External cluster validity indices (CVIs) are used to evaluate various clustering algorithms by comparing the obtained clustering result with the gold standard. These indices provide some values for the obtained clustering result, indicating how good or bad the obtained clustering result is compared to the gold standard. For an external CVI, it is desirable to always provide the values within its range; otherwise, the clustering result cannot be correctly interpreted. Thus, in this work, our objective is to theoretically obtain the scenarios where these indices are unable to provide valid values. 26 commonly used indices identified in the existing literature are considered for our analysis purposes. For all these CVIs, we are able to determine the scenarios where these indices provide undefined values. Using the Bell number and Bell polynomial notion, we have attempted to represent the number of such scenarios.

In this work, capacity based optimal best beam set selection methods in MIMO channel condition are presented. Proposed methods are suitable for existing Verizon 5G specifications \cite{VzW} and also scalable to upcoming 3GPP NR\cite{NR} mmWave communications systems. Proposed methods are aimed at improving the performance, robustness and computational efficiency of the user equipment (UE) system where beam measurements are intended. Two methods namely, a two stage capacity maximization (CM) based method and a hybrid power and capacity maximization based method are proposed herein. Capacity based solutions are preferred over received signal strength (RSS) based methods in view of its superior performance in presence of unknown interference and noise. Hybrid method exploits the sparse channels inherent in beamformed mmWave systems. Also, it is preferable as it complies with the evolving 5G communication requirements which need estimation of additional alternative MIMO transmit and receive beam pairs apart from the serving beams to achieve higher throughput by overcoming random beam blockage and misalignment. Efficacy and performance of the proposed beam-selection methods is compared with existing RSS based method. Also computational complexity of the proposed methods are analyzed and numerical results comparing them are presented. Advantages of the capacity based methods proposed are demonstrated through analysis and simulation results.

We define a new model of data streaming algorithms that employ a prover/helper to outsource difficult computations in a verifiable way. While for the verifier the usual time (per symbol read) and space constraints of the data streaming model are in place, the prover has unbounded space. Both parties cannot look into the future (i.e., do not know data arriving later). Previous work on such models either severely restricted the total communication between the prover and the verifier, or extended the computation by a long annotation that has to be streamed from the prover to the verifier offline after the original stream has ended, delaying the computation of the result. We argue that restricting the total communication severely is unnatural and investigate a model that only bounds the communication overhead, i.e., the amount of communication sent from the prover to the verifier per symbol of the data stream. This allows for vastly more communication between prover and verifier while maintaining the online nature of the model (in particular long annotations sent after the stream has ended are not allowed). Relaxing the communication requirement allows us to find simple algorithms for problems like the Longest Increasing Subsequence Problem (LIS), finding the Median, and for deciding whether the rank of a matrix is full or not. All our algorithms have a similar structure with phases whose length shrinks geometrically, and phase i being used to verify certain properties of the stream up to phase i-1 using re-streaming of parts of the previous stream. The challenge in each case is to tie the different phases together.