**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**156

# Search results for: Pythagorean Theorem

##### 156 Fermat’s Last Theorem a Simple Demonstration

**Authors:**
Jose William Porras Ferreira

**Abstract:**

This paper presents two solutions to the Fermat’s Last Theorem (FLT). The first one using some algebraic basis related to the Pythagorean theorem, expression of equations, an analysis of their behavior, when compared with power and power and using " the “Well Ordering Principle” of natural numbers it is demonstrated that in Fermat equation . The second one solution is using the connection between and power through the Pascal’s triangle or Newton’s binomial coefficients, where de Fermat equation do not fulfill the first coefficient, then it is impossible that:

z^{n}=x^{n}+y^{n} for n>2 and (x, y, z) E Z^{+} - {0}

**Keywords:**
Fermat’s Last Theorem,
Pythagorean Theorem,
Newton Binomial Coefficients,
Pascal’s Triangle,
Well Ordering Principle.

##### 155 A Quadratic Approach for Generating Pythagorean Triples

**Authors:**
P. K. Rahul Krishna,
S. Sandeep Kumar,
Jayanthi Sunder Raj

**Abstract:**

The article explores one of the important relations between numbers-the Pythagorean triples (triplets) which finds its application in distance measurement, construction of roads, towers, buildings and wherever Pythagoras theorem finds its application. The Pythagorean triples are numbers, that satisfy the condition “In a given set of three natural numbers, the sum of squares of two natural numbers is equal to the square of the other natural number”. There are numerous methods and equations to obtain the triplets, which have their own merits and demerits. Here, quadratic approach for generating triples uses the hypotenuse leg difference method. The advantage is that variables are few and finally only three independent variables are present.

**Keywords:**
Arithmetic progression,
hypotenuse leg difference method,
natural numbers,
Pythagorean triplets,
quadratic equation.

##### 154 Application of Rapidly Exploring Random Tree Star-Smart and G2 Quintic Pythagorean Hodograph Curves to the UAV Path Planning Problem

**Authors:**
Luiz G. Véras,
Felipe L. Medeiros,
Lamartine F. Guimarães

**Abstract:**

**Keywords:**
Path planning,
path smoothing,
Pythagorean
hodograph curve,
RRT*-Smart.

##### 153 Pythagorean-Platonic Lattice Method for Finding all Co-Prime Right Angle Triangles

**Authors:**
Anthony Overmars,
Sitalakshmi Venkatraman

**Abstract:**

This paper presents a method for determining all of the co-prime right angle triangles in the Euclidean field by looking at the intersection of the Pythagorean and Platonic right angle triangles and the corresponding lattice that this produces. The co-prime properties of each lattice point representing a unique right angle triangle are then considered. This paper proposes a conjunction between these two ancient disparaging theorists. This work has wide applications in information security where cryptography involves improved ways of finding tuples of prime numbers for secure communication systems. In particular, this paper has direct impact in enhancing the encryption and decryption algorithms in cryptography.

**Keywords:**
Pythagorean triples,
platonic triples,
right angle triangles,
co-prime numbers,
cryptography.

##### 152 On the Maximum Theorem: A Constructive Analysis

**Authors:**
Yasuhito Tanaka

**Abstract:**

**Keywords:**
Maximum theorem,
Constructive mathematics,
Sequentially
locally at most one maximum.

##### 151 Power Series Form for Solving Linear Fredholm Integral Equations of Second Order via Banach Fixed Point Theorem

**Authors:**
Adil AL-Rammahi

**Abstract:**

In this paper, a new method for solution of second order linear Fredholm integral equation in power series form was studied. The result is obtained by using Banach fixed point theorem.

**Keywords:**
Fredholm integral equation,
power series,
Banach fixed point theorem,
Linear Systems.

##### 150 Positive Solutions for Semipositone Discrete Eigenvalue Problems via Three Critical Points Theorem

**Authors:**
Benshi Zhu

**Abstract:**

**Keywords:**
Discrete eigenvalue problems,
positive solutions,
semipositone,
three critical points theorem

##### 149 Constructive Proof of Tychonoff’s Fixed Point Theorem for Sequentially Locally Non-Constant Functions

**Authors:**
Yasuhito Tanaka

**Abstract:**

We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions.

**Keywords:**
sequentially locally non-constant functions,
Tychonoff’s
fixed point theorem,
constructive mathematics.

##### 148 Weyl Type Theorem and the Fuglede Property

**Authors:**
M. H. M. Rashid

**Abstract:**

**Keywords:**
Fuglede Property,
Weyl’s theorem,
generalized
derivation,
Aluthge Transformation.

##### 147 Extremal Properties of Generalized Class of Close-to-convex Functions

**Authors:**
Norlyda Mohamed,
Daud Mohamad,
Shaharuddin Cik Soh

**Abstract:**

**Keywords:**
Argument of f ′(z) ,
Carathéodory Function,
Closeto-
convex Function,
Distortion Theorem,
Extremal Properties

##### 146 A Constructive Proof of the General Brouwer Fixed Point Theorem and Related Computational Results in General Non-Convex sets

**Authors:**
Menglong Su,
Shaoyun Shi,
Qing Xu

**Abstract:**

In this paper, by introducing twice continuously differentiable mappings, we develop an interior path following following method, which enables us to give a constructive proof of the general Brouwer fixed point theorem and thus to solve fixed point problems in a class of non-convex sets. Under suitable conditions, a smooth path can be proven to exist. This can lead to an implementable globally convergent algorithm. Several numerical examples are given to illustrate the results of this paper.

**Keywords:**
interior path following method,
general Brouwer fixed
point theorem,
non-convex sets,
globally convergent algorithm

##### 145 A Sandwich-type Theorem with Applications to Univalent Functions

**Authors:**
Sukhwinder Singh Billing,
Sushma Gupta,
Sukhjit Singh Dhaliwal

**Abstract:**

**Keywords:**
Univalent function,
Starlike function,
Differential subordination,
Differential superordination.

##### 144 On the Central Limit Theorems for Forward and Backward Martingales

**Authors:**
Yilun Shang

**Abstract:**

**Keywords:**
central limit theorem,
martingale difference sequence,
backward martingale.

##### 143 Cryptographic Attack on Lucas Based Cryptosystems Using Chinese Remainder Theorem

**Authors:**
Tze Jin Wong,
Lee Feng Koo,
Pang Hung Yiu

**Abstract:**

_{4,6}) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC

_{3}cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC

_{4,6}cryptosystem than LUC

_{3}and LUC cryptosystems. Current study concludes that LUC

_{4,6}cryptosystem is more secure than LUC and LUC

_{3}cryptosystems in sustaining against Lenstra’s attack.

**Keywords:**
Lucas sequence,
Dickson Polynomial,
faulty signature,
corresponding signature,
congruence.

##### 142 Reduction of Search Space by Applying Controlled Genetic Operators for Weight Constrained Shortest Path Problem

**Authors:**
A.K.M. Khaled Ahsan Talukder,
Taibun Nessa,
Kaushik Roy

**Abstract:**

**Keywords:**
Genetic Algorithm,
Evolutionary Optimization,
Multi Objective Optimization,
Non-linear Schema Theorem,
WCSPP.

##### 141 Recursive Wiener-Khintchine Theorem

**Authors:**
Khalid M. Aamir,
Mohammad A. Maud

**Abstract:**

Power Spectral Density (PSD) computed by taking the Fourier transform of auto-correlation functions (Wiener-Khintchine Theorem) gives better result, in case of noisy data, as compared to the Periodogram approach. However, the computational complexity of Wiener-Khintchine approach is more than that of the Periodogram approach. For the computation of short time Fourier transform (STFT), this problem becomes even more prominent where computation of PSD is required after every shift in the window under analysis. In this paper, recursive version of the Wiener-Khintchine theorem has been derived by using the sliding DFT approach meant for computation of STFT. The computational complexity of the proposed recursive Wiener-Khintchine algorithm, for a window size of N, is O(N).

**Keywords:**
Power Spectral Density (PSD),
Wiener-KhintchineTheorem,
Periodogram,
Short Time Fourier Transform (STFT),
TheSliding DFT.

##### 140 Some Applications of Gröbner bases

**Authors:**
Hassan Noori,
Abdolali Basiri,
Sajjad Rahmany

**Abstract:**

**Keywords:**
Gr¨obner bases,
Application of Gr¨obner bases,
Automatic
Geometric Theorem Proving,
Graph Coloring,
Cryptography.

##### 139 Quadrilateral Decomposition by Two-Ear Property Resulting in CAD Segmentation

**Authors:**
Maharavo Randrianarivony

**Abstract:**

**Keywords:**
Quadrangulation,
simply connected,
two-ear theorem.

##### 138 Existence of Positive Solutions for Second-Order Difference Equation with Discrete Boundary Value Problem

**Authors:**
Thanin Sitthiwirattham,
Jiraporn Reunsumrit

**Abstract:**

We study the existence of positive solutions to the three points difference-summation boundary value problem. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem due to Krasnoselskii in cones.

**Keywords:**
Positive solution,
Boundary value problem,
Fixed
point theorem,
Cone.

##### 137 Periodic Solutions for a Third-order p-Laplacian Functional Differential Equation

**Authors:**
Yanling Zhu,
Kai Wang

**Abstract:**

By means of Mawhin’s continuation theorem, we study a kind of third-order p-Laplacian functional differential equation with distributed delay in the form: ϕp(x (t)) = g t, 0 −τ x(t + s) dα(s) + e(t), some criteria to guarantee the existence of periodic solutions are obtained.

**Keywords:**
p–Laplacian,
distributed delay,
periodic solution,
Mawhin's continuation theorem.

##### 136 Lagrange-s Inversion Theorem and Infiltration

**Authors:**
Pushpa N. Rathie,
Prabhata K. Swamee,
André L. B. Cavalcante,
Luan Carlos de S. M. Ozelim

**Abstract:**

**Keywords:**
Green-Ampt Equation,
Lagrange's Inversion
Theorem,
Talsma-Parlange Equation,
Three-Parameter Infiltration
Equation

##### 135 Multiple Positive Periodic Solutions to a Predator-prey system with Harvesting Terms and Holling II Type Functional Response

**Authors:**
Pan Wang,
Yongkun Li

**Abstract:**

In this paper, a periodic predator-prey system with harvesting terms and Holling II type functional response is considered. Sufficient criteria for the existence of at least sixteen periodic solutions are established by using the well known continuation theorem due to Mawhin. An example is given to illustrate the main result.

**Keywords:**
Periodic solution,
predator-prey system,
harvesting terms,
continuation theorem.

##### 134 A new Configurable Decimation Filter using Pascal-s Triangle Theorem

**Authors:**
A. Chahardah Cherik,
E. Farshidi

**Abstract:**

**Keywords:**
Decimation filter,
sigma delta,
Pascal's triangle'stheorem,
memory

##### 133 PI Control for Second Order Delay System with Tuning Parameter Optimization

**Authors:**
R. Farkh,
K. Laabidi,
M. Ksouri

**Abstract:**

**Keywords:**
Genetic algorithm,
Hermit-Biehler theorem,
optimization,
PI controller,
second order delay system,
stability region.

##### 132 The Positive Solution for Singular Eigenvalue Problem of One-dimensional p-Laplace Operator

**Authors:**
Lv Yuhua

**Abstract:**

In this paper, by constructing a special cone and using fixed point theorem and fixed point index theorem of cone, we get the existence of positive solution for a class of singular eigenvalue value problems with p-Laplace operator, which improved and generalized the result of related paper.

**Keywords:**
Cone,
fixed point index,
eigenvalue problem,
p-Laplace operator,
positive solutions.

##### 131 Simplified Equations for Rigidity and Lateral Deflection for Reinforced Concrete Cantilever Shear Walls

**Authors:**
Anas M. Fares

**Abstract:**

**Keywords:**
Cantilever shear walls,
flexural deformation,
lateral deflection,
lateral loads,
reinforced concrete shear walls,
rigidity,
shear deformation,
virtual work theorem.

##### 130 The Boundary Theory between Laminar and Turbulent Flows

**Authors:**
Tomasz M. Jankowski

**Abstract:**

**Keywords:**
Freed gravitons,
free gravitons.

##### 129 Analysis of Permanence and Extinction of Enterprise Cluster Based On Ecology Theory

**Authors:**
Ping Liu,
Yongkun Li

**Abstract:**

This paper is concerned with the permanence and extinction problem of enterprises cluster constituted by m satellite enterprises and a dominant enterprise. We present the model involving impulsive effect based on ecology theory, which effectively describe the competition and cooperation of enterprises cluster in real economic environment. Applying comparison theorem of impulsive differential equation, we establish sufficient conditions which ultimately affect the fate of enterprises: permanence, extinction, and co-existence. Finally, we present numerical examples to explain the economical significance of mathematical results.

**Keywords:**
Enterprise cluster,
permanence,
extinction,
impulsive,
comparison theorem.

##### 128 A Sum Operator Method for Unique Positive Solution to a Class of Boundary Value Problem of Nonlinear Fractional Differential Equation

**Authors:**
Fengxia Zheng,
Chuanyun Gu

**Abstract:**

**Keywords:**
Fractional differential equation,
Boundary value
problem,
Positive solution,
Existence and uniqueness,
Fixed point
theorem of a sum operator.

##### 127 Existence of Solutions for a Nonlinear Fractional Differential Equation with Integral Boundary Condition

**Abstract:**

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form: Dαt x(t) = f(t, x(t),Dβ t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle.

**Keywords:**
Fractional differential equation,
Integral boundary condition,
Schauder fixed point theorem,
Banach contraction principle.