In how many ways you can triangulate a simple polygon with n vertices. Give both qualitative and quantitative answer.

1. The differential equation y''+4y=f(t) and y'(0)=y(0)=0

a. Find the transfer function and impulse response.

b. If f(t)=u(t)-u(t-1). Find the y(t) by convolution and Laplace techniques. u(t) is unit step function.

c. If f(t)= cos(t) ; find the y(t) by convolution and Laplace techniques.

2. The differential equation y''+3y'+2y=e^(-3t) and y'(0)=y(0)=0

a. Find the system transfer function and impulse response.

b. Find the y(t) by convolution and Laplace techniques.

3. y''+3y'+2y=f(t) and y'(0)=y(0)=0

Plot y(t) without any calculations and write down your reasons!

4. y''-4y'+3y=f(t) and y'(0)=y(0)=0

If f(t)=u(t)-u(t-1). Find the y(t) by convolution and Laplace techniques. u(t) is unit step function.

Define:

a. f(t)'s system input, and y(t)'s system output. How to define Transfer function? Find the transfer function.

b. Find y(t) by Laplace Transformation Technique.

Calculate the OC curve of the 2.5-sigma C-chart to control the number of defects per engine assembly at nine. Use c values of 3, 5, 7, 9, 10, 15, and 20.

Use the Function Design Recipe (FDR) to design, code and test the definition of a new function named distance. This function returns the distance between two points, given by the coordinates (x1, y1) and (x2, y2). The function parameters are the x and y values.

Need help. Thank you

Create a one page table to show the major similarities and differences between public and private sector. This should cover all areas not just finance e.g. HR, technology etc

Let p be the prime number (2^20)*(3^7)5 + 1 = 11466178561. Solve for x such that 2^x ≡ 2376886429 (mod p) Explain your method carefully.

4. (a) Use the Euclidean algorithm to find the greatest common divisor of 21 and 13, and the greatest common divisor of 34 and 21.

(b) It turns out that 21 and 13 is the smallest pair of numbers for which the Euclidean algorithm requires 6 steps (for every other pair a and b requiring 6 or more steps a > 21 and b > 13). Given this, what can you say about 34 and 21?

(c) Can you guess the smallest pair of numbers requiring 8 Euclidean algorithm steps?

(d) Is there a pattern here? Do the numbers which keep coming up have a name?

Describe a real-world prediction problem using urban data for which accuracy is paramount and interpretability may be less important, and for which it might be preferable to use random forests rather than decision trees. Argue why this is the case.

1.Is there evidence of multicollinearity? Perform Factor analysis by extracting four factors and name for those factors.

Problem needs to solved using R

Solve using Laplace Transform

ty'' + 2(t-1)y'-2y = 0 y(0)=0, y'(0)=0, Y(s) = C/(s^2+2s)^2 (C is arbitrary)

The input-output (consumption) matrix for a closed economy is given below. Solve the associated input- output model. Use r if you need a variable in your answer. [0.07 0.05 0.27] [0.17 0.48 0.19] [0.76 0.47 0.54]

Logic & Sets (Proofs question)

Show that complex numbers cannot be ordered in a way that satisfies our axioms.

Axioms for order:

1. if x is less than/equal to y and w is greater than zero, then wx is less than/equal to wy

2. for w, x, y, z w is less than/equal to x, y is less than/equal to z then w + y = x + z if and only iff w = x and y = z

Use the Frobenius method to solve: 2xy"-3y'+y=0. Find index r and recurrence relation. Compute the first 5 terms (a₀ – a₄) using the recurrence relation for each solution and index r.

Describe a real-world prediction problem using urban data for which interpretability of your models and results is essential, and for which it might be preferable to use decision trees rather than random forests. Argue why this is the case.