项目类别:
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了