Classical and Noncommutative Processes

经典过程和非交换过程

• 批准号: 0504198
• 负责人: Wlodzimierz Bryc
• 金额: $ 6.97万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504198&HistoricalAwards=false
• 关键词: Classical; Noncommutative; Processes

This research will advance the understanding of new mathematical connections between different theories that model physical phenomena of significant practical interest. Starting from intuitive formulas for the conditional means and conditional variances of stochastic processes, the proposer will study the class of associated orthogonal martingale polynomials and use them to construct new Markov processes. These polynomials will be described by a q-commutation equation which will provide a new link between the noncommutative and commutative probability, and will tie together theories as different as Hammersley's probabilistic models of long-range misorientation in the crystalline structure of metals and Frisch and Bourret parastochastic models of turbulence convection.The project will advance the understanding of connections between the classical probability which is the foundation of statistics, and the non-commutative probability which is a foundation of quantum physics. It will be advanced through personal research, through research with graduate students, and through cooperative activities with several mathematicians in USA and in Europe. The results will be disseminated broadly and in a timely manner through the use of the Internet, through presentations at conferences and workshops, and through research publications.

这项研究将增进对不同理论之间新的数学联系的理解，这些理论模拟了具有重大实际意义的物理现象。从随机过程的条件均值和条件方差的直观公式开始，提议者将研究相关的正交鞅多项式的类别，并使用它们来构造新的马尔可夫过程。 这些多项式将通过 q 交换方程来描述，该方程将在非交换概率和交换概率之间提供新的联系，并将不同的理论联系在一起，例如哈默斯利的金属晶体结构中的远程错误取向概率模型和弗里施和布尔雷湍流对流的顺随机模型。该项目将增进对作为统计基础的经典概率与概率之间联系的理解。 非交换概率是量子物理学的基础。 它将通过个人研究、与研究生的研究以及与美国和欧洲的几位数学家的合作活动来推进。 研究结果将通过使用互联网、在会议和研讨会上的演讲以及研究出版物进行广泛而及时的传播。