14. Extra Credit: Cayley’s Theorem is an important one in advanced algebra. It says that “Every algebraic group is isomorphic to some permutation group.” Demonstrate this to be true by finding a permutation group (Sn, ∘ ) that is isomorphic to (ℤ3, +) for some n.

(a)Show that S = {a+b √ 5 | a, b ∈ Q} is a subring of the real numbers (with the usual + and × of real numbers). Explain why S is a field.

(b) Prove that if r is an element of a ring R and r 3 = 0, then 1 − r is a unit in R.

(c) Write down all the nilpotent elements of Z24, stating the index of nilpotence in each case. Verify the statement in part (b) holds in Z24.

(ii) The market consists of the following stocks. Their prices and number of shares are as follows: Stock Price Number of Shares Outstanding A $10 100,000 B 20 10,000 C 30 200,000 D 40 50,000 (a) What is the percentage increase in the market if a S&P 500 type of measure of the market (value-weighted average) is used? b. The price of Stock C doubles to $60. What is the percentage increase in the market if a S&P 500 type of measure of the market (value-weighted average) is used? c. Repeat question (b) but use a Value Line type of measure of the market (i.e., a geometric average) to determine the percentage increase. d. Suppose the price of stock B doubled instead of stock C. How would the market have fared using the aggregate measures employed in (b) and (c)? Why are your answers different?

Solve the Cauchy-Euler equation

x⁴y'''' - 4x²y'' + 8xy' - 8y = 12xlnx

x > 0

Solve the following initial value problems

y'' + y = 2/cos x , y(0) = y'(0) = 2

x^3 y''' − 6xy' + 12y = 20x^4, x > 0, y(1) = 8/3 , y'(1) = 50/3 , y''(1) = 14

x^2 y'' − 2xy' + 2y = x^2, x > 0, y(1) = 3, y'(1) = 5

brief one paragraph summary report about a Road Plan and Profile (PnP) Drawing.

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such that (V, E0 ) and (V, E00) are spanning trees which are not isomorphic.

Consider the region bounded between y = 3 + 2x - x^2 and y = e^x + 2 . Include a sketch of the region (labeling key points) and use it to set up an integral that will give you the volume of the solid of revolution that is obtained by revolving the shaded region around the x-axis, using the... (a) Washer Method (b) Shell Method (c) Choose the integral that would be simplest to integrate by hand and integrate and evaluate that integral, showing all steps.

How many bit strings of length fifteen

a) Contain at least four 0s?

b) Contain at most four 0s?

c)Contain exactly four 0s?

d) Begin with four 0s?

Fill in the blanks

USING THE TERMS LISTED BELOW, (10 TERMS LISTED) CHOOSE THE BEST ANSWERS TO FILL IN THE BLANKS.

Paragraph #1: Mark, an investor, examines the financial statements of Disney. He calculates gross profit of services and does a year-to-year variance comparison of service revenues and costs of services.

Paragraph #2: Mark then compares the net income taking into account the non-cash activity of Disney (adding it back) as well as the other expenses that he feels do not represent management discretion.

Paragraph #3: Through this calculation, Mark is able to compare the income figure that includes expenses that can be managed versus those that are somewhat "fixed" or contractual (not adding back any non-cash activity).

Paragraph #1 is an example of using _ __ information to analyze the financial results of Disney.  

Paragraph #2 is an example of using _ __ to analyze the financial results of Disney.

Paragraph #3 is an example of using _ _ to analyze the financial results of Disney.

matching at the line historical cost   
accruals below the line GAAP
EBIT above the line consistency
EBITDA gross margin conservatism

4. Consider the linear program in problem 3. The value of the optimal solution is 48. Suppose the right-hand side for constraint 1 is increased from 9 to 10.

(problem 3 linear program)

Min 8X+12Y s.t.

1X+3Y≥9

2X+2Y≥10

6X+2Y≥18

A,B≥0

A) Use the graphical solution procedure to find the new optimal solution.

b) Use the solution to part (a) to determine the shadow price for constraint 1.

c) The sensitivity report for the linear program in Problem 3 provides the following right-hand-side range information:

What does the right-hand-side range information for constraint 1 tell you about the shadow price for constraint 1?

d) The shadow price for constraint 2 is 3. Using this shadow price and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2?

Please show the steps to solving the problem. Thank you.

assume that U(x, y) = xy, Px = 1, Py = 4 and B = 120.

Using the Bordered Hessian matrix, verify that the second-order conditions for a maximum are satisfied. Show steps.

1. Does the Diophantine equation 3x−4y = 0 have an integer solution? If so, can you list all integer solutions?

2. Does the equation 3x − 4y = 5 have an integer solution? If so, can you list all the integer solutions?

Solve the following equation using Green’s functions. Set up the integral for the particular solution and evaluate

y''+9y=x+sinx y(0)=1,y'(0)=2