There are n people who can shake hands with one another (where n > 1). Use pigeonhole principle to show that there is always a pair of people who will shake hands with the same number of people.

Hint. Pigeonhole principle does not immediately apply to this problem. Solve the problem for two cases:

(1) There is no person who shakes hands with everyone else (all handshake numbers are strictly less than n − 1). Easy case.

(2) There is a person who shakes hands with everyone else (hand- shake number is n − 1). What can you say about handshake numbers for everyone else?

Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1, Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).

How does Willson, F. (2002). Shapes, Numbers, Patterns, And The Divine Proportion In God's Creation. explain that a Spirals in sunflowers demonstrates the divine proportion?

Explain why Willson said…“The only rational conclusion is that the Creator of the universe is a personal, intelligent Being, who created these things as a visible fingerprint of His invisible, yet personal existence.”

Suppose that there are three Suppliers (Supplier A, Supplier B, Supplier C) and we are interested to evaluate the performance of the suppliers by using AHP method. Some of the criteria for supplier selection process can be defined as follows:

Cost, Quality, and Timeliness.

1. We would like to add more factors as criteria to the three above mentioned criteria. What else factor can be defined as criteria? Define at least two more criteria.
2. By considering only the above three criteria (Cost, Quality, and Timeliness), you first introduce three different suppliers that you may know and further rank these three alternative suppliers and select the best one using AHP method. The pairwise comparison matrix for each of the three suppliers for each criterion and the comparison matrix for the three criteria can be given by you based on your judgment and experiences.   

Make up an example to show that Dijkstra’s algorithm fails if negative edge lengths are allowed.

Group theory

Consider the group GL₂(Z_p) of invertible 2X2 matrices with entries in the field Zp, where p is an odd prime.

Zp is an abelian group under addition, the group of unites of Zp is Zp^x, which is an abelian group under multiplication. We say (Z_p , +, ·) is a field.

1. 1. Show that the subset D₂(Z_p) of diagonal matrices in GL₂(Z_p) is an abelian subgroup of order (p - 1)².
2. 2. For A, B ∈GL₂(Z_p), show that A and B are in the same right D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈ Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
   3. For A, B∈GL₂(Z_p), show that A and B are in the same left D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
   4. Find a matrix   A such that the left and right D₂(Z_p) cosets containing A are different, that is, so that D₂(Z_p)A ≠ AD₂(Z_p).
   5. Find a non-identity matrix B such that the left and right D₂(Z_p) cosets containing B are the same, that is, so that D₂(Z_p)A = AD₂(Z_p).

3. The effect of financial leverage on ROE

Companies that use debt in their capital structure are said to be using financial leverage. Using leverage can increase shareholder returns, but leverage also increases the risk that shareholders bear.

Consider the following case:

Water and Power Co. is a small company and is considering a project that will require $500,000 in assets. The project will be financed with 100% equity. The company faces a tax rate of 25%. What will be the ROE (return on equity) for this project if it produces an EBIT (earnings before interest and taxes) of $140,000?

23.10%

21.00%

15.75%

22.05%

Determine what the project’s ROE will be if its EBIT is –$40,000. When calculating the tax effects, assume that Water and Power Co. as a whole will have a large, positive income this year.

-6.0%

-7.20%

-6.90%

-5.70%

Water and Power Co. is also considering financing the project with 50% equity and 50% debt. The interest rate on the company’s debt will be 13%. What will be the project’s ROE if it produces an EBIT of $140,000?

33.86%

32.25%

35.48%

27.41%

What will be the project’s ROE if it produces an EBIT of –$40,000 and it finances 50% of the project with equity and 50% with debt? When calculating the tax effects, assume that Water and Power Co. as a whole will have a large, positive income this year.

-28.27%

-27.19%

-23.92%

-21.75%

The use of financial leverage _______ the expected ROE, _________ the probability of a large loss, and consequently __________ the risk borne by stockholders. The greater the firm’s chance of bankruptcy, the__________    its optimal debt ratio will be.___________    manager is more likely to use debt in an effort to boost profits.

The CPI was 214.537 in 2009 and 232.957 in 2013 and the Tuition per credit hour was $99.65 in 2009 and $116.15 in 2013

(i) Find the absolute change for the CPI from 2009 to 2013.
Express your answer rounded to the nearest thousandth.

(ii) Find the relative change for the CPI from 2009 to 2013.
Express your answer rounded correctly to the nearest tenth of a percent.
%

(iii) Find the absolute change for the Tuition from 2009 to 2013.
Express your answer rounded to the nearest cent.
$

(iv) Find the relative change for the Tuition from 2009 to 2013.
Express your answer rounded correctly to the nearest tenth of a percent.
%

Which has increased more in this time period?  ? CPI Tuition

If you haven't answered the question correctly in 3 attempts, you can get a hint

Let A and B be two non empty bounded subsets of R:

1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B

2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A

hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)

(Connected Spaces)
(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.

Prove: An (n × n) matrix A is not invertible ⇐⇒ one of the eigenvalues of A is λ = 0

. Determine whether K4 (the complete graph on 4 vertices contains the following: i) A walk that is not a trail. ii) A trail that is not closed and is not a path. iii) A closed trail that is not a cycle.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.