Open Access Short Research Article

Open Access Original Research Article

Transient Solution for Single Server Machine Interference Problem with Additional Server for Long Queues under N-Policy Vacations

S. A. Ojobor, S. E. Omosigho

Journal of Advances in Mathematics and Computer Science, Page 1-15
DOI: 10.9734/BJMCS/2016/24515

We consider the transient state single server machine interference problem with additional server for long queues under N-policy vacations. There are M operating machines with two repairmen. The first repairman is always available for serving the failed machines but go on a single vacation when there are no failed machines in the system. The second repairman is always on vacation but only comes back from vacation to attend to broken down machines if there are more than or equal to N broken down machine in queue in the system (N-policy vacations). Otherwise he goes for another vacation. The number of servers available for service in this system is two. The service discipline is first in first out (FIFO). The Chapman-Kolmogorov differential equations obtained for the model is solved through ODE45 in MATLAB. The transient probabilities obtained for the model are used to compute the expected number of failed machines E[F], expected number of operating machine E[O], expected length of vacation the servers has E[V], the machine availability at time t (M.A.(t)) and variance of the number of broken down machines σ2(t) for the systems. We investigate the effect of CPU time and different parameters on the availability of the machine for the single server machine interference problem with additional server for long queues. We found that with the same service rate μ, failure rate λ and vacations length θ, as the number of failed machines that trigger repairman 2 in the system increases the variance is less than one. This is caused by the additional repairman. The additional repairman reduces the waiting time of failed machines in the system. 

Open Access Original Research Article

New Extension of Kannan and Chatterjea Fixed Point Theorems on Complete Metric Spaces

Aleksa Malčeski, Samoil Malčeski, Katerina Anevska, Risto Malčeski

Journal of Advances in Mathematics and Computer Science, Page 1-10
DOI: 10.9734/BJMCS/2016/25864

The theory of fixed point is applied in many fields of mathematics as well as in other studies. That is the main reason for producing new results, as well as for generalizing already known theorems about fixed point. This paper considers several generalizations of R. Kannan [1] and S. K. Chatterjea [2] theorems about fixed point.

Open Access Original Research Article

Construction of Picture Maze along Set of Image Dot Vertices

Tomio Kurokawa

Journal of Advances in Mathematics and Computer Science, Page 1-14
DOI: 10.9734/BJMCS/2016/26350

There are three interests in this study. First is to make an analogy between graph theory and image processing. Given a number of image items (dots) on a picture, this paper shows a way to make a path (closed path) connecting and traversing all of them but each only once, imitating to construct a path (closed path) on a plane by considering a dot as a vertex and a connecting line as an edge. This looks like constructing a Hamiltonian path on a plane and drawing a plane graph connecting all vertices with no crossing among the edges. Second is to construct the maze based on the obtained path with some detouring within each of image items (dots). This path could be very long and complex. Owing to this path, the maze is constructed in an organized manner. Third is to apply this path making procedure to different pictures with different image shapes like symbols or English letters --- possibly generating a complex picture maze. Pictures or some meaningful message appear when the maze is solved. Algorithm and its proof are given with a number of successful experimental results.

Open Access Original Research Article

A Proposed Method for Numerical Integration

Felix O. Mettle, Enoch N. B. Quaye, Louis Asiedu, Kwasi A. Darkwah

Journal of Advances in Mathematics and Computer Science, Page 1-15
DOI: 10.9734/BJMCS/2016/23048

The main objective of this paper is to propose a numerical integration method that provides improved estimates as compared to the Newton-Cotes methods of integration. The method is an extension of trapezoidal rule where after segmentation, the top part of each segment was further subdivided into rectangles and/or squares and triangles (approximate). The area of each segment is then obtained as the sum of areas of these geometric shapes and the area of the down part of the segments which is usually a rectangle. The process resulted in an improved formula for numerical integration which we derived in the paper. The proposed method was compared with some Newton-Cotes methods of integration and it outperformed. With the proposed method, one can provide estimates with predetermined desired absolute relative true errors.

Open Access Original Research Article

Formulation of Time Dependent Bloch NMR Equations for Computational Analyses of Nano Particles in Porous Media

A. A. Adeleke

Journal of Advances in Mathematics and Computer Science, Page 1-26
DOI: 10.9734/BJMCS/2016/26359

Nuclear Magnetic Resonance (NMR) has been very useful in the study of pore size distribution of porous materials and in molecular recognition. Important properties of the porous media have been shown to be very much dependent on the T1 and T2 relaxation times. The NMR transverse magnetization carries information on the pores’ properties. This has been demonstrated by many experiments on porous media but analytical expressions showing the direct relationships between the pore features and the NMR parameters have been quite scarce in literature.  In this study, formulation of time dependent Bloch NMR equation for computational analyses of nano particles in porous media has been presented. Since the nano particle is expected to be imaged in a nano-porous medium, we apply the transformation that makes the NMR transverse magnetization My (t) expressible in term of My (p) with porosity p(t) . Two new parameters which validate the transformation are properly defined in terms of the porosity, T1 and T2 relaxation parameters. The results obtained in this study can have applications in functional magnetic resonance imaging (fMRI), Petroleum exploration and well design, geological engineering and could be a frontier towards a very robust way of describing porousity and permeability in systems transporting particles of specific shape and form.

Open Access Original Research Article

Fibonacci p-codes and Codes of the “Golden” p-proportions: New Informational and Arithmetical Foundations of Computer Science and Digital Metrology for Mission-Critical Applications

Alexey Stakhov

Journal of Advances in Mathematics and Computer Science, Page 1-49
DOI: 10.9734/BJMCS/2016/25969

In the 70s and 80s years of the past century, the new positional numeral systems, called Fibonaссi p-codes and codes of the golden p-proportions, as new informational and arithmetical foundations of computer science and digital metrology, was considered as one of the most important directions of Soviet computer science and digital metrology for mission-critical applications. The main advantage of this direction was to improve the information reliability of computer and measuring systems. This direction has been patented abroad widely (more than 60 patents of US, Japan, England, France, Germany, Canada and other countries). Theoretical basis of this direction have been described in author’s books "Introduction into the Algorithmic Measurement Theory" (1977), and "Codes of the Golden Proportion" (1984). Unfortunately, these books didn’t be translated into English and, therefore, the Soviet scientific achievements in this field were virtually unknown for Western and world experts in computer science and digital metrology. Under author’s leadership, a number of interesting engineering developments have been carried out. Some of them (self-correcting 18-bit analog-to-digital and digital-to- analog converters) exceeded the world level. Unfortunately, after the collapse of the Soviet Union in 1991, government funding of these developments was stopped. However, theoretical developments in this area continued. The purpose of this article is to state the history and the main scientific and engineering achievements in this field, as one of the important direction in the improvement of informational reliability of computer and measuring systems for mission-critical applications.