PeripateticShippingLines,Inc.,isashipping company that owns n ships and provides service to n ports. Each of its ships has a schedule that says, for each day of the month, which of the ports it’s currently visiting, or whether it’s out at sea. (You can assume the “month” here has m days, for some m > n.) Each ship visits each port for exactly one day during the month. For safety reasons, PSL Inc. has the following strict requirement:

(†) No two ships can be in the same port on the same day.

The company wants to perform maintenance on all the ships this month, via the following scheme. They want to truncate each ship’s schedule: for each ship Si, there will be some day when it arrives in its scheduled port and simply remains there for the rest of the month (for maintenance). This means that Si will not visit the remaining ports on its schedule (if any) that month, but this is okay. So the truncation ofSi’s schedule will simply consist of its original schedule up to a certain specified day on which it is in a port P; the remainder of the truncated schedule simply has it remain in port P.

Now the company’s question to you is the following: Given the sched- ule for each ship, find a truncation of each so that condition (†) continues to hold: no two ships are ever in the same port on the same day.

Show that such a set of truncations can always be found, and give an algorithm to find them.

Example. Suppose we have two ships and two ports, and the “month” has four days. Suppose the first ship’s schedule is

port P1; at sea; port P2; at seaand the second ship’s schedule isat sea; port P1; at sea; port P2

Then the (only) way to choose truncations would be to have the first ship remain in port P2 starting on day 3, and have the second ship remain in port P1 starting on day 2.

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not defined. V = { f : R --> R | f(1) = 0 }

Use method of Frobenius to find one solution of Bessel's equation of order p: x^2y^''+xy^'+(x^2-p^2)y=0

If you don't have a calculator, you may want to approximate (243.034)^3/5 by 243^3/5. Use the Men Value Theorem to estimate the error in this approximation.

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not defined. V = { (x1, x2, x3) ∈ R^3 | x1 > or equal to 0, x2 > or equal to 0, x3 > or equal to 0}

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not identified. V = R^2 , < X1 , X2 > + < Y1 , Y2 > = < X1 + Y1 , 0> c< X1 , X2 > = < cX1 , cX2 >

Determine whether the set with the definition of addition of vectors and scalar multiplication is a vector space. If it is, demonstrate algebraically that it satisfies the 8 vector axioms. If it's not, identify and show algebraically every axioms which is violated. Assume the usual addition and scalar multiplication if it's not defined. V = {all polynomials with real coefficients with degree > or equal to 3 and the zero polynomial}

In the real projective plane, list all possible + / - (positive and negative) patterns for three coordinates. Then match these triples to regions of the Fundamental Triangle. Also, describe the location of the usual four Euclidean quadrants in the Fundamental Triangle.

1. Use the method of exhaustion to prove the following statement: “For every prime number p between 30 and 58, 10 does not divide p − 9.”

2. Prove that 0.17461461 . . . is rational (digits 461 in the fractional part are periodically repeated forever).

Solve d) and e). I have provided answer for c) too required in e)

Winkler Furniture manufactures two different types of china cabinets: a French provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

Formulate as an LP problem and obtain the revenue

c. What is the total Revenue at the optimal solution?

d. Bob Winkler wants to add this requirement to his production policy: To produce at least as many French Provencial cabinets as Danish Modern. How many French Provencial and Danish Modern Cabinets should Bob produce?

e. What is the impact on revenue of the solution in part d compared to the result in part c?

Solution for c)

Optimal Solution:

X1 = 60 X2 = 90

Revenue

Revenue =28*60+25*90= $3,930

#1 (a) Express, if possible,b= (7,8,9) as a linear combination of u1= (2,1,4),u2= (1,−1,3),u3= (3,2,5)

(b) What is Span{u1,u2,u3}?

(c) Is the set{u1,u2,u3}linearly independent?

#2 Considerv1= (1,2,1,1),v2= (1,1,3,1) and v3= (3,5,5,3). Find a homogeneoussystem the solution of which is Span{v1,v2,v3}. (Hint: Consider x= (x1, x2, x3, x4) where x=sv1+tv2+rv3 and look for conditions on x1, . . . , x4 where this system is consistent)

3.2.1. Find the Fourier series of the following functions:

(g) | sin x |

(h) x cos x.

Let V = {P(x) ∈ P10(R) : P'(−4) = 0 and P''(2) = 0}. If V= M_3×n(R), find n.

Recall P2(t) is the set of polynomials of order less than or equal to 2. Consider the the set of vectors in P2(t).

B={t^2,(t−1)^2,(t+1)^2}

(a) Show B is a basis for P2(t).

(b) If E={1,t,t^2}is the standard basis, calculate the change of basis matrices PE→B and PB→E

(c) Given v= 2t^2−5t+ 3, find its components in B

Given a sequence of closed intervals arranged so that each interval is a subinterval of the one preceding it and so that the lengths of the intervals shrink to zero, then there is exactly one point that belongs to every interval of the sequence. (This is known as the nested interval property)