A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G), u =v such
that there is a trail from u to v which goes over every edge of G. The following
sequence of questions is towards a proof of the following:
Theorem 1. A connected graph G is semi-Eulerian (but not Eulerian) if and only
if it has exactly two vertices of odd degree.
Let G be semi-Eulerian with a trail t starting at a vertex u₀ and ending at a vertex
v₀ Let G ⁰ be a graph obtained by adding an edge e₀ joining u₀ and v₀ , so that
G ⁰ − e = G.
(a) Prove that given a semi-Eulerian trail t in G from v₀ to u₀ , it is possible to
construct a Eulerian trail in G₀.
(b) Prove that given an Eulerian trail in G 0 it is possible to construct a semi-Eulerian
trail in G.
(c) Prove Theorem 1.

Use the following information answer the questions below.

Company Name :- XEROX CORP

Total liabilities ($millions)         $ 13090

Total assets ($millions)               $ 18051

Net operating cash flows ($millions)                            $ 1095

Income from continuing operations ($millions)           $ 1349

Pension plan assumptions

      Discount rate 4.1 %      Expected asset return 5.8 %    Rate of salary increase 21  %

Pension plan disclosures

Projected benefit obligation ($millions)                       $ 1120

Fair value of pension plan assets ($millions)               $ 1123

Accumulated benefit obligation ($millions)                 $ 2230

Cash contributions to the pension plan during the year ($millions)                                 $ 1350

% of pension assets invested in equities 23 %

% of pension assets invested in bonds 28 %

OPEB disclosures

Expected postemployment benefit obligation ($millions)                                     $ 2305

Accumulated postemployment benefit obligation ($millions)                                     $ 4421

Fair value of plan assets ($millions)   $ 2230

From 2016 and previous financial statements

Actual percentage return on pension plan assets ($ actual return / $ beginning of year fair value of plan assets)

2016 5.6 %    2015 4.55 %   2014 6.3 %   2013 4.2  %   2012 3.98 %    2011 2.22 %

1. What dollar amount does your firm report for its defined-benefit pension plan and its OPEB plan on its year-end 2016 balance sheet? Prove this amount by reconciling from the related off-balance sheet footnote accounts.

2. What dollar amount does your firm report for its defined-benefit pension plan and its OPEB plan on its year-ending 2016 income statement?

3. Are your firm's defined-benefit pension plan and/or its OPEB plan material to your firm's financial position and financial performance? Explain using selected data from your firm's financial statements.

4. Are your firm's defined-benefit pension plan and its OPEB plan well-funded or poorly funded as of year-end 2016? Explain.

5.  Is your firm's 2016 expected return on plan assets assumption reasonable? Explain.

NOTE: At a minimum compare your firm's expected return on plan assets in 2016 with its average (geometric mean) return over the past 6 years. See details on computing the mean return below.

6. Defined-benefit pension plans and OPEB plans typically generate temporary differences between financial books and tax books. Discuss the significance of your firm's defined-benefit pension plans and OPEB plans to its year-end 2016 deferred tax accounts. Use dollar amounts from the tax footnote.

*Compute average (geometric mean) return

Suppose that in 3 successive years the return on an investment is 5%, 20%, and -4%. The geometric mean, or compound annual rate of return, is computed as:

             [(1.05)*(1.20)*0.96)]^1/3 -1 = 0.065 = 6.5%

Let the function c(v) model the gas consumption (in liters/km) of a car going at velocity v (in kilometers/hour). In other words, c(v) tells you how many liters of gas the car uses to go 1 km, if it is going at velocity v.

You find that (80) 0.04 and '(80) 0.0004

1. Let the function d(v) model the distance the same car goes on 1 L of gas at velocity v.

a. Express the relationship between c(v) and d(v) in an equation. [4 pts]

b. Find d(80) and d’(80). (Hint: Find the general d’(v) first.) [4 pts]

c. Interpret your result for d’(80) in a sentence. (That is, “When the car is travelling at 80 kph ….” ) [4 pts] (Even if you couldn’t get part b, you can still tell me what d’(80) means about the car.) [5 pts]

Calculate a geometric series

1.For v greater than 0 and less than 1, what is the Sum(v^i) for i = 1 to 100?

2.For v greater than 0 and less than 1, what is the Sum(v^i) for i = 1 to infinity?

3.Let v = 1/(1+r). State the answer to question 2 in terms of r.

You have the following information on return on the stocks of Target (T), Macy's (M),and Best Buy (B)

Target (T) Macy's (M) Best Buy (B)

$6 $3 $11

$7 $8 $5

$3 $5 $8

$4 $8 $4

What are the variances and standard deviations of the returns on the three stocks?

What is the covariance of the returns on T and M?

What is the covariance of M and B?

5. Suppose you have a data set:

Experiment 56 880

Control       15 398

Experiment 30 479

Control       18 293

Experiment 35 365

Control       22 984

Control        13 230

Control        20 432

Experiment 34 560

Variables are treatment, weight (g), area (m²), please create a SAS program that:

(A). Reading the data and create a SAS data set;

(b). Create a new variable wt_per_area, using wt_per_area=weight/area and change the unit of weight from g to kg;

(c). Sort the data set by treatment;

(d). Calculate the mean, standard error of weight, area and wt_per_area;

(e). Calculate the mean and standard error for treatment=”experiment” and “control”.

Let ?V be the set of vectors in ?2R2 with the following definition of addition and scalar multiplication:
Addition: [?1?2]⊕[?1?2]=[0?2+?2][x1x2]⊕[y1y2]=[0x2+y2]
Scalar Multiplication: ?⊙[?1?2]=[0??2]α⊙[x1x2]=[0αx2]
Determine which of the Vector Space Axioms are satisfied.

A1. ?⊕?=?⊕?x⊕y=y⊕x for any ?x and ?y in ?V
? YES NO

A2. (?⊕?)⊕?=?⊕(?⊕?)(x⊕y)⊕z=x⊕(y⊕z) for any ?,?x,y and ?z in ?V
? YES NO

A3. There exists an element 00 in ?V such that ?⊕0=?x⊕0=x for each ?∈?x∈V
? YES NO

A4. For each ?∈?x∈V, there exists an element  −?−x in ?V such that ?⊕(−?)=0x⊕(−x)=0
? YES NO

A5.  ?⊙(?⊕?)=(?⊙?)⊕(?⊙?)α⊙(x⊕y)=(α⊙x)⊕(α⊙y) for each scalar ?α and any ?x and ?y ?V
? YES NO

A6. (?+?)⊙?=(?⊙?)⊕(?⊙?)(α+β)⊙x=(α⊙x)⊕(β⊙x) for any scalars ?α and  ?β and any ?∈?x∈V
? YES NO

A7. (??)⊙?=?⊙(?⊙?)(αβ)⊙x=α⊙(β⊙x) for any scalars ?α and  ?β and any ?∈?x∈V
? YES NO

A8. 1⊙?=?1⊙x=x for all ?∈?x∈V
? YES NO

1(a) If ut − kuxx = f, vt − kvxx = g, f ≤ g, and u ≤ v at x = 0, x = l and t = 0, prove that u ≤ v for 0 ≤ x ≤ l, 0 ≤ t < ∞.

(b) If vt − vxx ≥ cos x for −π/2 ≤ x ≤ π/2, 0 < t < ∞, and if v(−π/2, t) ≥ 0, v(π/2, t) ≥ 0 and v(x, 0) ≥ cos x, use part (a) to show that v(x, t) ≥ (1 − e −t ) cos x.

This question is in reference to BFS and DFS for data structures and algorithms

Consider a graph algorithm with a growth function on V and E: f(V, E). How would you convert f(V,E) to f'(V) such that f(V,E)=O(g(n))=f(V)? (That is, convert a growth function of two variables to be of one variable in such a way that the Big-Oh bound for the one variable function will hold for the two variable function.) Explain the steps in creating f', and explain why your idea works.

Let G be a connected graph and let e be a cut edge in G. Let K be the subgraph of G defined by:

V(K) = V(G) and

E(K) = E(G) - {e}

Prove that K has exactly two connected components. First prove that e cannot be a loop. Thus the endpoint set of e is of the form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a path in K from v to ṽ, or there is a path in K from w to ṽ