Various directions of obtaining
novel structured matrices are discussed. A new class of matrices, called “the L-family”
matrices are introduced and their properties are studied.

Abstract:
Recent advances in wireless communications are diffusing into many new applications. The tiny sensor node, which consists of sensing, data processing and communicating components, led to the idea of sensor networks. A sensor network composed of a large number of sensor nodes that are densely deployed either inside the phenomenon or very close to it. The applications envisioned for sensor networks vary from monitoring inhospitable habitats and disaster areas to operating indoors for intrusion detection and equipment monitoring. In most cases the network designer would have little control over the exact deployment of the network. Nowadays Vehicular Networks are drawing lots of attention due to the wide variety of applications that they can provide. These applications include traffic monitoring, positioning, security etc. A lot of research work is being conducted to define the standard for vehicular communication. These include frequency allocation, standards for physical and link layers, routing algorithms, security issues and new applications. In this paper we discuss the disadvantages of the traffic monitoring by traditional methods and by using GPS equipped sensors. Then we propose a new routing protocol for a fixed topology containing both stationary and mobile nodes. We also try to optimize the energy of the sensor nodes. We simulate our routing algorithm in MATLAB and evaluate it for different possible cases.

Abstract:
Zhong et al. formulated the problem of disclosing credentials (associated with privacy) for building trust in an open environment like Internet. Thus, one form of privacy-trust problem is formulated in [17]. In [17], an entropy measure is defined to quantify privacy loss. In this research paper, by a proper formulation (modelling), the privacy-trust tradeoff is attacked from the Information Theoretic view point. It is shown that as long as privacy loss (or more accurately a measure related to privacy loss) is below a limit, through the use of properly coded credentials, it is possible to achieve arbitrarily good trust. The approach presented here will lead to several practical schemes in trading privacy for trust.

Abstract:
In this research paper, structured bi-matrix variate, matrix quadratic equations are considered. Some lemmas related to determining the eigenvalues of unknown matrices are proved. Also, a method of determining the diagonalizabe unknown matrices is provided. The results are generalized to multi-matrix variate, matrix polynomial equations. Briefly generalization to tensor variate polynomial equations is discussed. It is hoped that the results lead to important contributions in "non-commutative algebra".

Abstract:
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical polynomials and arithmetical power series are introduced. Using these concepts, an interesting Theorem relating arithmetical power series and infinite products has been proved. Also arithmetical polynomials are related to probabilistic number theory. Furthermore some results related to the Waring problem are discussed.

Abstract:
In this research paper, it is proved that an approximation to Gibbs-Shannon entropy measure naturally leads to Tsallis entropy for the real parameter q =2 . Several interesting measures based on the input as well as output of a discrete memoryless channel are provided and some of the properties of those measures are discussed. It is expected that these results will be of utility in Information Theoretic research.

Abstract:
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 some 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.

Abstract:
In this research paper, weighted / unweighted, directed / undirected graphs are associated with interesting Discrete Time Markov Chains (DTMCs) as well as Continuous Time Markov Chains (CTMCs). The equilibrium / transient behaviour of such Markov chains is studied. Also entropy dynamics (Shannon entropy) of certain structured Markov chains is investigated. Finally certain structured graphs and the associated Markov chains are studied.

Abstract:
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. Some results related to stable states/vectors, anti-stable states/vectors (over the hypercube) are discussed. Some results related to the computation of global optimum stable state (an NP hard problem) are discussed. It is hoped that the results shed light on resolving the P \neq NP problem.

Abstract:
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.