ITR: Analysis of the Capacity Improvement for Wireless Networks with Multiple Transmit and Receive Antennas

众所周知，有时连续和离散之间存在着惊人的分歧。为了统一微分方程和差分方程的研究，十年前Stefan Hilger提出了时标理论。它不仅能够同时处理这两种情况，而且还可以处理介于连续和离散之间的许多其他情况，这些情况可能对应用程序很重要。出于这个原因，它也可能是值得思考的后果，一个更好的发展时间尺度理论的研究生，特别是本科教育。由于这一理论是非常新的，而且实际上对分析的每一个领域都很重要，所以现在在这一问题上所做的工作将是基础性的。如果国家科学基金会资助这项建议，我们将以各种方式扩展时标上的动力方程理论。我们希望在时间尺度上进行线性动力系统的研究，这将导致更好地理解高阶动力方程。对于这些任务的特点不共轭是一个有趣的，这个问题，我们也想解决高阶Sturm-Liouville动力方程，或更一般的线性哈密顿动力系统的时间尺度。这样的系统也将是有趣的，如果一个本征值参数参与，我们想得到一个存在定理，一个扩展定理，瑞利原理，和更一般的振动结果，这些本征值问题。另一个有趣的任务，这是密切相关的上述问题，将建立一个理论的变分分析的时间尺度。给出强和弱局部极小的充分必要条件将是这样一个项目的主要关注点，它也将有助于建立一个Weierstrab理论的变分问题的时间尺度。

It is very well known that there is sometimes a striking divergence between the continuous and the discrete. The theory of time scales has been initiated by Stefan Hilger ten years ago in order to unify study for differential and difference equations. Not only is it able to treat these two cases simultaneously, but it also can handle numerous other cases \in between" the continuous and the discrete, and those cases might be important for applications. For this reason it also could be worth to think about the consequences of a better developed time scales theory for graduate and in particular for undergraduate education. As the theory is very new and virtually important to every area in analysis, work on this subject done now will be fundamental. If NSF funds this proposal we would expand the theory of dynamic equations on time scales in various ways. We would like to conduct study of linear dynamic systems on time scales which would result in a better understanding of higher order dynamic equations. For those the task of characterizing disconjugacy is an interesting one, and this question we also want to address for higher order Sturm-Liouville dynamic equations or, more general, linear Hamiltonian dynamic systems on time scales. Such systems also would beinteresting if an eigenvalue parameter was involved, and we would like to derive an existence theorem, an expansion theorem, Rayleigh's Principle, and more general oscillation results for these eigenvalue problems. Another interesting task, which is intimately connected to the above problems, would be to establish a theory of variational analysis on time scales. To give necessary and sufficient conditions for strong and weak local minima would be the main concern of such a project, and it would also be of use to establish a Weierstrab Theory for variational problems on time scales.