To prove time delay trajectory for the following figure traces an ellipse for 0 < τ < π 2 Concept Introduction: The general format of the ellipse equation is Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 A, B, C, D, E, F are constants and (x,y) is a set of points. The condition for non-degenerate real ellipse is, B 2 − 4AC < 0 and C | A B 2 D 2 B 2 C E 2 D 2 E 2 F | = C Δ < 0 If C Δ > 0 , ellipse is imaginary, and if Δ = 0 , then ellipse is a point.

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In this exercise, we will investigate a technique to prove that a language is notregular. This tool is called the pumping lemma.The pumping lemma says that if M = (S, I, f, s0, F ) is a DFA with p states (i.e., p = |S|) and if the wordw is in L(M ) (the language generated by M ) and w has length greater than or equal to p, then w may bedivided into three pieces, w = xyz, satisfying the following conditions:1. For each i ∈ N, xy^i z ∈ L(M ).2. |y| > 0 (i.e., y contains at least one character).3. |xy| ≤ p (i.e., the string xy has at most p characters). Use the pumping lemma to show the following language is not regular (HINT: Use proof by contradictionto assume the language is regular and apply the pumping lemma to the language):L = {0^k1^k | k ∈ N}

Answer 1

Question

Chapter 12.4, Problem 1E

Interpretation Introduction

Interpretation:

To prove time delay trajectory for the following figure traces an ellipse for 0<τ<π2

Concept Introduction:

The general format of the ellipse equation is

Ax2+ Bxy + Cy2+ Dx + Ey + F = 0

A, B, C, D, E, F are constants and (x,y) is a set of points.

The condition for non-degenerate real ellipse is,

B2−4AC<0 and

C|AB2D2B2CE2D2E2F|=CΔ<0

If CΔ>0, ellipse is imaginary, and if Δ=0, then ellipse is a point.

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