Motivated by the application problem of sensor fusion the author introduced the concept of graded set. It is reasoned that in classification problem arising in an information system (represented by information table), a novel set called Granular set
naturally arises. It is realized that in any hierarchical classification problem, Granular set naturally arises. Also when the target set of objects forms a graded set the lower and upper approximations of target sets form a graded set. This generalizes the concept of rough set. It is hoped that a detailed theory of granular/ graded sets finds several applications.
In this paper, we quantitatively (mathematically) reason the energy savings achieved by the Leveling and Sectoring protocol. Due to the energy constraints on the sensor nodes (in terms of supply of energy) energy awareness has become crucial in netwo
rking protocol stack. The understanding of routing protocols along with energy awareness in a network would help in energy opti-mization with efficient routing .We provide analytical modelling of the energy wastage in the absence of Leveling and Sectoring protocol by considering the network in the form of binary tree, nested tree and Q-ary tree. The simulation results reflect the energy wastage in the absence of Levelling and Sectoring based hybrid protocol.
In this research paper, the problem of optimization of a quadratic form over the convex hull generated by the corners of hypercube is attempted and solved. It is reasoned that under some conditions, the optimum occurs at the corners of hypercube. Res
ults related to the computation of global optimum stable state (an NP hard problem) are discussed. An algorithm is proposed. It is hoped that the results shed light on resolving the P not equal to NP problem.
In this research paper, state space representation of concurrent, linearly coupled dynamical systems is discussed. It is reasoned that the Tensor State Space Representation (TSSR) proposed in [Rama1] is directly applicable in such a problem. Also som
e discussion on linearly coupled, concurrent systems evolving on multiple time scales is included. Briefly new ideas related to distributed signal processing in cyber physical systems are included.
In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer n} and generalize the result to all powers of 2. We also
tabulate and show their relationship with other whole numbers up to 10. A number of minor results are also proved. Based on these results, approaches to determine the cardinality of Mersenne primes are discussed.
