Journal of Advances in Mathematics and Computer Science

On Relation between the Joint Essential Spectrum and the Joint Essential Numerical Range of Aluthge Transform

2021-09-16T02:14:03+00:00

Associated with every commuting m-tuples of operators on a complex Hilbert space X is its Aluthge transform. In this paper we show that every commuting m-tuples of operators on a complex Hilbert space X and its Aluthge transform have the same joint essential spectrum. Further, it is shown that the joint essential spectrum of Aluthge transform is contained in the joint essential numerical range of Aluthge transform.

Nonexistence of Global Solutions to A Semilinear Wave Equation with Scale Invariant Damping

2021-09-20T03:14:49+00:00

We obtain a blowup result for solutions to a semilinear wave equation with scale-invariant dissipation. We perform a change of variables that transforms our starting equation into a Generalized Tricomi equation, then apply Kato’s lemma, we can prove a blowup result for solutions to the transformed equation under some assumptions on the initial data. In the critical case, we use the fundamental solutions of the Generalized Tricomi equation to modify Kato’s lemma to deal with it.

A New Approach to Detecting and Correcting Single and Multiple Errors in Wireless Sensor Networks

2021-09-22T15:22:42+00:00

Transmission errors are commonplace in communication systems. Wireless sensor networks like many other communication systems are susceptible to various forms of errors arising from sheer noise, heat and interference in sensor circuitry and from other forms of distortions. Research efforts in WSN have attempted to guarantee reliable and accurate data transmission from a target environment in the midst of these unwanted exposures. Many techniques have appeared and employed over the years to deal with the issue of transmission errors in communication systems. In this paper we present a new approach for single and multiple error control in WSN relying on the inherent fault tolerant feature of the Redundant Residue Number System. As an off shoot of Residue Number System, RRNS's fault tolerant capabilities help in building robust systems required for reliable data transmission in WSN systems. The Chinese Remainder Theorem and the Manhattan Distance Heuristics are used during the integer recovery process when detecting and correcting error digit(s) in a transmitted data. The proposed method performs considerably better in terms of data retrieval time than similar approaches by needing a smaller number of iterations to recover an originally transmitted data from its erroneous form. The approach in this work is also less computationally intensive compared to recent techniques during the error correction steps. Evidence of utility of the technique is illustrated in numerical examples.