Analytical Approach to Obtain Some New Traveling Wave Solutions of Coupled Systems of Nonlinear Equations | Chapter 11 | Advances in Mathematics and Computer Science Vol. 2

The second order nonlinear ordinary differential equation is executed as an auxiliary equation. For illustration, a new extension of so called (G’/G) method is considered to investigate the generalized Hirota-Satsuma coupled KdV equations for producing some new analytical solutions. The obtained solutions belong to hyperbolic functions, trigonometric functions and rational forms which show the wider applicability of this new extended method for handling other nonlinear evolution equations. The numerical results are also described in the figures.

Author(s) Details

Dr. Hasibun Naher
Department of Mathematics and Natural Sciences, BRAC University, 66 Mohakhali, Dhaka 1212, Bangladesh.

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Translation of Weighted Voting Concepts to the Boolean Domain: The Case of the Banzhaf Index | Chapter 10 | Advances in Mathematics and Computer Science Vol. 2

This chapter is a tutorial exposition on how to translate concepts of voting systems to the Boolean domain, and consequently on how to use Boolean tools in the computation of a prominent index of voting powers, viz., the Banzhaf voting index. We discuss Boolean representations for yes-no voting systems, in general, and for weighted voting systems, in particular. Our main observation is that non-minimal winning coalitions are related to minimal ones via partial-order structures and also as particular subordinate loops that cover the all-1 cell in the Karnaugh map. We review the method of computing the total Banzhaf indices by the Conventional Karnaugh Map (CKM). Then we extend this method to handle larger problems via the Variable-Entered Karnaugh Map (VEKM). The map methods are demonstrated by two classical weighted voting systems.

Author(s) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah 21589, Saudi Arabia.

Omar Mohammed Ba-Rukab
Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, P.O.Box 344, Rabigh 21911, Saudi Arabia.

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Augmented Lagrangian Method for One Dimensional Optimal Control Problems Governed by Delay Differential Equation | Chapter 09 | Advances in Mathematics and Computer Science Vol. 2

In this research, numerical solutions of continuous optimal control problems governed by linear damping evolution with delay and real coefficients are presented. The necessary conditions obtained from the knowledge of calculus of variation for optimal control problem constrained by delay differential equation is a linear two-point boundary value problem involving both delay and advance terms. Clearly, this coupling that exists between the state variable and the control variable is not amenable to analytical solution hence a direct numerical approach is adopted. We propose an augmented discretized continuous algorithm via quadratic programming, which is capable of handling optimal control problems constrained by delay differential equations. The discretization of the problem using trapezoidal rule (a one-step second order numerical scheme) and Crank-Nicholson with quadratic formulation amenable to quadratic programming technique for solution of the optimal control problems are considered. A control operator (penalized matrix), through the augmented Lagrangian method, is constructed. Important properties of the operator as regards sequential quadratic programming techniques for determining the optimal point are shown..

Author(s) Details

O. C. Akeremale
Department of Mathematics, Federal University Lafia, P.M.B. 146, Lafia, Nasarawa State, Nigeria.

Dr. O. Olotu
Department of Mathematical Sciences, Federal University of Technology, P.M.B. 704, Akure, Ondo State, Nigeria.

A. Olaiju
Department of Mathematics and  Statistics, Federal Polytechnics, Ilaro, P.M.B. 50, Ogun State, Nigeria.

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Nontrivial Weak Solutions of a Quasilinear Equation Involving p-Laplace Operator | Chapter 08 | Advances in Mathematics and Computer Science Vol. 2

In the present paper, using direct variational approach, and the monotone operator method, theexistence of nontrivial solutions for a quasilinear elliptic equation involving the p-Laplacian is obtained.

Author(s) Details

Mustafa Avci
Department of Science, Grande Prairie Regional College, Grande Prairie, AB, Canada.

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Performance Evaluation for Local User Terminals Dedicated for Search and Recues Services from LEO Satellites | Chapter 07 | Advances in Mathematics and Computer Science Vol. 2

Aims: NOAA’s (National Oceanic and Atmospheric Administration) – LEO environmental satellites provide continuous coverage of Earth, supplying high-resolution global meteorological, oceanic and space observation data. These satellites are part of the international Search and Rescue Satellite Aided Tracking (SARSAT) system. SARSAT is designed to provide distress alert and location data in order to assist on search and rescue operation. SARSAT system detects and locates distress beacons (406MHz) activated at distress location. System calculates the distress event location using Doppler processing techniques. Each satellite pass transmits information about distress location. Passes with too short communication duration, are considered as missed passes. Communication duration analysis, among SARSAT satellites and local user terminal (LUT) dedicated for search and rescue services are provided. LUT implementation process and the mask record are also given.

Study Design:  Simulation.

Place and Duration of Study: NOAA Satellite Operations Facility, Suitland, MD, USA, October, 28 – December 24, 2009. 

Methodology: Communication analysis, are based on the hypothetical terminal assumed to be implemented in Kosovo (LUTKOS).  Four hypothetical beacons indicating random distress locations are considered. Satellite orbital altitude of 860 km, orbital time of 102 minutes and inclination of 98.7º. Communication duration is considered for period 1-30, October 2009. 57560 satellite passes were considered to conclude about missed passes.

Results: For Doppler processing at least four events are required. Duration of 250s is considered as the lower margin, providing at least four Doppler events. The highest events density was in between 300s to 700s, sufficient for distress location determination. Only 3% of total considered passes were below 250s. The ratio of missed passes over total passes for the whole ground segment results as 0.17%, or in average 0.021% per month. 

Conclusion: Through LUTKOS simulation, it is confirmed communication reliability and proper functionality of LUTKOS with a single SARSAT satellite.

Author(s) Details

Dr. Shkelzen Cakaj
Post and Telecommunication of Kosovo, Prishtina, Kosovo.
Faculty of Electrical and Computing Engineering, University of Prishtina, Kosovo.
Faculty of Information Technology, Polytechnic University of Tirana, Albania.

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Computations of the Eight Basic Measures in Diagnostic Testing | Chapter 06 | Advances in Mathematics and Computer Science Vol. 2

Diagnostic testing concerning categorical or dichotomized variables is ubiquitous in many fields including, in particular, the field of clinical or epidemiological testing. Typically, results are aggregated in two-by-two contingency-table format, from which a surprisingly huge number of indicators or measures are obtained. In this chapter, we study the eight most prominent such measures, using their medical context. Each of these measures is given as a conditional probability as well as a quotient of certain natural frequencies. Despite its fundamental theoretical importance, the conditional probability interpretation does not seem to be appealing to medical students and practitioners. This paper attempts a partial remedy of this situation by visually representing conditional probability formulas first in terms of two-variable Karnaugh maps and later in terms of simplified acyclic (Mason) Signal Flow Graph (SFGs), resembling those used in digital communications or DNA replication. These graphs can be used, among other things, as parallels to trinomial graphs that function as a generative model for the ternary problems of conditional probabilities, which were earlier envisioned by Pedro Huerta and coworkers. The arithmetic or algebraic reading or solving of a typical conditional-probability problem is facilitated and guided by embedding the problem on the SFG that parallels a trinomial graph. Potential extensions of this work include utilization of more powerful features of SFGs, interrelations with Bayesian Networks, and reformulation via Boolean-based probability methods.

Author(s) Details

Ali Muhammad Ali Rushdi
Department of  Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.

Fayez Ahmad Talmees
Department of  Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.

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Analytical Modeling of the Thermoelectric Effect in Photovoltaic Cells: Combined Solar Photovoltaic and Thermoelectric Generator System (PV+TEG) | Chapter 05 | Advances in Mathematics and Computer Science Vol. 2

Aims: Analytical modeling of the combined systems photovoltaic-thermoelectric (PV + TEG). The advantage of these systems is double:

  • On the one hand, they allow to cool the photovoltaic cells (PV), which avoids the loss of electrical efficiency observed in the devices,
  • On the other hand, recover this lost energy in the form of heat, and transform it into electrical energy thanks to the thermoelectric modules operating in Seebeck mode.

Study Design: Laboratory of Radiation Physics LPR, FAST-UAC, 01 BP 526, Cotonou, Benin. Department of Physics (FAST) and Doctoral Formation Materials Science (FDSM), University of Abomey-Calavi, Benin.

Methodology: We considered the temperature distribution in the semiconductor plate of the Thermoelectric Generator System (TEG). We resolved the thermal conductivity equation described by:

Where  a^2 is the thermal diffusivity, Q(x, y, z) is the heat flow going from the PV to the TEG module which is dissipated through the latter; using constants variation method. We assumed that the temperature along the y-axis is considered uniform.

Results: The results obtained show that, the temperature distribution in the form of a traveling wave is maintained by external heating. This depends on both the hot and cold side temperature and the temperature span.

Conclusion: The heat flux available at the hot side of the TEG is assumed to be what remains of the absorbed radiation of the PV power production.

Author(s) Details

Géraud F. Hounkpatin
Département de Physique (FAST) et Formation Doctorale Sciences des Matériaux (FDSM), Université d’Abomey-Calavi, Bénin.
Laboratoire de Physique du Rayonnement LPR, FAST-UAC, 01 BP 526 Cotonou, Bénin.

Macaire Agbomahéna
Laboratoire de Physique du Rayonnement LPR, FAST-UAC, 01 BP 526 Cotonou, Bénin.
Laboratoire de Caractérisation Thermophysique des Matériaux et Appropriation Energétique (Labo CTMAE/EPAC/UAC), Abomey-Calavi, Bénin.

Basile B. Kounouhéwa
Département de Physique (FAST) et Formation Doctorale Sciences des Matériaux (FDSM), Université d’Abomey-Calavi, Bénin.
Laboratoire de Physique du Rayonnement LPR, FAST-UAC, 01 BP 526 Cotonou, Bénin.
Centre Béninois de la Recherche Scientifique et Technique (CBRST), 03 BP 1665 Cotonou, Bénin.

Vianou I. Madogni
Département de Physique (FAST) et Formation Doctorale Sciences des Matériaux (FDSM), Université d’Abomey-Calavi, Bénin.
Laboratoire de Physique du Rayonnement LPR, FAST-UAC, 01 BP 526 Cotonou, Bénin.

Antoine Vianou
Laboratoire de Caractérisation Thermophysique des Matériaux et Appropriation Energétique (Labo CTMAE/EPAC/UAC), Abomey-Calavi, Bénin.

Cossi N. Awanou
Département de Physique (FAST) et Formation Doctorale Sciences des Matériaux (FDSM), Université d’Abomey-Calavi, Bénin.
Laboratoire de Physique du Rayonnement LPR, FAST-UAC, 01 BP 526 Cotonou, Bénin.

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Relations between Various Magnitudes Which Serve as Signs of a Right-angled Triangle | Chapter 04 | Advances in Mathematics and Computer Science Vol. 2

Relations between different magnitudes in a triangle, such as the radius of the circumcircle of the triangle, the radius of the incircle of the triangle, the radii of the excircles the triangle (which are tangent to one side from the outside and the continuations of the two other sides), as well as their relations with the side lengths the triangle and certain segments in the triangle, are a fascinating subject. We present a proven scheme of 15 relations between the sides of a triangle, the radii of incircles and excircles of the triangle, the area and half the perimeter of the triangle, which serve as signs of a right-angled triangle, together with the 16th relation which is the Pythagorean Theorem. The relations are investigated dynamically using the computerized software.

Author(s) Details

Prof. Moshe Stupel
“Gordon” and “Shaanan”-Academic College, Haifa, Israel and “Shaanan”-Academic College, Haifa, Israel.

Dr. Avi Sigler
“Shaanan”-Academic College, Haifa, Israel.

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Some Geometric Properties of a Generalized Difference Sequence Space Involving Lacunary Sequence | Chapter 03 | Advances in Mathematics and Computer Science Vol. 2

In this paper, we define a new generalized difference sequence space l(p, θ, ∆m, s) involving lacunarysequence where p = (pr) is a bounded sequence of positive real numbers with pr >1 for all r ∈ N and s ≥ 0.  Then, we examine the uniform Opial property, k-NUC property and Banach-Saksproperty of type p for this space.

Author(s) Details

Sushomita Mohanta
Department of Mathematics, Utkal University, Vani-Vihar, Bhubaneswar-751004, India.

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Comparative Analysis of Four Heuristic Functions that Optimizes the A* Search Algorithm | Chapter 02 | Advances in Mathematics and Computer Science Vol. 2

In  this  chapter are compared  four  heuristic  functions  with  high  efficiency  for an optimum solving  of  8-puzzle. The analysis is realized among Chebyshev distance, Hamming distance and Manhattan distance using A* search algorithm  implemented in  Java.  The  two  heuristic  functions defined  using  Chebyshev  distance  are  more informed than Hamming and Manhattan heuristics. This chapter also presents necessary stages in object oriented development  of  an  interactive  software dedicated to  simulate  the  A*  search  algorithm  for  8-puzzle.  The modelling  of  software  is achieved through  specific  UML  diagrams representing phases  of analysis,  design and implementation, the system thus being described in a clear and practical manner. In order to confirm that second Chebyshev  heuristic  is  more  efficient  was  used space  complexity  performance  criteria.  The  space  complexity was measured  by number  of generated  nodes from search tree, by  number  of  expanded  nodes  and by effective branching factor. From experimental results obtained by using second Chebyshev heuristic, improvements were observed  regarding  space  complexity  of  A* algorithm  versus  Hamming,  Manhattan  and  first  Chebyshev heuristics. Analysing the results presented in graphics, it can be asserted that number of steps made for obtaining the  solution  is  the  same  for  similar  configurations,  determining  the optimal  solutions  for  all  four  examined heuristics. But, investigating generated nodes number in the search tree associated with the A* algorithm using second Chebyshev  heuristic,  it  can  be  observed  that  this  number is  strictly  less  than number  obtain  by  using other  three  heuristics.  Calculating  approximately  effective branching  factor  for  all  four  heuristics,  there  were obtained values illustrated in figures. The values of b* appropriate to function hC2 are more appropriate to value 1 than values of b* appropriate to functions hM, hH, and hC1, so the A* algorithm using hC2 heuristic drives to an optimal  solution  in  a  way  that  appears  to  be  linear. According  to  these  experimental  values,  results  the superiority  of  second Chebyshev  heuristic  from  Manhattan,  Hamming  and  first  Chebyshev  heuristics.  In this case we can say that hC2 heuristic dominates hM, hH, and hC1 heuristics, from space complexity point of view.

Author(s) Details

Anca-Elena Iordan
Department of Electrical Engineering and Industrial Informatics, Engineering Faculty of Hunedoara, Polytechnic University Timisoara, Romania.

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