Euler’s method

Consider the initial-value problem y′ = −2y, y(0) = 1. The analytic solution is y(x) = e−2x . (a) Approximate y(0.1) using one step of Euler’s method. (b) Find a bound for the local truncation error in y1 . (c) Compare the error in y1 with your error bound. (d) Approximate y(0.1) using two steps of Euler’s method. (e) Verify that the global truncation error for Euler’s method is O(h) by comparing the errors in parts (a) and (d).

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for the transition matrix P= 0.8 0.2 0.0 , solve the equation SP=S to find the stationary matrix S and the limiting matrix P.

0.5 0.1 0.4

0.0 0.6 0.4

discrete structures problems

1.Find a limit to show that x(In(x²))³ is O(x²). Simplify when possible to avoid doing more work than you have to. You will need to use L'Hôpital's rule at least once.

2.Suppose that f is o(g). What is lim(f(n)/g(n)) as n→ ∞?

3.Suppose that algorithm has run-time proportional to log n and takes 1 millisecond to process an array of size 3,000. How many milliseconds will it take to process an array of size 27,000,000,000 ? Hint: what simple relationship is there between the first number and the second number?

Want a mini project for a topic under Dynamical systems for an MSc student in Applied Mathematical Modelling

A data set contains the yearly tuitions in for undergraduate programs in arts and humanities at 66 universities and colleges. Tuition fees are different for domestic and international students. Suppose the mean tuition charged to domestic students was ​$5146, with a standard deviation of ​$944. For international​ students, suppose the mean was $14,504​, with a standard deviation of ​$3175. Which would be more​ unusual: a university or college with a domestic student tuition fee of ​$3000 or one with an international student tuition fee of ​$8500​? Explain.

Complete the statement below.

1. The domestic student tuition fee has a​ z-score of _______ and the international student tuition has a​ z-score of __________. Thus, the _____________ (domestic or international) is more unusual than the ___________ (domestic or international)

Provide an example of a proof by mathematical induction. Indicate whether the proof uses weak induction or strong induction. Clearly state the inductive hypothesis. Provide a justification at each step of the proof and highlight which step makes use of the inductive hypothesis.

Four sophomores, five juniors and six seniors have volunteered to serve on a committee consisting of five people. In how many ways can the membership of the committee be selected under the following restrictions respectively?

(a) Any of the volunteers can serve on the committee.

(b) Only seniors can serve on the committee.

(c) At least one senior must serve on the committee.

(d) At least two juniors and at least two seniors must serve on the committee.

(e) At least one person from each class must serve on the committee, but no more than two members of the same class can serve on the committee.

Find the solution to the problem of initial value 2y'''' +3y'''--16''+15y'-- 4y=0 subjected to y(0)= -- 2, y'(0)=6, y''(0)=3, y'''(0)=1/2

PART 1

The weighted voting systems for the voters A, B, C, ... are given in the form

{q: w₁, w₂, w₃, w₄, ..., w_n}.

The weight of voter A is w₁, the weight of voter B is w₂, the weight of voter C is w₃, and so on.

Consider the weighted voting system {78: 4, 74, 77}.

(a) Compute the Banzhaf power index for each voter in this system. (Round your answers to the nearest hundredth.)

(b) Voter B has a weight of 74 compared to only 4 for voter A, yet the results of part (a) show that voter A and voter B both have the same Banzhaf power index. Explain why it seems reasonable, in this voting system, to assign voters A and B the same Banzhaf power index. Select one of the following below.

Despite the varied weights, this is a minority system. Any one of the three voters can stop a quota.

Despite the varied weights, this is a dictator system. Voter C controls the outcome, while voters A and B are dummy voters.    

Despite the varied weights, in this system, all of the voters are needed for a quota.

Despite the varied weights, in this system, all voters are dummy voters. No voter is critical to a successful outcome.

Despite the varied weights, this is a majority system. Any two of the three voters are needed for a quota.

PART 2

The weighted voting systems for the voters A, B, C, ... are given in the form

At the beginning of each football season, the coaching staff at Vista High School must vote to decide which players to select for the team. They use the weighted voting system {7: 6, 5, 1}. In this voting system, the head coach A has a weight of 6, the assistant coach B has a weight of 5, and the junior varsity coach C has a weight of 1.

(a) Compute the Banzhaf power index for each of the coaches. (Round your answers to the nearest hundredth.)

(b) Explain why it seems reasonable that the assistant coach and the junior varsity coach have the same Banzhaf power index in this voting system. Select one of the following below.

As to forming a winning coalition, the two minor coaches are the same.

Winning coalitions often include support of different weight.    

The weightings for the minor coaches are different, so are their critical votes.

q: w₁, w₂, w₃, w₄, ..., w

PART 3

The weight of voter A is w₁, the weight of voter B is w₂, the weight of voter C is w₃, and so on.

Calculate, if possible, the Banzhaf power index for each voter. Round to the nearest hundredth. (If not possible, enter IMPOSSIBLE.)

{18: 18, 5, 2, 2, 1, 1}

R is included in (R-{0} )x(R-{0} )

R = {(x,y) : xy >0}

Show that R is an equivalent relation and find f its equivalent classes

Consider the following statements,

1. [ 0 , 1 ] × [ 0 , 1 ] with the dictionary order is complete.
2. [ 0 , 1 ] × [ 0 , 1 ) with the dictionary order is complete.
3. [ 0 , 1 ) × [ 0 , 1 ] with the dictionary order is complete.

Where the dictionary order on R × R is given by ( a , b ) < ( x , y ) if either a < x or a = x and b < y.

For each of these three statements either (i) prove it is true or (ii) provide a counterexample to show that it is false.

Adrian invested $1,600 at the beginning of every 6 months in an RRSP for 11 years. For the first 7 years it earned interest at a rate of 4.50% compounded semi-annually and for the next 4 years it earned interest at a rate of 5.60% compounded semi-annually.

a. Calculate the accumulated value of his investment after the first 7 years.

b. Calculate the accumulated value of her investment at the end of 11 years.

c. Calculate the amount of interest earned from the investment.

If an undamped spring-mass system with a mass that weighs 24 lb and a spring constant 9 lbin is suddenly set in motion at t=0 by an external force of 180cos(8t) lb, determine the position of the mass at any time. Assume that g=32 fts2. Solve for u in feet.

Let x, y be integers, and n be a natural number. Prove that x ^(2n) − y ^(2n) is divisible by x + y