Advanced Computational Mathematics

a) Define linearly dependent and independent sets. Check whether the vectors are $\alpha_1 = (1,2,3)$, $\alpha_2 = (0,1,0)$ and $\alpha_3 = (0,1,0)$ are linearly dependent or linearly dependent

b) Define each of the following.

1. Hash function
2. Heaviside's unit function and error function
3. Modular arithmetic

a) Prove that $H_n(-x) = (-1)^n H_n(x)$

b) Define T: V₃ → V₃ by the rule

Show that this is a linear transformation.

a) Find the Fourier transform of $e^{-|x|}$.

b) Solve the Laplace equation $u_{xx} + u_{yy} = 0$, for the mesh with boundary values shown in the following figure.

a) Solve $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ subject to initial condition $u = \sin \pi t$ at $t=0$ for $0 \le x \le 1$ and $u=0$ at $x=0$ and $x=1$ for $t > 0$, by Gauss-Seidel iterative method.

b) Write the properties of DFT, WFT and Haar transform.

a) Out of 800 families with four children each, how many families would be expected to have

1. 2 boys and 2 girls
2. at least one boy
3. No girl
4. at most two girls

Assume equal probabilities for boys and girls

b) Find the mean and variance of Poisson's distribution.

a) Obtain the steady state difference equation for the queuing model {M/M/1: (N/FCFS)} and show that

b) Show that normal distribution as the limiting case of Binomial distribution when $p=q$.

a) Explain the Markov Chain. Draw transition diagram and write down the properties of Markov Chain.

b) Customers at a box office window, being managed by a single man, arrive according to a Poisson input process with a mean rate of 30 per hour. The time required to serve a customer has an exponential distribution with a mean of 2 minutes. Find the average waiting time of customers.

a) Let A and B be fuzzy sets defined on a universal set X.

Then prove that: $|A| + |B| = |A \cup B| + |A \cap B|$

b) Write the MATLAB statements required to calculate $y(t)$ from the equation

for values of $t$ between $-9$ and $9$ in step of $0.5$.