1. Explain the outcome of 3^4^5. In particular, what is the order of execution of the two exponentiation operations?

2. Write (5^4^3)−1 as a product of prime numbers.

3. The greatest common divisor of two integers a and b can be written as a linear combination (with integer coefficients k and ℓ) of a and b: gcd(a,b)=ka+ℓb.

   In Sage this is achieved with the command xgcd. Look in the help page of this command to write the greatest common divisor of 12214 and 2012 as an integer linear combination of these two numbers.

   Use Sage to verify your result.

4. What is the difference in Sage between 1/3+1/3+1/31/3+1/3+1/3 and 1.0/3+1.0/3+1.0/31.0/3+1.0/3+1.0/3? Explain.

1. Give your own example of a plane figure and its quadrature.
2. How does the quadrature of the triangle depend on the quadrature of the rectangle?
3. Draw a lune like Hippocrates.’ What is the relationship between the large semicircle and the semicircle that contains the lune?

"We want to verify that IP(·) and IP^-1(·) are truely inverse operations. We consider a vector x = (x1, x2, . . . ,x64) of 64 bit. Show that IPfive bits of x, i.e. for xi, i = 1,2,3,4,5.

Sketch the region of continuity for f (x; y) on a set of axes and sketch the region of

continuity for df/dy (x. y) on a separate set of axes. Apply Picard’s Theorem to determine whether the

solution exists and whether it is unique.

a) y'  = 2x²y + 3xy² ; y(1) = 2

b) y' = sqrt(2x - 3y) ; y(3) = 2

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.

1.let {v=(1,2,3,5,9),v2=(3,1,2,8,9),v3=(2,-5,5,9,4)} and {u1=(0,1,1,1,2),u2=(0,2,-2,-2,0)} be basis of subspaces V and U of R5 respectively.find a basis and the dimension of V+U and V intersection U.

2.does a matrix have a right inverse ?if so find one A=[2,-3,-7,11;3,-1,-7,13;1,2,0,2]

3.find the interpolating polynomial that passes through the point (1,2),)(-1,-8) and (2,1)

Consider the equation uux + uy = 0 with the initial condition

u(x, 0) = h(x) = ⇢ 0 for x > 0

u_o for x < 0,   with u_o< 0.

Show that there is a second weak solution with a shock along the line x = u_o y / 2

The solution in both mathematical and graphical presentation before and after the shock.

Find the adjoint of matrix A, the determinant of matrix A, and the determinant of the adjoint A.

A= 1 1 0 2

2 1 1 0

0 2 1 1

1 0 2 1

A SEIRS model with stochastic transmission :project proposal

National governments issue debt securities known as sovereign bonds, which can be denominated in either local currency or global reserve currencies, like the U.S. dollar or euro. First define what these bonds are. Why are these issued? Then discuss the issues that can arise when investors invest in these types of bonds. What are the advantages and disadvantages of these bonds? Are there unique issues that can arise only with this type of bond? Would you invest in sovereign bonds?

Show that radical 3, radical 5, radical 7, radical 24, and radical 31 are not rational numbers

Consider the initial value problem given below.

y'=x+4cos(xy), Y(0)=0

Use the improved​ Euler's method subroutine with step size h=0.3 to approximate the solution to the initial value problem at points x= 0.0,0.3,0.6.....3.0

4. (Applying LU and LUP decompositions) In this problem, we'll use the LU/LUP decomposition to solve a linear system of equations.

a) For A = [12 -8 13 -1 13;14 11 -5 -5 -7;1 -8 -9 10 8;-11 10 -8 3 8;-11 -8 4 2 -4] find matrices P, L, and U so that PA = LU using Matlab's lu function. Based on your results: did Matlab use pivoting during the lu-computation?

b) For b = [4;-4;-5;3;7] solve Ax = b using the LU decomposition as follows. Solving Ax = b is the same as solving PAx = Pb. (With P from a). Since PA = LU, we need to solve LUx = Pb, and we can split that into two triangular systems as follows: Ly = Pb, and Ux = y. Solve both of these systems using Matlab's linsolve, state x and y explicitly.

c) Compare the quality of the x you found in b to the solution of Ax = b you get from using linsolve. (As in 2d, work with the differences Ax - b).

d) You want so solve Ax = b for various vectors b, so you collect them into a single matrix B. So your goal is to find a matrix X so that AX = B (one column in X for each column in B). Working with P, L, and U from parts a/b we see that this amounts to solving two systems: LY = PB and UX = Y. For B = [18 -7 -14 10 -14 -2 13 12 -15 -15;-15 -14 4 -2 13 -16 15 -3 -15 14;3 12 -10 -17 2 19 -17 17 15 5;-1 -8 6 -11 20 -20 -4 -13 3 -6;-20 1 8 17 -17 11 -10 -10 2 1] solve these two equations using Matlab's linsolve.

First find Y in LY = PB, and then use that to find X in UX = Y. Check that AX - B is close to the zero matrix.

For this last problem, work with format short (or even format compact) so that the matrices don't use up too much screenspace.

2. (Solving linear systems)

Consider the linear system Ax = b

with A =[14 9 14 6 -10;-11 -11 5 8 6;15 -2 -14 8 -15;14 13 11 -3 -7;0 9 13 5 -14], and . b = [-4;8;6;0;10].

a) Verify that the linear system has a unique solution. Hint: use rref, rank, det, or any other Matlab method. Briefly explain the answer please.

You'll now solve Ax = b in three different ways. Store the three different solutions in four different variables (x1,x2,x3,x4, say), we need to compare them in parts c and d.

b1) Using rref, determine the solution x of the system Ax = b, and store it in variable x1. Hint: x1 is the final column of the reduced row echelon form of M = (A|b). Remember horzcat.

b2) Using Matlab's linsolve method, find a solution to the system Ax = b. Store the solution in x2.

b3) Using Matlab's inverse method (^(-1) or inv), solve the system as x = A^(-1)b, store the result in x3.

b4) Use the rref method from class to find an inverse matrix B of A. Use that to calculate x = Bb. Store the result in x4.

c) Compare the solutions x1, x2, x3,x4. Do any two of them agree? Does that contradict a) or not?

d) Try ranking the solutions by quality. To do so, compute Ax-b, for the various values of x, and see which x produces the smallest difference between Ax and b.

Q: (LU decomposition) Find the LU decomposition of A = [-3 2 5 1; 12 -4 -20 -2; -6 0 15 1; -9 6 35 4]. You can use the compact method which works within a single matrix or you can build L and U separately. State L and U explicitly, and verify (in Matlab) that A = L*U. Hint: Matlab's built-in lu function isn't useful, since it pivots.