Prove that every nontrivial finite group has a composition series

Let 2a > -1, If the area of the region of the plane defined by {(x,y) : x ≥ 0, 2y-x ≥ 0, ax+y-3 ≤ 0} is equal to 3, then the value of a, lies in
Answer key: (0.75, 1.5)

9.2 Give 3 examples of equivalence relations and describe the equivalence classes.

9.3 Let R be an equivalence relation on a set S. Prove that two equivalence classes are either equal or do not intersect. Conclude that S is a disjoint union of all equivalence classes.

Describe various spaces associated with an m × n matrix A, such as null space, row space. column space and eigenspace. What are the relationships among them? How does the concept of a linear transformation and its properties relate to matrices and those spaces of the matrices?

Let V be a Hilbert space. Let f(x) = ∥x∥ for x ∈ V. Using the definition of Frechet differentiation, show that ∇f(x) = x for all x ̸= 0. Furthermore, show that f(x) is not Frechet differentiable at x = 0.

17. Integrate the function sin(x) from 0 to π using Riemann and Simpson integration methods with N subintervals, where N increases by factors of 2 from 2 to 256 (i.e. N = 2, N = 4, etc.). Plot the relative error ∆I/I = [| estimated value - true value |/(true value)], assuming the

highest value of N gives the ‘true’ value. Which method converges the fastest?

Solve the recurrence equations by Substitution

a) T(n) = 4T (n/2) + n, T (1) = 1

b) T(n) = 4T (n/2) + n² , T (1) = 1

c) T(n) = 4T (n/2) + n³ , T (1) = 1

A telephone sales force can model its contact with customers as a Markov chain. The six states of the chain are as follows:
State 1 Sale completed during most recent call
State 2 Sale lost during most recent call
State 3 New customer with no history
State 4 During most recent call, customer’s interest level low State 5 During most recent call, customer’s interest level medium State 6 During most recent call, customer’s interest level high
Based on past phone calls, the following transition matrix has been estimated:
1100000 20 1 0 0 0 0 3 0.10 0.30 0 0.25 0.20 0.15
P  4 0.05 0.45 0 0.20 0.20 0.10
5 0.15 0.10 0 0.15 0.25 0.35
6 0.20 0.05 0 0.15 0.30 0.30 
a) For a new customer, determine the average number of calls made before the customer buys the product or the sale is lost.
b) What fraction of new customers will buy the product?
c) What fraction of customers currently having a high degree of interest will buy the
product?
d) Suppose a call costs 20 TL and a sale earns 200 TL in revenue. Determine the “value”
of each type of customer.

Discrete math

1) Using MatLab (any language is fine just say what language is used).

function d=lemma_gcd(a,b)
%This program will return the greatest common divisor for input variables a
%and b where a and b are integers such that they are not both 0. It uses
%Lemma 4.8.3 on page 225.

%Note: gcd(a,b)=gcd(-a,b)=gcd(a,-b)=gcd(-a,-b), so
a=abs(a);
b=abs(b);

%This following section sets up the arrays we will use to store the
%changing values of our variables as a sequence. "c" will be the count
%variabe, stored in "C", "a" and "b" will also bechanging and stored in "A"
%and "B" arrays through the iterations of the loops.

c=0;
C(1)=c; %Stores c=0 in the array, C
%Next find the initial a and b to kick the program into gear

if a==0 & b==0
fprintf('You cannot input these variables\n')
d=inf;
else
if floor(a)~=a
fprintf('You cannot input these variables\n')
d=inf;
if floor(b)~=b
fprintf('You cannot input these variables\n')
d=inf;
  
  
  
  
  
  
  
  
end
end
end

A company manufactures three types of luggage: economy, standard, and deluxe. The
company produces 1000 pieces each month. The cost for making each type of luggage is
$20 for the economy model, $30 for the standard model, and $40 for the deluxe model.
The manufacturer has a budget of $25,700. Each economy piece requires 6 hours of
labor, the standard model requires 8 hours of labor and the deluxe requires 12 hours of
labor. The manufacturer has a total of 7400 hours of labor available each month. If the
manufacturer sells all of the luggage produced, exhausts his entire budget and uses all
available hours of labor, how many pieces of each can be produced?

a) Write the system of equations for this information. Identify variables.

b) Write the augmented matrix for this information.

c) Solve using the Gauss-Jordan elimination method. Label each answer.

Use your knowledge of second-order systems forced by a sinusoidal function to solve (a) y '''' + y ′′ = sin x.

Hint: Try to integrate twice immediately. Extra hint: Your solution should involve four constants.

(b) Instead consider y'''' + y ′′ = x sin x. Note that ∫ x s sin s s = sin x − x cos x + c, where c is a constant.