Use induction to prove that the union of n countable sets is countable where n is a positive integer. (can use the fact that union of two countable sets is countable)

Show that the partition problem is polynomially reducible to the decision version of the knapsack problem. Please give details. This problem has been "solved" before but the answer given made no sense at all.

1. (11 pts) All Boots is a retailer of boots. It sources a kind of waterproof hunting boots from an Asian supplier for $40 each and sells them to customers for $108 each. Leftover boots at the end of season will be sold to an outlet mall at $30 each. Given the $108 retail price, All Boots forecasts the demand distribution as follows:

Now suppose All Boots found a reliable vendor in the United States that can produce boots very quickly but at a higher price than All Boots’ Asian supplier. Hence, in addition to boots from Asia, All Boots can buy an unlimited quantity of additional boots from this American vendor at $65 each after demand is know.

a) Suppose All Boots orders 500 boots from the Asian supplier (Note that the first order quantity of 500 units is given, which may not be the optimal order quantity). What is the probability that All Boots will order from the American supplier once demand is known, i.e., the probability of placing a second order? (Hint: given the 1st order quantity of 500 units, with what demand outcomes will All Boots need to place a second order?)

b) Again assume that All Boots orders 500 boots from the Asian supplier. On average, how many boots should the American supplier expect that All Boots will order, i.e., the expected second order quantity? Parts c) and d) are separate from parts a)

c) Given the opportunity to order from the American supplier at $65 per boot, what order quantity from its Asian supplier now maximizes All Boots’ expected profit, i.e., optimal first order quantity?

d) Given the order quantity in part c) (not the quantity in parts a and b), what is All Boots’ expected profit? [Hint: expected profit = maximum profit – mismatch cost, where maximum profit = (p-c)* ?]

1.) use partial fractions to decompose each into a fraction with a linear factor in the denominator:

a.) 2/(x+1)(x+2)

b.) 2/(y)(100-y)

c.) y/(y)(100-y)

d.) 5/x(x+1)(x-2)

e.) 2x+3/x(x+1)(x-2)

f.) x^2/x(x+1)(x-2)

2. Consider the ODE model for population growth:

a. Use separation of variables to determine the solution.

b. What is the value of y(1)?

c. What is the value of y(10)?

d. At what time will the population reach 100? At what time will it reach 1000?

3. Consider the logistic growth model for population growth:

a. Use separation of variables to determine the solution.

b. What is the value of y(1)? c. What is the value of y(10)?

d. At what time will the population reach 100? At what time will it reach 1000?

4. Consider the solutions to the previous two problems

a. What does the first model predict about solutions as t increases?

b. What does the second model predict about solutions as t goes to infinity?

c. Use MATLAB’s ODE45 command to generate plots of the solutions, give a plot of the two functions together on a single set of axes.

d. How are values similar or different for this model in comparison to the previous one?

In analyzing hits by bombs in a past war, a city was subdivided into 552 regions, each with an area of 0.25-km². A total of 447 bombs hit the combined area of 552 regions. The Poisson distribution applies because we are dealing with the occurrences of an event (bomb hits) over some interval (a region with area of 0.25-km².

Find the mean number of hits per region:
mean =

Find the standard deviation of hits per region:
standard deviation =

If a region is randomly selected, find the probability that it was hit exactly twice.
(Report answer accurate to 4 decimal places.)
P(X=2)=P(X=2)=

Based on the probability found above, how many of the 552 regions are expected to be hit exactly twice?
(Round answer to a whole number.)
ans =

If a region is randomly selected, find the probability that it was hit at most twice.
(Report answer accurate to 4 decimal places.)
P(X≤2)=P(X≤2)=

Give a description of why the law of sines holds. That is, "prove" the law of sines.

Which statement is false about the DTFT of a sequence x(n)?

(a) It always exists (b) It is continuous in w (c) It is periodic (d) It can be complex

The principle that there can be no output before there is any input to a system is called

(a) linearity (b) shift invariance (c) relativity (d) causality

Is the transform T[x(n)] =x²(n) linear? Justify your answer using sequences x1(n)=δ(n-0) and x2(n)=δ(n-1) with a = 1 and b = -2.

a-) Is the following statements TRUE or FALSE? Prove it or give a counterexample.

i) If f(x) : Rⁿ → R is a convex function, then for all α ∈ R, the set {x : f(x) ≤ α} is a convex set.

ii) If {x : f(x) ≤ α} is a convex set for all α ∈ R, then f(x) is a convex function.

b-) Prove that if x^* is a vector such that ∇g(x^* ) = 0 and ∇² g(x^*) is positive definite, then x^* is a local minimizer for g(x).

1. ) Determine if the set form the basis in R³

(1,1,2); (1,2,5); 5,3,4)

1. Write the vector v = (2,3,-5); as a linear combination of u₁= (1,2,-3), u₂= (2,3,-4) and u₃= (1,3,-5)

maximize z = 2x₁+3x₂

subject to   x₁+3X₂ 6

                  3x₁+2x₂ 6

                 x₁,x₂

This can be simply done by drawing all the lines in the x-y plane and looking at the corner points.

Our points of interest are the corner points and we will check where we get the maximum value for our objective function by putting all the four corner points. (2,0), (0,2), (0,0), (6/7, 12/7)

We get maximum at = (6/7, 12/7) and the maximum value is = 6.8571

1.Implement simplex algorithm.

2.What is the sequence of extreme points in the simplex algorithm?

How would I do this problem?

A mass of 1 kg is attached to a spring whose constant is 5 N/m. Initially the mass is released 1 m below the equilibrium position with a downward velocity of 5 m/s, and the subsequent motion takes place in a medium that offers a damping force numerically equal to two times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force equal to f(t) = 12 cos (2t) + 3 sin (2t)

Let T be a connected graph and z ∈ℤ between the closed interval of 1 and the least degree of a vertex in T. Let a z - matching be a A ⊆ E s.t. there aren’t vertices with more than z edges in A. Let a z - cover be a X ⊆ E s.t. all vertices belong to at least z edges in X.

Let:

δ (T) = Max {|A| : A is a z - matching}

μ (T) = Min {|X| : X is a z - cover}

Show that δ (T) + μ (T) = zn

For the following questions, say you were given a line and a plane as below

                Line: r(t)= < x(t), y(t), z(t) > = < t-2, t+1, 3t > ,        Plane:   : a(x+2) + b(y-3) - 4z = 2

                a)            What relationship would have to exist between scalars a and b for the line not to intersect with the plane?

                Hint 1) Plug in x, y, z given in the line into the plane equation.

                Hint 2) Say you were solving this equation and got the following. What would that indicate?

                                i)             t = 2:      ________________________________________________________

                                ii)            6 = –5:   ________________________________________________________

                                iii)           4 = 4:     ________________________________________________________

                                So what must be true about the coefficient of t? What also can’t be true about the constant term?

                b)            What relationship would have to exist between scalars a and b for the plane to contain the line?

                c)            Assuming the line is not contained in the plane but does intersect it, give an expression for the time t that the line intersects the plane. Also give the point of intersection.
                                (Both answers will have scalars a and b in them)

Prove by induction that:

1) x^n - 1 is divisible by x-1

2)2n < 3^n for all natural numbers n