4. Show that the set A = {fm,b : R → R | m does not equal 0 and fm,b(x) = mx + b, m, b ∈ R} forms a group under composition of functions. (The set A is called the set of affine functions from R to R.)
In: Advanced Math
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