MTH232 - National Open University of Nigeria

... In unit 1 of MTH 112 Differentialcalculus, you have learnt that if a relation y = y(x) involving two variables x and y exist then we call x the independent variable and y the dependent variable. Further, suppose we are given a relation of the type f(x,t 1, t2, ……., tn) = 0 involving (n + 1) variable ...

... In unit 1 of MTH 112 Differentialcalculus, you have learnt that if a relation y = y(x) involving two variables x and y exist then we call x the independent variable and y the dependent variable. Further, suppose we are given a relation of the type f(x,t 1, t2, ……., tn) = 0 involving (n + 1) variable ...

Introduction to Lie Groups

... theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and analysis. We focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of ...

... theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and analysis. We focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of ...

differential equations and linear algebra manual

... Solution (a) Since this equation solves a sin y term, it is not linear in y. (b) and (c) Since these equations have no nonlinear terms in y, they are linear. 11 The product rule gives what derivative for et e−t ? This function is constant. At t = 0 this constant is 1. Then et e−t = 1 for all t. Solu ...

... Solution (a) Since this equation solves a sin y term, it is not linear in y. (b) and (c) Since these equations have no nonlinear terms in y, they are linear. 11 The product rule gives what derivative for et e−t ? This function is constant. At t = 0 this constant is 1. Then et e−t = 1 for all t. Solu ...

MODULAR ARITHMETIC 1. Introduction

... In the real numbers, every nonzero number has a multiplicative inverse. This is not generally true in modular arithmetic: if a 6≡ 0 mod m it need not follow that we can solve ax ≡ 1 mod m. (For instance, 4 6≡ 0 mod 6 and 4 mod 6 has no multiplicative inverse.) The correct test for invertibility in Z ...

... In the real numbers, every nonzero number has a multiplicative inverse. This is not generally true in modular arithmetic: if a 6≡ 0 mod m it need not follow that we can solve ax ≡ 1 mod m. (For instance, 4 6≡ 0 mod 6 and 4 mod 6 has no multiplicative inverse.) The correct test for invertibility in Z ...

Mathematics of Cryptography

... answers in which both x and y are nonnegative integers). We divide both sides by 5 to get 4x + y = 20. We then solve the equation 4s + t = 1. We can find s = 0 and t = 1 using the extended Euclidean algorithm. The particular solutions are x0 = 0 × 20 = 0 and y0 = 1 × 20 = 20. The general solutions w ...

... answers in which both x and y are nonnegative integers). We divide both sides by 5 to get 4x + y = 20. We then solve the equation 4s + t = 1. We can find s = 0 and t = 1 using the extended Euclidean algorithm. The particular solutions are x0 = 0 × 20 = 0 and y0 = 1 × 20 = 20. The general solutions w ...

(pdf)

... Observation 4.4. Let A be an adjacency matrix of a labeled graph G, with n vertices. Then the entry aij of Ak , where k ≥ 1, is the number of walks of length k from vertex j to vertex i. Proof by Induction. The statement holds for k = 1 by the definition of an adjacency matrix. Assume the statement ...

... Observation 4.4. Let A be an adjacency matrix of a labeled graph G, with n vertices. Then the entry aij of Ak , where k ≥ 1, is the number of walks of length k from vertex j to vertex i. Proof by Induction. The statement holds for k = 1 by the definition of an adjacency matrix. Assume the statement ...

2y + 4x – 6 = 0 2y = -4x + 6 Y = -2x + 3 b

... B belongs to (BC) → yB = -3/5 xB + b 3 = -3/5(-3) + b → b= 6/5 → (BC): y = -3/5 x + 6/5 → (x 5) → 5y + 3x = 6 (CD): y = ax + b a(CD) = (yD – yC) / (xD – xC) = (3 – 0) / (7 -2) = 3/5 C belongs to (CD) → yC = 3/5 xC + b 0 = 3/5(2) + b 0 – 6/5 = b → b = - 6/5 → (CD): y = 3/5 x - 6/5 → (x 5) → 3x – 5y = ...

... B belongs to (BC) → yB = -3/5 xB + b 3 = -3/5(-3) + b → b= 6/5 → (BC): y = -3/5 x + 6/5 → (x 5) → 5y + 3x = 6 (CD): y = ax + b a(CD) = (yD – yC) / (xD – xC) = (3 – 0) / (7 -2) = 3/5 C belongs to (CD) → yC = 3/5 xC + b 0 = 3/5(2) + b 0 – 6/5 = b → b = - 6/5 → (CD): y = 3/5 x - 6/5 → (x 5) → 3x – 5y = ...

Trace of Positive Integer Power of Real 2 × 2 Matrices

... Traces of powers of matrices arise in several fields of mathematics, more specifically, Network Analysis, Numbertheory, Dynamical systems, Matrix theory, and Differential equations [1]. When analyzing a complex network, an important problem is to compute the total number of triangles of a connected ...

... Traces of powers of matrices arise in several fields of mathematics, more specifically, Network Analysis, Numbertheory, Dynamical systems, Matrix theory, and Differential equations [1]. When analyzing a complex network, an important problem is to compute the total number of triangles of a connected ...