CAREER: Complexity, Reality, and Rationality in Large Nonlinear Equation Solving

职业：大型非线性方程求解的复杂性、现实性和合理性

• 批准号: 0349309
• 负责人: J Maurice Rojas
• 金额: $ 40万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0349309&HistoricalAwards=false
• 关键词: CAREER; Complexity; Reality; Rationality; Large

In this Career project, the investigator studies problemsthat involve an interplay between algebraic geometry, numbertheory, and complexity theory. Specific projects include: (1) A new generation of fast randomized algorithms for real root counting, allowing analytic as well as algebraic equations, (2) Further explorations into algorithmic fewnomial theory over the real and p-adic numbers, (3) A deeper analysis of numerical conditioning of random sparse polynomial systems, (4) Sharpened quantitative results for rational solutions of straight-line programs, (5) Further exploration of number-theoretic approaches to the P vs. NP question. Solving equations is ubiquitous in applications, cuttingacross many areas of engineering and science: A brief listincludes drug design, faster and more reliable methods for radarimaging and geometric modelling, and more reliable and efficientapproaches to robotics and autonomous vehicles. The investigatordevelops methods for problems related to the solution ofpolynomial equations, with an eye toward these applications. Healso actively recruits students from schools in poorer areas ofTexas (with large African-American and Hispanic populations) tohelp disadvantaged students and channel more bright young peopleinto the computational sciences. He also continues work onrelated software, so that the broader public can benefit from thediscoveries of this project.

在这个职业项目中，研究人员研究问题的问题涉及代数几何，数字理论和复杂性理论之间的相互作用。 Specific projects include: (1) A new generation of fast randomized algorithms for real root counting, allowing analytic as well as algebraic equations, (2) Further explorations into algorithmic fewnomial theory over the real and p-adic numbers, (3) A deeper analysis of numerical conditioning of random sparse polynomial systems, (4) Sharpened quantitative results for rational solutions of straight-line programs, (5) Further exploration PS. NP问题的数字理论方法。 求解方程在应用中无处不在，切割工程和科学的许多领域：简短的列表包括药物设计，更快，更可靠的方法，用于雷达和几何建模，以及对机器人和自动驾驶汽车的更可靠，更有效的方法。 研究与多种方程式有关的问题的研究方法，以关注这些应用。 Healso积极招募来自较贫穷地区的学校的学生（有大量的非裔美国人和西班牙裔人口）TOHELP处于弱势学生，并引导年轻的年轻人在计算科学中更加聪明。 他还继续进行工作的软件，以便更广泛的公众可以从该项目的范围内受益。