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

众所周知，连续型和离散型之间有时会有惊人的分歧。时间尺度理论是由斯蒂芬·希尔格在十年前提出的，目的是统一对微分方程组和差分方程组的研究。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，我们也有必要思考一个更完善的时间尺度理论对研究生，尤其是对本科教育的影响。由于这一理论是非常新的，实际上对分析的每个领域都很重要，所以现在在这个问题上所做的工作将是根本的。如果NSF资助这项提议，我们将以各种方式扩展时间尺度上的动态方程理论。我们想在时间尺度上研究线性动力学系统，这将导致对高阶动力学方程的更好的理解。对于这些人来说，刻画不共轭是一个有趣的任务，我们也想要解决高阶Sturm-Liouville动力学方程或更一般的时间尺度上的线性哈密顿动力系统的这个问题。如果涉及到特征值参数，这类系统也会很有趣，我们想要得到这些特征值问题的存在定理、展开定理、瑞利原理和更一般的振荡结果。与上述问题密切相关的另一项有趣的任务是建立时间尺度上的变分分析理论。给出强局部极小值和弱局部极小值的充要条件将是这类项目的主要目的，这对于建立时间尺度上的变分问题的魏尔斯特拉布理论也是有用的。

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.