Integration and Preparation Theorems
积分和准备定理
基本信息
- 批准号:1101248
- 负责人:
- 金额:$ 2.09万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2011
- 资助国家:美国
- 起止时间:2011-09-01 至 2013-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
This project consists of two lines of investigation. The first part of the project studies the integration theory of real constructible functions, which by definition are sums of products of real globally subanalytic functions and their logarithms. The significance of the constructible functions is that they form the smallest class of functions that extends the globally subanalytic functions and is stable under integration. Properties of Lebesgue spaces and also multivariate harmonic analysis will be studied in the context of constructible functions, including asymptotic estimates of oscillatory integrals and possibly questions concerning the integrability of Fourier transforms of constructible functions. The main tool employed to study integration of constructible functions is the subanalytic preparation theorem. The second part of the project aims to show through a purely analytic proof that the subanalytic preparation theorem holds in a more general quasianalytic setting. An important motivation for doing so is to obtain a more informative proof of the preparation theorem that could be used to study the decidability of expansions of the real field by functions from quasianalytic classes and power functions in order to combine the principal investigator's previous work on decidability, which dealt only with functions from quasianalytic classes, and the work of Jones and Servi on decidability, which dealt only with power functions.The origins of this project stem from two very classical questions which are pervasive throughout much of mathematics and its applications to science and engineering: 1) how to solve equations and inequalities, and related to this, how to determine the truth or falsity of statements built up through the use of equations and inequalities and also logical operations; 2) how to compute and study properties of functions defined by integral formulas. The subanalytic preparation theorem shows that, at least in theory, a wide class of equations (which includes all polynomial equations) can be solved by radicals in a more liberal sense that allows the use of locally defined analytic functions in addition to arithmetic operations and radicals. One aim of this project is to obtain a new algorithmic proof of the preparation theorem which, incidentally, would also generalize the theorem to wider classes of functions. In addition to studying equations, the preparation theorem is an important tool for studying the asymptotic behavior and integrals of constructible functions, which is a class of functions that contains, in particular, all algebraic functions. The Fourier transform is an important operation used in many areas of mathematics and its applications, and it is defined by an integral formula. Another aim of the project is to the lay groundwork for a theory of integration that could be used to show that Fourier transforms of constructible functions have simple properties.
This project consists of two lines of investigation. The first part of the project studies the integration theory of real constructible functions, which by definition are sums of products of real globally subanalytic functions and their logarithms. The significance of the constructible functions is that they form the smallest class of functions that extends the globally subanalytic functions and is stable under integration. Properties of Lebesgue spaces and also multivariate harmonic analysis will be studied in the context of constructible functions, including asymptotic estimates of oscillatory integrals and possibly questions concerning the integrability of Fourier transforms of constructible functions. The main tool employed to study integration of constructible functions is the subanalytic preparation theorem. The second part of the project aims to show through a purely analytic proof that the subanalytic preparation theorem holds in a more general quasianalytic setting. An important motivation for doing so is to obtain a more informative proof of the preparation theorem that could be used to study the decidability of expansions of the real field by functions from quasianalytic classes and power functions in order to combine the principal investigator's previous work on decidability, which dealt only with functions from quasianalytic classes, and the work of Jones and Servi on decidability, which dealt only with power functions.The origins of this project stem from two very classical questions which are pervasive throughout much of mathematics and its applications to science and engineering: 1) how to solve equations and inequalities, and related to this, how to determine the truth or falsity of statements built up through the use of equations and inequalities and also logical operations; 2) how to compute and study properties of functions defined by integral formulas. The subanalytic preparation theorem shows that, at least in theory, a wide class of equations (which includes all polynomial equations) can be solved by radicals in a more liberal sense that allows the use of locally defined analytic functions in addition to arithmetic operations and radicals. One aim of this project is to obtain a new algorithmic proof of the preparation theorem which, incidentally, would also generalize the theorem to wider classes of functions. In addition to studying equations, the preparation theorem is an important tool for studying the asymptotic behavior and integrals of constructible functions, which is a class of functions that contains, in particular, all algebraic functions. The Fourier transform is an important operation used in many areas of mathematics and its applications, and it is defined by an integral formula. Another aim of the project is to the lay groundwork for a theory of integration that could be used to show that Fourier transforms of constructible functions have simple properties.
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Daniel Miller其他文献
Transparency in Patient Access to Dermatopathologic Reports.
患者获取皮肤病理报告的透明度。
- DOI:
10.1001/jamadermatol.2019.4121 - 发表时间:
2020 - 期刊:
- 影响因子:10.9
- 作者:
Daniel Miller - 通讯作者:
Daniel Miller
‘And who is my neighbor?’: reading animal ethics through the lens of the Good Samaritan
“谁是我的邻居?”:通过好撒玛利亚人的视角解读动物伦理
- DOI:
- 发表时间:
2010 - 期刊:
- 影响因子:0
- 作者:
Daniel Miller - 通讯作者:
Daniel Miller
Contemporary Comparative Anthropology – The Why We Post Project
当代比较人类学——我们为什么发布项目
- DOI:
10.1080/00141844.2017.1397044 - 发表时间:
2019 - 期刊:
- 影响因子:1.3
- 作者:
Daniel Miller;Elisabetta Costa;L. Haapio;N. Haynes;Jolynna Sinanan;T. Mcdonald;R. Nicolescu;J. Spyer;S. Venkatraman;Xinyuan Wang - 通讯作者:
Xinyuan Wang
Tele-ICU Implementation and Risk-Adjusted Mortality Differences Between Daytime and Nighttime Coverage.
远程 ICU 的实施和日间和夜间覆盖范围之间的风险调整死亡率差异。
- DOI:
- 发表时间:
2020 - 期刊:
- 影响因子:9.6
- 作者:
Mario V Fusaro;C. Becker;Daniel Miller;I. F. Hassan;C. Scurlock - 通讯作者:
C. Scurlock
Daniel Miller的其他文献
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{{ truncateString('Daniel Miller', 18)}}的其他基金
Collaborative Research: Understanding the Molecular Recognition Behavior of Hollow Helices
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2108597 - 财政年份:2021
- 资助金额:
$ 2.09万 - 项目类别:
Standard Grant
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1414851 - 财政年份:2014
- 资助金额:
$ 2.09万 - 项目类别:
Fellowship Award
Shooting the Viliui Sakha Expedition for the Documentary Moving Day
为纪录片搬家日拍摄维留伊萨哈探险队
- 批准号:
1042313 - 财政年份:2010
- 资助金额:
$ 2.09万 - 项目类别:
Standard Grant
Filming "Breath of Life - Silent No More" California Indian Language Restoration Workshop at the University of California at Berkeley
加州大学伯克利分校拍摄《生命的呼吸——不再沉默》加州印第安语言修复工作坊
- 批准号:
0438121 - 财政年份:2004
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$ 2.09万 - 项目类别:
Standard Grant
SFC Travel Award (in Indian Currency) to Complete a Formal Research Proposal on Transfer Reactions With the University of Bangalore
SFC 旅游奖(以印度货币计)与班加罗尔大学完成关于转移反应的正式研究提案
- 批准号:
8319746 - 财政年份:1984
- 资助金额:
$ 2.09万 - 项目类别:
Standard Grant
A Focal-Plane Detector and Fast Electronics System For the IUCF Dual Magnetic Spectrometers (Physics)
用于 IUCF 双磁谱仪的焦平面探测器和快速电子系统(物理)
- 批准号:
8406361 - 财政年份:1984
- 资助金额:
$ 2.09万 - 项目类别:
Continuing Grant
Nuclear Structure and Nuclear Processes at Intermediate Energies (Physics)
中能核结构和核过程(物理)
- 批准号:
8114339 - 财政年份:1981
- 资助金额:
$ 2.09万 - 项目类别:
Continuing Grant
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