In this paper, we give a new approach to the study of Weyl-type theorems. Precisely, we introduce the concepts of spectral valued and spectral partitioning functions. Using two natural order relations on the set of spectral valued functions, we reduce the question of relationship between Weyl-type theorems to the study of the set difference between the parts of the spectrum that are involved. This study solves completely the question of relationship between two spectral valued functions, comparable for one or the other order relation. Then several known results about Weyl-type theorems become corollaries of the results obtained.
The architecture and working of the Artificial Neural Networks are an inspiration from the human brain. The brain due to its highly parallel nature and immense computational powers still remains the motivation for researchers. A single system-single processor approach is a highly unlikely way to model a neural network for large computational needs. Many approaches have been proposed that adopt a parallel implementation of ANNs. These methods do not consider the difference in processing powers of the constituting units and hence workload distribution among the nodes is not optimal. Human brain not always has equal processing power among the neurons. A person having disability in some part of brain may be able to perform every task with reduced capabilities. Disabilities weaken the processing of some parts. This inspires us to make a self-adaptive system of ANN that would optimally distribute computation among the nodes. The self-adaptive nature of the algorithm makes it possible for the algorithm to taper dynamic changes in node performance. We used data, node and layer partitioning in a hierarchical manner in order to evolve the most optimal architecture comprising of the best features of these partitioning techniques. The adaptive hierarchical architecture enables performance optimisation in whatever condition and problem the algorithm is used. The system was implemented and tested on 20 systems working in parallel. Besides, the computational speed-up, the algorithm was able to monitor changes in performance and adapt accordingly.