# Square Difference Prime Labeling –More Results on Path Related Graphs | Chapter 12 | Advances in Mathematics and Computer Science Vol. 4

Vertices of the graph G are labeled with first p-1 whole numbers, where p is the number of vertices of the graph and edges are labeled with absolute difference of the squares of the labels of the end vertices. If the greatest common divisor of the labels of all edges incident on a vertex of degree greater than one is one then the graph admits square difference prime labeling.

Here we investigate, strong duplicate graph of path, splitting graph of path, tortoise graph of path and some more path related graphs for square difference prime labeling.

Author(s) Details

B. S. Sunoj
Department of Mathematics, Government Polytechnic College, Attingal, India.

T. K. Mathew Varkey
Department of Mathematics, TKM College of Engineering, Kollam, India.

# Proposition of a Recursive Formula to Calculate the Higher Order Derivative of a Composite Function without Using the Resolution of the Diophantine Equation | Chapter 11 | Advances in Mathematics and Computer Science Vol. 4

The formula of Fa`a Di Bruno provides a powerful tool to calculate the higher order derivative of a composite function. Unfortunately it has three weaknesses: it is not a recursive formula, it totally depends on the resolution of the diophantine equation and a change in the order of the derivative requires the total change of the calculation. With these weaknesses and the absence of a formula to program, Fa`a Di Bruno’s formula is less useful for formal computation.

Other complicated techniques based on ﬁnite diﬀerence calculation (see ) are recursive, however the complexity of the calculation algorithm is very high. There is as well some techniques based on graphs (see ) to calculate the coeﬃcients to a certain order, but without giving the general formula.

In our work we propose a new formula to calculate the higher order derivative of a composite function gof. It is of great interest, because it is recursive and it is not based on the resolution of the diophantine equation. We complete this work by giving an expression that allows to ﬁnd directly the n-th derivative of a composite function.

Author(s) Details

Dr. El Khomssi Mohammed
Modelling and Scientiﬁc Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.

Dr. Chaachoui Ghizlane
Modelling and Scientiﬁc Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.

Dr. Ez-Zriouli Rachid
Modelling and Scientiﬁc Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.

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# Reliability Measurement of Square Model to Develop IVR Software | Chapter 10 | Advances in Mathematics and Computer Science Vol. 4

Square model is a software process model to develop Interactive Voice Response (IVR) software, used in call center industry. In this article, we measured the reliability of Square Model in the presence of covariate information. For this, we derived a failure intensity function which is used to measure the reliability of model using fix number of calls, failure intensity parameter value, failures / CPU hour and mean failures observed empirically for all components to estimate failure intensity of IVR software over various time periods when it is assumed that the software is changed after each time period and that software metrics information is available after each update.

Author(s) Details

Devesh Kumar Srivastava
SCIT, Manipal University Jaipur, India.

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# Bayesian Approach to Reliability Estimation | Chapter 09 | Advances in Mathematics and Computer Science Vol. 4

In this chapter Bayes estimators of the reliability function R(t) under Type II censoring of Weibull distribution have been obtained by taking non-informative and beta prior distributions. The loss functions used are squared error, linex, precautionary and entropy.

Author(s) Details

Dr. Arun Kumar Rao
Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur, India.

Dr. Himanshu Pandey
Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur, India.

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# Image Segmentation Using N – Cut Based Graph Partitioning | Chapter 08 | Advances in Mathematics and Computer Science Vol. 4

The process of image segmentation is one of the most important steps in computer vision for image retrieval, visual summary, image-based modeling and in many other processes. The goal of segmentation is typically to locate certain objects of interest. In this paper, we have studied and investigated graph based normalized cut segmentation methods and proposed a technique for adding flexibility to the parameters for performance improvement. These methods are examined analytically and tested their performance for the standard images. The results obtained for the important metrics show that these methods perform better than others approach and are computationally efficient, and useful for precise image segmentation.

Author(s) Details

RMD Sinhgad School of Engineering, Pune – 411058, India.

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# Finding the Location of the Axes, the Vertices and the Foci of a Parabola, an Ellipse and a Hyperbola Using a Straightedge and a Compass | Chapter 07 | Advances in Mathematics and Computer Science Vol. 4

We present geometrical constructions using a straightedge and a compass in order to find the location of the axes, the vertices and the foci of a parabola, an ellipse and a hyperbola from their plots. The constructions are based on familiarity with theorems and special properties characterizing these loci, which therefore can be used for implementing and applying knowledge acquired during the studies of analytical geometry.

Author(s) Details

Moshe Stupel
Shaanan  Academic College of  Education, and Gordon Academic College of Education, Haifa, Israel.

Avi Sigler
Shaanan – Academic College of Education, Haifa, Israel.

Shula Weissman
Gordon Academic College, Haifa, Israel.

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# Improving Method of Evaluating Semantic Filtering for Human Computer Interaction in an Adaptive Collaborative Learning Environment | Chapter 06 | Advances in Mathematics and Computer Science Vol. 4

Human Computer Interaction Semantic filtering techniques are used in learning environment to track problems in collaborative systems. However, as noted in Adigun et al. , when sharing and dynamism are promoted, a problem of redundancy and integrity appeared not to have been well addressed. An improved ASF-based method of evaluating semantic filtering for social network systems in a collaborative learning environment is developed, which assisted participants to achieve greater levels of performance with information sharing from other collaborators, as well as in reusing ideas across the period of collaboration.

Author(s) Details

Department of Information and Communication Technology, Osun State University, Osogbo, Nigeria.

A. O. Osofisan
Department Computer Science, University of Ibadan, Ibadan, Nigeria.

O. Longe
Department of Computer Science, University of Ibadan, Ibadan, Nigeria.

M. O. Kolawole
Department of Electrical and Electronics Engineering, Federal University of Technology, Akure, Nigeria.

# A zeta function Computation of Casimir Energy | Chapter 05 | Advances in Mathematics and Computer Science Vol. 4

A computation of Casimir energy via spectral zeta function is considered in this Chapter. The original computations deriving the Casimir energy and force consists of first taking limits of the spectral zeta function and afterwards analytically extending the result. This process of computation presents a weakness in Hendrik Casimir’s original argument since limit and analytic continuation do not commute. A case of the Laplacian on a parallelepiped box representing the space as the vacuum between two plates modelled with Dirichlet and periodic Neumann boundary conditions is constructed to address this anomaly. It involves the derivation of the regularised zeta function in terms of the Riemann zeta function on the parallelepiped. The values of the Casimir energy and Casimir force obtained from our derivation agree with those of Hendrik Casimir.

Author(s) Details

Dr. Louis Omenyi
Department of Mathematics/Computer Science/Statistics/Informatics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria.

# Some S-degree Based Topological Indices of Silicates (SiO2) Layer | Chapter 04 | Advances in Mathematics and Computer Science Vol. 4

A topological index is a graph invariant, which is generated as a real number from a set of finite graphs. A large number of topological indices have been developed based on vertex degree, eccentricity, etc. and studied extensively. In this paper we propose a new version of the inverse sum indeg index (ISI), sum-connectivity index (SCI), redefined third Zagreb index and  (a,b)- Zagreb index based on S-degrees of vertices a graph and discuss them in connection with SiO2 layer structure in a graph theoretic perspective.

Author(s) Details

Budheswar Deka
Department of Mathematics, Dibrugarh University, Dibrugarh-786004, India.

Abhigyan Mahanta
Department of Mathematics, Dibrugarh University, Dibrugarh-786004, India.

A. Bharali
Department of Mathematics, Dibrugarh University, Dibrugarh-786004, India.

# A Queuing Model to Analyze Data Center Performances in a Cloud Computing Environment | Chapter 03 | Advances in Mathematics and Computer Science Vol. 4

In the last decades cloud computing has been the focus of a lot of research in both academic and industrial fields, however, implementation-related issues have been developed and have received more attention than performance analysis which is an important aspect of cloud computing and it is of crucial interest for both cloud providers and cloud users. Successful development of cloud computing paradigm necessitates accurate performance evaluation of cloud data centers. Because of the nature of cloud centers and the diversity of user requests, an exact modeling of cloud centers is not practicable; in this work we report an approximate analytical model based on an approximate Markov chain model for performance evaluation of a cloud computing center. Due to the nature of the cloud environment, we considered, based on queuing theory, a MMPP task arrivals, a general service time for requests as well as large number of physical servers and a finite capacity. This makes our model more flexible in terms of scalability and diversity of service time. We used this model in order to evaluate the performance analysis of cloud server farms and we solved it to obtain accurate estimation of the complete probability distribution of the request response time and other important performance indicators such as: the Mean number of Tasks in the System, the distribution of Waiting Time, the Probability of Immediate Service, the Blocking Probability and Buffer Size

Author(s) Details

Mohamed Hanini
FST, Hassan 1st University, Settat, Morocco.
IR2M laboratory, FST, Hassan 1st University, Settat, Morocco.

Fatima Oumellal
FST, Hassan 1st University, Settat, Morocco.

Abdelkrim Haqiq
FST, Hassan 1st University, Settat, Morocco.
IR2M laboratory, FST, Hassan 1st University, Settat, Morocco.