Solve your IGNOU Doubts

Question:

Let C1 and C2 be two binary codes with generator matrices

$G_1=\begin{bmatrix}1&0&0&1\\0&1&0&0\\0&0&1&1\end{bmatrix},\quad G_2=\begin{bmatrix}1&0&0&1\\0&1&1&0\end{bmatrix}$

respectively.

a) Find the minimum distance of both the codes.

b) Find the generator matrix of the code

C = {(u|u+v)|u ∈ C1,v ∈ C2}

obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .

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Question:

For each of the linear codes, find the degree, a generator matrix and a parity check matrix.

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Question:

Find the minimum distance for each of the codes.

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Question:

Which of the following binary codes are linear?

i) $\mathcal{C}=\{(0,0,0,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)\}$

ii) $\mathcal{C}=\{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}$

Justify your answer.

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Question:

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If the weight of each element in the generating matrix of a linear code is at least r, the mininum distance of the code is at least r.

ii) There is no linear self orthogonal code of odd length.

There is no 3-cyclotomic coset modulo 121 of size 25.

iv) There is no duadic code of length 15 over F₂.

v) There is no LDPC code with parameters n = 16, c = 3 and r = 5.

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Question:

A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 1 every 6 minutes and service time follows exponential distribution with a mean of 10 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.

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Question:

Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter- arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?

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Question:

For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, p_o, = 0.056, calculate

i) The average time a customer is in the system

ii) The average number of customers in the system

iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?

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Question:

A company has three factories that supply to three markets. The transportation costs from each factory to each market are given in the table. Capacities of the factories and market requirements are shown. Find the minimum transportation cost.

	M₁	M₂	M₃	a_i
F₁	2	1	3	20
F₂	1	2	3	30
F₃	2	1	2	10
b_j	10	10	20	40/60

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Question:

The owner of a readymade garments store sells two types of shirts: Zee-shirts and Button-down shirts. He makes a profit of Rs. 5 and Rs. 10 per shirt on Zee-shirts and Button-down shirt, respectively. He has two tailors, A and B at his disposal to stitch the shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively. Both these shirts are to be stitched by both the tailors. Tailors A and B spend 2 hours and 5 hours, respectively in stitching one Zee-shirts, and 4 hours and 3 hours, respectively in stitching a Button-down shirt. How many shirts of both types should be stitched in order to maximize daily profit?

a) Formulate and solve this problem as an LP problem.

b) If the optimal solution is not integer-valued, use Gomory technique to derive the optimal integer solution.

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Question:

Do the stability analysis of the following model formulated to study the effect of toxicant on one competing species where the environment toxicant concentration is being taken to change w.r.t. time.

$\begin{align*}\frac{dN_1}{dt}&=r_1 N_1-\alpha_1 N_1 N_2-d_1 C_0 N_1\\\frac{dN_2}{dt}&=r_2 N_2-\alpha_2 N_1 N_2\\\frac{dC_0}{dt}&=k_1 P-g_1 C_0-m_1 C_0\\\frac{dP}{dt}&=Q-hP-kPN_1+gC_0N_2\end{align*}$

along with the initial conditions.

$N_1(0)=N_{10},\quad N_2(0)=N_{20},\quad C(0)=0,\quad P(0)=P_0>0$

Here,

N₁ (t)=Density of prey population

N₂(t)=Density of predator population

C_o(t) = Concentration of the toxicant in the individual of the prey population

P=Constant environmental toxicant concentration.

a,, a, are the predation rates, r,r, are the growth rates or birth rates, d, is the death rate due to C, m, is the depuration rate, Q, h, k, g are positive rate constants.

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Question:

The population dynamics of a species is governed by the discrete model

$x_{n+1}=x_n\exp\left[r\left(1-\frac{x_n}{K}\right)\right]$

where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviors of the population for $r=2+\varepsilon,\quad\text{where}0<\varepsilon\ll 1.$ . Since a species becomes extinct if x_n ≤1 1 for any n> 1, show using iterations, that irrespective of the size of r> 1 the species could become extinct if the carrying capacity k $k<r\exp[1+e^{r-1}-2r]$

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Question:

Do the stability analysis of the following competing species system of equations with diffusion and advection

$\begin{align*}\frac{\partial N_1}{\partial t}&=a_1 N_1-b_1 N_1 N_2+D_1\frac{\partial^2 N_1}{\partial x^2}-V_1\frac{\partial N_1}{\partial x}\\\frac{\partial N_2}{\partial t}&=-d_1 N_2+c_1 N_1 N_2+D_2\frac{\partial^2 N_2}{\partial x^2}-V_2\frac{\partial N_2}{\partial x},\quad 0\leq x\leq L\end{align*}$

where Vand V, are advection velocities in x direction of the two populations with densities N, and N, respectively. a, is the growth rate, b, is the predation rate, d, is the death rate, C, is the conversion rate. D, and D, are diffusion coefficients. The initial and boundary conditions are:

$N_i(x,0)=f_i(x)>0,\quad 0\leq x\leq L,\quad i=1,2,\dots$

$N_i=\overline{N}\text{at}x=0\text{and}x=L\quad\forall t,\quad i=1,2,\dots$

where $\overline{N_{i}$ are the equilibrium solutions of the given system of equations. Interpret the solution obtained and also write the limitations of the model.

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Question:

A model for insect populations leads to the difference equations

$N_{k+1}=\frac{\lambda N_k}{1+aN_k}$

where λ and a are positive constants.

i) Write the equation in the form $N_{k+1}=N_k+R(N_k)N_k$ , and hence identify the growth rate.

ii) Plot the graph of R(N_k) as a function of N₂

iii) Express the intrinsic growth rater and the carrying capacity K, for this model, in terms of the parameters, a and λ.

iv) Find the steady-state solution of this model and analyse the solution.

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Question:

In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.

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Question:

Assume that the return distribution on the two securities X and Y be as given below:

	Market 1	Market 2	Market 3
Probability	2.0	5.0	3.0
Security X	− 20%	18%	40%
Security Y	−10%	20%	15%

which security is more risky in the Markowitz sense. Also find the correlation coefficient of securities X and Y.

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Question:

Let P=(w₁,w₂) be a portfolio of two securities. If variance of P is minimum then find the value of w₁and w₂ in the following situations.

i) ρ12 = −1

ii)σ1 = σ2

iii)ρ12 = − ,5.0 σ1 = 5.1 and σ2 = .5.2

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Question:

Five securities have the following expected returns

A=20%, B=15%, C=25%, D=22%, E=18%. Calculate the expected returns for a portfolio consisting of all five securities under the following conditions

i) The portfolio weights are of equal percentage in each

ii) The portfolio weights are 32% in A and remaining are equally divided among other four securities.

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Question:

Consider the following data

x	2	3	4	5	6
y	8.3	16.5	30.2	65.2	125.6

Use a best fit line to estimate the value of y when x = 4.5.

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Question:

ompanies located on the banks of a river, dumping their chemicals waste into river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeds certain specified concentration limits. State, giving reasons, the type of modeling you will use to find a policy for discharging the waste to ensure that the concentration level never exceeds the specified limits. Also state four essentials and two non-essentials for the problem.

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