Advances in Mathematics and Computer Science Vol. 2

Weierstrass’s Global Division Theorem and Continuity of Linear Operators in H-spaces

Eugeny Smirnov — 2019-09-19

We introduce here new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Author has introduced this concepts in 2002 but progress in different areas of mathematics (algebraic geometry, differential equations, category theory, ets) defined the need to expand fundamental concepts. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Fréchet spaces and is stable under the operations of forming countable inductive and projective limits, closed subspaces and factor-spaces. Besides, for H-space the strengthened variant of the closed graph theorem holds true. In the present article generalization of Weierstrass’s preparation theorem and the division theorem for germs of holomorphic functions at a point of n-dimensional complex space are considered. The author formulates the global theorem about division in terms of existence and continuity of the linear operator.

Comparative Analysis of Four Heuristic Functions that Optimizes the A* Search Algorithm

Anca-Elena Iordan — 2019-09-19

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 h_C2 are more appropriate to value 1 than values of b* appropriate to functions h_M, h_H, and hC1, so the A* algorithm using h_C2 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 h_C2 heuristic dominates h_M, h_H, and h_C1 heuristics, from space complexity point of view.

Some Geometric Properties of a Generalized Difference Sequence Space Involving Lacunary Sequence

Sushomita Mohanta — 2019-09-19

In this paper, we deﬁne a new generalized diﬀerence sequence space involving lacunary sequence where is a bounded sequence of positive real numbers with and Then, we examine the uniform Opial property, k-NUC property and Banach-Saks property of type p for this space.

Relations between Various Magnitudes Which Serve as Signs of a Right-angled Triangle

Moshe Stupel — 2019-09-19

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 16^th relation which is the Pythagorean Theorem. The relations are investigated dynamically using the computerized software.

Analytical Modeling of the Thermoelectric Effect in Photovoltaic Cells: Combined Solar Photovoltaic and Thermoelectric Generator System (PV+TEG)

Géraud F. Hounkpatin — 2019-09-19

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 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.

Computations of the Eight Basic Measures in Diagnostic Testing

Ali Muhammad Ali Rushdi — 2019-09-19

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.

Performance Evaluation for Local User Terminals Dedicated for Search and Rescue Services from LEO Satellites

Shkelzen Cakaj — 2019-09-19

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.

Nontrivial Weak Solutions of a Quasilinear Equation Involving p-Laplace Operator

Mustafa Avci — 2019-09-19

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

Augmented Lagrangian Method for One Dimensional Optimal Control Problems Governed by Delay Differential Equation

O. C. Akeremale — 2019-09-19

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.

Translation of Weighted Voting Concepts to the Boolean Domain: The Case of the Banzhaf Index

Ali Muhammad Ali Rushdi — 2019-09-19

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.

Analytical Approach to Obtain Some New Traveling Wave Solutions of Coupled Systems of Nonlinear Equations

Hasibun Naher — 2019-09-19

The second order nonlinear ordinary differential equation is executed as an auxiliary equation. For illustration, a new extension of so called 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.