For each of the following groups, find all the cyclic subgroups: (a) Z8, (b) Z10, (c) Z x10, (d) S4.
In: Advanced Math
You bought 500 Forever stamps just before the price went up in January of 2014 at $0.46/stamp, a $0.03 savings per stamp. If you could have paid off a credit card charging 12% per year instead of investing in stamps, how fast must you use the stamps to breakeven? Assume the $0.49 price never changes before you run out of stamps. (Hint use 12% /year as your effective interest rate per year)
a. 3 months
b. 6 months
c. 10 months
d. 12 months
e. 36 months
In: Advanced Math
We saw in class that each LP can be transformed into an equivalent LP in any of the following two forms below: (1) maxcTx: Ax=b, x≥0 (2) maxcTx: Ax≤b. Can we always transform any LP in an LP of the form Prove your answer correct. maxcTx: Ax=b?
In: Advanced Math
For the wave equation, utt = c2uxx, with the following boundary and initial conditions,
u(x, 0) = 0
ut(x, 0) = 0.1x(π − x)
u(0,t) = u(π,t) = 0
(a) Solve the problem using the separation of variables.
(b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).
In: Advanced Math
Project proposal "Exploring the role of small and medium enterprises in Zimbabwe".
In: Advanced Math
Two chemicals A and B are combined to form a chemical C. The rate of the reaction is proportional to the product of the instantaneous amounts of A and B not converted to chemical C. Initially there are 23 grams of A and 47 grams of B, and for each gram of B, 1.3 grams of A is used. It has been observed that 17.5 grams of C is formed in 15 minutes. How much is formed in 40 minutes? What is the limiting amount of C after a long time?
In: Advanced Math
Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z
If z = − 2x − 3y, then w = 4x + 6y
Using the calculation above, mark the statements below that must
be true.
A. Span(w, x, y) = Span(w, y)
B. Span(x, y, z) = Span(w, z)
C. Span(w, x, z) = Span(x, y)
D. Span(w, z) = Span(y, z)
E. Span(x, z) = Span(x, y, z)
In: Advanced Math
1. Prove or disprove: if f : R → R is injective and g : R → R is surjective then f ◦ g : R → R is bijective.
2. Suppose n and k are two positive integers. Pick a uniformly random lattice path from (0, 0) to (n, k). What is the probability that the first step is ‘up’?
In: Advanced Math
Determine the edge-connectivity of the petersen graph
In: Advanced Math
Given x = [0, 0.05, 0.1, 0.15, 0.20, ... , 0.95, 1] and f(x) = [1, 1.0053, 1.0212, 1.0475, 1.0841, 1.1308, 1.1873, 1.2532, 1.3282, 1.4117, 1.5033, 1.6023, 1.7083, 1.8205, 1.9382, 2.0607, 2.1873, 2.3172, 2.4495, 2.5835, 2.7183], write a Matlab script that computes the 1st and 2nd derivatives of O(h^2).
In: Advanced Math
Compute the first partial derivatives of h(x,y,z)=(1+9x+4y)^z
In: Advanced Math
Solve the differential equation 2x^2y"-x(x-1)y'-y = 0 using the Frobenius Method
In: Advanced Math
Find all integer solutions to the equation:
a) 105x + 83y = 1
b) 105x + 83y = 8
In: Advanced Math
3. In R4 , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R4? In other words, can you write down any vector (a, b, c, d) ∈ R4 as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ?
4. In the vector space P2 of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span of the set {x + 2, x2 − 1}.
5. Let S be the subspace of P2 defined by S := {ax2 + bx + 2a + 3b : a, b ∈ R}, for different choices of real numbers a and b (you don’t need to show here that S is indeed a subspace, and can assume. But is a good practice problem). Find a basis, and hence dimension for S.
In: Advanced Math
Description: A terrible zombie apocalypse is ravaging across planet earth. The number of zombies is growing proportionally to the number of zombies present. After one month since the first infection, 100 million people had been infected and by the time another month had passed, there were 400 million infected.
1) Construct a differential equation which models the
number of zombies. Explain the reasoning behind how you constructed
the model you chose to use. Discuss any potential shortcomings of
your model. Describe any initial conditions associated with your
model.
In: Advanced Math