Applications of Frames to Problems in Mathematics and Engineering II

卡萨扎1307685 调查员和他的学生工作的四个基本问题有关的应用希尔伯特空间框架。 这些问题都需要开发全新的框架设计技术。 (1)融合帧在分布式处理、无线传感器网络等领域有着广泛的应用。 研究人员和他的学生构建了专门为应用设计的融合框架。 (2)现在我们知道，1959年的凯迪逊-辛格问题相当于纯数学、应用数学和工程学中十几个研究领域中的基本未解决问题。 最近令人惊讶的积极解决问题提出了一些基本的问题，这一结果的实施，所有这些领域的研究。 这里的进展从根本上影响了十几个重要的研究领域。 (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，特别是量子信息理论、量子态层析成像、量子密码学和量子力学的基础问题。 (4)研究者对框架理论中著名的Paulsen问题进行了研究，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，用于解决工程问题。 重点之一是设计无线传感器网络。 这是一种新兴技术，有可能探测运输中的化学、生物、放射性和核武器。 另一个应用是分布式处理，其中数据系统对于中央处理来说太大了。 这种数据过载现在出现在工程、物理学、天文学、医学、生物学和工业应用中，这些应用需要对不断增加的数据量进行实时处理。 框架也可以应用于纯数学中的一些重要问题。

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.