6. For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general.

1. For which n does the complete graph Kn have a matching?
2. Prove that if a graph has a matching, then |V||V| is even.
3. Is the converse true? That is, do all graphs with |V||V| even have a matching?
4. What if we also require the matching condition? Prove or disprove: If a graph with an even number of vertices satisfies |N(S)|≥|S||N(S)|≥|S| for all S⊆V,S⊆V,then the graph has a matching.

Please keep straight to the point and short if possible, I give good ratings on good legible writings and correctness. THANKS!!

Redesign topsort so it selects the LAST NODE in each iteration, rather than the first.

def topsort(G):

count = dict((u, 0) for u in G) # The in-degree for each node

for u in G:

for v in G[u]:

count[v] += 1 # Count every in-edge

Q = [u for u in G if count[u] == 0] # Valid initial nodes

S = [] # The result

while Q: # While we have start nodes...

u = Q.pop() # Pick one

S.append(u)    # Use it as first of the rest

for v in G[u]:

count[v] -= 1 # "Uncount" its out-edges

if count[v] == 0: # New valid start nodes?

Q.append(v) # Deal with them next

return S

Question 1
(a) What is the objective of sampling? [2]
(b) What is the difference between a poll and a survey? [4]

(c) What is the difference between probability sampling and non-probability sampling? Give exam-
ples of both. [8]

(d) What are the sources of errors in surveys? Explain in detail giving examples. How can the
errors be reduced? [12]
(e) List all possible simple random samples of size n = 2 that can be selected from the population
{0, 1, 2, 3, 4}, and calculate s^2

for the any two samples and show that s^2=(N/N−1)σ^2




. Also calculate

σ^2 for the population and V ar( ̄y) for the two samples.

Type into Google: "globalfoundries sand to silicon", and a Youtube video will come up (accessed on 3/17/2020). The address is:

youtube.com/watch?v=UvluuAIiA50

Watch more than once if needed.

The video will go over various parts of the process steps in making a silicon computer chip integrated circuit (IC). On this page (and on the back if needed):

1. List (in order) the major steps in the fabrication process described in the video set. I do not expect you to have a detailed description of each step – just write down the "highlights".

2. Explain why the ICs are fabricated in the "clean rooms" depicted in the video.

3. A representation of the silicon crystal is used in some of the animations in the video. Explain what is wrong with the representation.

A 36 V battery of negligible internal resistance is connected to a 47 kΩ and a 21 kΩ resistor in series.

a).

What reading will a voltmeter, of internal resistance 95 kΩ , give when used to measure the voltage across the first resistor?

Express your answer using two significant figures.

V47 kΩ=

b).

What is the percent inaccuracy due to meter resistance for each case?

Express your answer using two significant figures.

V not ideal−V ideal/V ideal=

c).

What reading will a voltmeter, of internal resistance 95 kΩ , give when used to measure the voltage across the second resistor?

Express your answer using two significant figures.

V21 kΩ=

d).

What is the percent inaccuracy due to meter resistance for each case?

Express your answer using two significant figures.

V not ideal−V ideal/V ideal=

Turing machine A that does the following:

• On its first and second tape, A receives two strings w and v, w, v ∈ {0, 1}? , representing two integer numbers. When machine A is started, the tape heads are located on the left-most position, on the most significant bits of w and v.

• If none of the inputs w and v is the empty word, the Turing machine A writes the binary representation of the sum of the two integer numbers w and v on the third tape and then accepts the input. If one of w or v is the empty word, the Turing machine does not write anything on the third tape and rejects the input. If machine A outputs something, it outputs the binary representation of the sum in such a way that the left-most bit contains the most significant bit of the sum. Be careful: w and v may be of different length.

Q: Define a 3-tape Turing machine M that implements multiplication, in a manner similar to how A implements addition

3. Sketch a T-v diagram and find the specified state for the following states

a. Propane at critical temp & press find T(°C), p(bar) and v(m3 /kg)

b. Water v=.05 m3 /kg, T=140°C find p(bar)

c. Water at p=30 MPa, T=100°C find v(m3 /kg)

Suppose that over a certain region of space the electrical potential V is given by the following equation.

V(x, y, z) = 4x² − 4xy + xyz

(a) Find the rate of change of the potential at P(3, 6, 6) in the direction of the vector v = i + j − k.
???
(b) In which direction does V change most rapidly at P?

???
(c) What is the maximum rate of change at P?

???