Applications of Frames to Problems in Mathematics and Engineering II

研究者和他的学生们研究关于希尔伯特空间框架应用的四个基本问题。这些问题都需要开发全新的框架设计技术。(1)融合框架在分布式处理、无线传感器网络和其他领域有许多应用。研究者和他的学生构建融合框架专门设计的应用。(2)现在人们知道，1959年的卡迪逊-辛格问题相当于纯数学、应用数学和工程等十几个研究领域中尚未解决的基本问题。最近对这个问题的令人惊讶的积极解决方案提出了一些关于将这一结果应用于所有这些研究领域的基本问题。这方面的进展对十几个重要的研究领域产生了根本性的影响。(3)研究者和他的学生推导出新的格拉斯曼框架类。这些框架在通信理论、图论和量子物理学中有着广泛的应用，尤其是量子信息论、量子态断层扫描、量子密码学和量子力学中的基础问题。(4)研究者攻击了框架理论中著名的Paulsen问题，该问题对算子理论和其他纯数学领域具有重要意义。研究者构建和设计融合框架的主机应用到工程问题。其中一个重点是设计无线传感器网络。这是一项新兴技术，具有检测运输中的化学、生物、放射性和核武器的潜力。另一个应用是分布式处理，其中数据系统太大而无法进行中央处理。这种数据过载现在出现在工程、物理、天文学、医学、生物学和工业应用中，这些应用需要实时处理不断增加的数据量。框架也应用于纯数学中的许多重要问题。

Casazza1307685 The investigator and his students work on four fundamental problems concerning applications of Hilbert space frames. These problems all require developing completely new techniques for frame design. (1) Fusion frames have a host of applications in distributed processing, wireless sensor networks, and other areas. The investigator and his students construct fusion frames designed specifically for applications. (2) It is now known that the 1959 Kadison-Singer Problem is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics, and engineering. The recent surprising positive solution to the problem raises a number of fundamental questions about the implementation of this result to all these areas of research. Advances here affect a dozen important research areas in a fundamental way. (3) The investigator and his students derive new classes of Grassmannian frames. These frames have broad application to communication theory, graph theory, and quantum physics -- especially quantum information theory, quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. (4) The investigator attacks the famous Paulsen Problem in frame theory, which has important implications for operator theory and other areas of pure mathematics. The investigator constructs and designs fusion frames for a host of applications to problems in engineering. One emphasis is on designing wireless sensor networks. which are an emerging technology with the potential to detect chemical, biological, radiological, and nuclear weapons in transport. Another application is to distributed processing where data systems are too large for central processing. This overload of data now appears in engineering, physics, astronomy, medicine, biology, and industrial applications that demand real-time processing on as ever-increasing amounts of data. Frames also have application to a number of important problems in pure mathematics.