Open Access Original Research Article

O. S. Cyprian

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.

Open Access Original Research Article

Changwang Xiao

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.

Open Access Original Research Article

Yakubu Abdul-Wahab Nawusu, Alhassan Abdul-Barik, Salifu Abdul-Mumin

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.

Open Access Original Research Article

Pradeep Kothiyal

Sarason did pioneer work on the reflexivity and purpose of this paper is to discuss the reflexivity of different class of contractions. Among contractions it is now known that C_{11} contractions with finite defect indices, C._{o} contractions with unequal defect indices and C_{1}. contractions with at least one finite defect indices are reflexive. More over the characterization of reflexive operators among c_{o} contractions and completely non unitary weak contractions with finite defect indices has been reduced to that of S (F), the compression of the shift on H^{2} ⊖ F H^{2}, F is inner. The present work is mainly focused on the reflexivity of contractions whose characteristic function is constant. This class of operator include many other isometries, co-isometries and their direct sum. We shall also discuss the reflexivity of hyponormal contractions, reflexivity of C_{1}. contractions and weak contractions. It is already known that normal operators isometries, quasinormal and sub-normal operators are reflexive. We partially generalize these results by showing that certain hyponormal operators with double commutant property are reflexive. In addition, reflexivity of operators which are direct sum of a unitary operator and C._{o} contractions with unequal defect indices,is proved Each of this kind of operator is reflexive and satisfies the double commutant property with some restrictions.

Open Access Original Research Article

Paride O. Lolika, Steady Mushayabasa

To understand the effects of animal movement on transmission and control of brucellosis infection, a reaction diffusion partial differential equation (PDE) brucellosis model that incorporates wild and domesticated animals under homogeneous Neumann boundary conditions is proposed and analysed. We computed the reproductive number for the brucellosis model in the absence of spatial movement and we established that, the associated model has a globally asymptotically stable disease-free equilibrium whenever the reproductive number is less or equal to unity. However, if the reproductive number is greater than unity an endemic equilibrium point which is globally asymptotically stable exists. We performed sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. For the model with spatial movement the disease threshold is studied by using the basic reproductive number. Additionally we investigate the existence of a Turing stability and travelling waves. Our results shows that incorporating diffusive spatial spread does not produce a Turing instability when the reproductive number R_{0}^{ODE} associated with the ODE model is less than unity. Finally the results suggest that minimizing interaction between buffalo and cattle population can be essential to manage brucellosis spillover between domesticated and wildlife animals. Numerical simulations are carried out to support analytical findings.

Open Access Original Research Article

Jaykov Foukzon

In this paper intuitionistic set theory INC^{#}_{∞#} in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number e^{e} is transcendental; (ii) the both numbers e + π and e − π are irrational.

Open Access Original Research Article

Olivier Denis

Some fundamental mathematical researches have been carried out about mathematical certainties based on ancient Egyptian mathematical sources and their problems following ancient Egyptian Wisdom set of knowledge building the new scientific paradigm following the rediscovery of the true value of PI and following the new approach of Global Dimensional Mathematics [1].

Some fundamental mathematical researches on the foundations of Egyptian mathematics covering the mathematical problem of The *Akhmin wooden tablets [2]*, the tenth and the fourteenth problem of The *Moscow Mathematical Papyrus* [3] as well as the forty-first and fiftieth problem from The *Rhind* *Mathematical* *Papyrus [3] *have been carried out, without forgotten, the resolution of the fundamental question of the quadrature of the circle which is now effective.

In the disclosure of Egyptian mathematics, the new approach to fundamental mathematical notions is established, adding the cornerstone to building the core of the new approach to Egyptian mathematics, mathematics and science in general.

The Egyptian mathematics disclosure solves, following the Egyptian approach to mathematics and following ancient Egyptian Wisdom set of knowledge, unsolved ancient Egyptian mathematical problems, such as finding the complete solution and decoding the glyph of the eye of Horus, as well as the problem of the truncated pyramid which has found a solution like the half basket problem found one. The question of the quadrature of the circle shatters the mathematical conceptions with all the consequences that we can only begin to understand.

The Egyptian mathematics disclosure forms the basis for building the new scientific approach based on ancestral Egyptian mathematical problems, the true rediscovered value of PI and the new original Global Dimensional Mathematics opening up a still unknown perspective on the world of science in general.

Open Access Original Research Article

Hezron Saka Were, Maurice Oduor Owino, Moses Ndiritu Gichuki

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))^{5} = (0); (Z(R))^{4} ̸= (0) and determine the structures of the unit groups of R for all its characteristics.