Applications of Frames to Internet Coding, Wireless Communication, Speech Recognition and Quantum Information Theory

希尔伯特空间框架的各种工业应用将被构建。 首先，将为基于分组的通信系统（如互联网）构建帧。 这里，将解决擦除错误，即，一些系数的丢失。目标是提高数据传输的整体速度和准确性。 接下来，将以提高速度和可靠性为目标，探索用于无线通信系统的特定帧。这些系统和基于分组的系统之间的误差性质的差异需要不同的构造。 此外，语音识别技术的框架将被构建，目的是快速识别，区分和提取重叠语音中的单独语音。语音识别严重依赖于相位中的噪声。 因此，噪声相位和相位信息丢失的具体问题将得到解决。 最后，Grassmannian（即等角框架）框架将在量子物理学，特别是量子信息理论中以符号和/或数值方式导出。 其目标是开发量子态层析成像、量子密码学和量子力学基础问题所需的最佳分布式帧。 对于每一个项目，新的框架类别将被发现，表征和构建。将产生提高互联网速度和准确性所需的数学。 这也将使信息能够准确地传递，即使许多原始信息在传输中丢失。在相关的工作中，将解决与无线通信系统（如手机）相关的具体问题。在此，希望生产出不衰减并具有清晰信号的蜂窝电话。 此外，将探讨在重叠语音中分离语音所需的数学。 目标是最终使汽车、电器等能够直接快速地响应语音通信。 所有这些项目都对国土安全和民用有重要的应用。

Hilbert space frames for a variety of industrial applications will be constructed. First, frames will be constructed for packet-based communication systems, such as the internet. Here, errors of erasure, that is, the loss of some coefficients will be addressed. The goal is to improve the overall speed and accuracy of data transmission. Next, specific frames for wireless communication systems will be explored with the goal of improving both speed and reliability. The differences in the nature of the error between these systems and packet-based systems requires different constructions. Also, frames for speech-recognition technology will be constructed with the aim to quickly recognize, distinguish and extract separate voices found in overlapped speech. Speech recognition is heavily dependent upon noise in the phase. So the specific issues of noisy phase and the loss of phase information will be addressed. Finally, Grassmannian (i.e. equiangular frames) frames will be derived symbolically and/or numerically for quantum physics, especially quantum information theory. The goal is to develop optimally distributed frames needed for quantum state tomography, quantum cryptography and foundationalissues in quantum mechanics. For each of these projects new classes of frames will be discovered, characterized and constructed.Mathematics necessary to improve the speed and accuracy of the internet will be produced. This will also allow messages to be delivered accurately even if much of the original information is lost in transmission. In related work, specific problems related to wireless communications systems - such as cell phones will be addressed. Here it is desired to produce cell phones which do not fade and have crisp clear signals. Also the mathematics needed to separate voices in overlapped speech will be explored. The goal is to eventually have cars, appliances etc. which respond directly and quickly to voice communications. All of these projects have important applications to homeland security as well as civilian use.