MAT 204 Discrete Structures – Assignment #10

Number theory is the branch of mathematics concerned with the integers. Traditionally, number theory was a pure branch of mathematics – known for its abstract nature rather than its applications. The great English mathematician, G.H. Hardy (1877 – 1947), used number theory as an example of a beautiful, but impractical, branch of mathematics. However, in the late 1900s, number theory became extremely useful in cryptosystems – systems used for secure communications.

Find the following for each pair of integers:

(a) The prime factorization;

(b) The greatest common divisor;

(c) The least common multiple;

(d) Verify that gcd (m, n) * lcm(m, n) = mn.

(i) 315, 825

(ii) 2091, 4807

f(x) = (x^2 )0 < x < 1, (2−x), 1 < x < 2

A) Solve this integral, writing An as an expression in terms of n. Write down the
values of A1,A2,A3,A4,A5 correct to 8 significant figures.

b) Use MATLAB to find the coefficients of the first five harmonics and compare the
results with those from part (e). Your solution should include a copy of the m-file
fnc.m which you use to obtain the coefficients

c) Using MATLAB, plot the function and its approximating five-term Fourier series.

Find the general solution of the linear system
x ̇1 = x1, x ̇2 = ax2
Where a is a constant. Draw the phase planes for a = −1, 0, 1. Comment on the changes of the phase plane

. Three Dice of a Kind

Consider the following game: You roll six 6-sided dice d1,…,d6 and you win if some number appears 3 or more times. For example, if you roll:

(3,3,5,4,6,6)

then you lose. If you roll

(4,1,3,6,4,4)

then you win.

1. What is the probability that you win this game?

2. Drinking Warm Beer

We put 10 bottles of Molson Export and 3 bottles of Labatt 50 into the trunk of our black car on a hot summer day. We reach into the the cooler, pull out a random bottle b1 and drink it. Then we reach into the cooler, pull out a second bottle b2 and drink it.

1. Describe the sample space S for this experiment.
2. For each outcome ω∈S, determine Pr(ω).
3. Let A be the event "b1 is a bottle of Molson Export" and let B be the event "b2 is a bottle of Labatt 50". Determine Pr(A) and Pr(B).
4. Are the events A and B independent? In other words, is Pr(A∣B)=Pr(A)?

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

4. (10pts) Consider the damped forced harmonic oscillator with mass 1 kg, damping coefficient 2, spring constant 3, and external force in the form of an instantaneous hammer strike (Section 6.4) at time t = 4 seconds. The mass is initially displaced 2 meters in the positive direction and an initial velocity of 1 m/s is applied. Model this situation with an initial value problem and solve it using the method of Laplace transforms.

1. Consider the undamped forced harmonic oscillator with mass 1 kg, damping coefficient 0, spring constant 4, and external force h(t) = 3cos(t). The mass is initially at rest in the equilibrium position. You must understand that you can model this as: y’’ = -4y +3cost; y(0) = 0; y’(0) = 0.
   1. (5pts) Using the method of Laplace transforms, solve this initial value problem.
   2. Check your solution solves the IVP.
      1. (4pts) Be sure to check that your solution satisfies both the differential equation and
      2. (1pt) the two initial conditions.

a) Verify that y1 and y2 are fundamental solutions of the given homogenous second-order linear differential equation

b) find the general solution for the given differential equation

c) find a particular solution that satisfies the specified initial conditions for the given differential equation

y'' - y = 0 y1 = e^x, y2 = e^-x : y(0) = 0, y'(0) = 5

Determine if the following subsets are subspaces:
1. The set of grade 7 polynomials
2. The set of polynomials of degree 5 such that P (0) = 0
3. The set of continuous functions such that f (0) = 2

Expand in Fourier series:

f(x) = x|x|, -L<x<L, L>0

f(x) = cosx(sinx)^2 , -pi<x<pi

f(x) = (sinx)^3, -pi<x<pi

Expand in Fourier series:

Expand in fourier sine and fourier cosine series of: f(x) = x(L-x),    0<x<L

Expand in fourier cosine series: f(x) = sinx, 0<x<pi

Expand in fourier series f(x) = 2pi*x-x^2, 0<x<2pi, assuming that f is periodic of period 2pi, that is, f(x+2pi)=f(x)

write the theory and formulas for solving the systems of equations using the Laplace transform. Must contain bibliography