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

将构建用于各种工业应用的希尔伯特空间框架。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，即某些系数的丢失。目标是提高数据传输的整体速度和准确性。接下来，将探讨用于无线通信系统的特定帧，目标是提高速度和可靠性。这些系统和基于分组的系统之间的差错性质的差异需要不同的构造。此外，还将构建语音识别技术的框架，目的是快速识别、区分和提取重叠语音中的独立声音。语音识别在很大程度上依赖于相位中的噪声。因此，将解决噪声相位和相位信息丢失的具体问题。最后，对于量子物理，特别是量子信息论，将从符号和/或数值的角度导出Grassman标架(即等角标架)。其目标是开发量子态层析、量子密码学和量子力学基础问题所需的最佳分布框架。对于这些项目中的每一个，都将发现、表征和构建新的帧类别。将产生提高互联网速度和准确性所需的数学知识。这还将使消息能够准确地传递，即使许多原始信息在传输过程中丢失。在相关工作中，将解决与无线通信系统相关的具体问题--例如手机。在这里，人们希望生产不褪色且信号清晰的手机。此外，还将探讨在重叠语音中分离声音所需的数学运算。其目标是最终拥有能够直接快速响应语音通信的汽车、家用电器等。所有这些项目在国土安全和民用方面都有重要的应用。

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.