Calculate the SVD of matrix A =

by hand and find the rank 1 approximation of A

Calculate the SVD of matrix A =

by hand and find the rank 1 approximation of A

v

To illustrate how to conduct rate-of-change calculations, we will use the following example. Note that this is just an example; the data in the table below do not match the data collected in this experiment.

Using these numbers, you need to calculate the rate of change in the relative frequency of stickleback with a complete pelvis per 1,000 years.

Step 1. Calculate the relative frequency of stickleback with a complete pelvis in each layer using this formula:

In this example, layer 1 had a total of 20 fish and 15 had a complete pelvis; the relative frequency of fish with a complete pelvis is 15/20 = 0.75. In other words, 75% of fish in that layer had a complete pelvis.

For layer 2 the relative frequency of fish with a complete pelvis is 0.5.

Step 2. Calculate the rate of change in relative frequencies between layer 1 and layer 2—a span of 3,000 years.

To do that, you subtract the number of the older layer (layer 1) from that of the more recent neighboring layer (layer 2).

Thus, the change in relative frequency of stickleback with a complete pelvis between layer 1 and layer 2 = 0.5-0.75 = -0.25. (Note that it is a negative number because the relative

frequency of fish with a complete pelvis decreased.)

Step 3. Calculate the rate of change for 1,000-year increments. To do this, you must divide each rate of change by 3 because there are 3 1,000-year increments between layers 1 and 2, and between layers 2 and 3, and so on.

So, the rate of change in relative frequency of stickleback with a complete pelvis between layer 1 and layer 2 per 1,000 years = -0.25/3 = -0.083. In other words, for every thousand years between layer 1 and layer 2 there is an average 8.3% decrease in the relative frequency of fish with the complete pelvis.

First 3,000 years
(From layer 1 to layer 2)

?

Next 3,000 years
(From layer 2 to layer 3)

?

Next 3,000 years
(From layer 3 to layer 4)

Next 3,000 years
(From layer 4 to layer 5)

?

Next 3,000 years
(From layer 5 to layer 6)

?

Rate of change per
thousand years

?

Show that the normalizer of a 5-Sylow subgroup of A_5 has order 10, and is a maximal subgroup of A_5.

Consider the missile allocation problem (MAP) with discretized time. (a) For MAP, an extreme case may be to maximize the probability of shooting down only those ASMs targeting the high value ships, ignoring the rest. How can you modify MAP model to accomplish that situation? (b) The probability of no leaker may be a very small figure, when there is a large number of attacking ASMs. For such cases, modify the objective function for maximizing the expected number of ASMs shot down.

This is a question about Ordinary Differential Equations.

For solving linear differential equations, I have seen people use the method of integrating factors and the method of variation of parameters.

Is it true that either of these 2 methods can be used to solve any linear differential equation?

If so, could you show me an example where a linear differential equation is solved using both of these methods.

If not, could you explain using examples as to why this is the case? For example, does it depend on the complicatedness of the equation etc.

Thank you.

Produce a 2/3 page explanation on

Applications of PDEs in Engineering,Science and Economics.

In your explanation you should give three detailed examples. One ex-

ample should involve a PDE from an area of Engineering, one example

from Science and one from Economics. In at least one example you

should discuss the solution to your PDE and how this has practical

implications

What is the lower bound on a bivariate CDF with uniform binary marginals? What is the upper bound on a bivariate CDF with uniform binary marginals? What do these bounds mean? What PDFs do they correspond to?

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by:

α(x1,x2)=(αx1,αx2)

(x1,x2)⊕(y1,y2)=(x1 +y1,0)

We use the symbol⊕to denote the addition operation for this system in order to avoid confusion with the usual addition x+y of row vectors.

Show that S, together with the ordinary scalar multiplication and the addition operation⊕, is not a vector space.

Test ALL of the eight axioms and report which axioms fail to hold.

Prove that the cross ratio of four complex numbers z1, z2, z3, z4 is real if and only if the points z1, z2, z3, z4 lie on a line or a circle. Then, compute the cross ratio of 1+√ 3, 1−3i, −1 − i and 1 + i and determine whether they lie on a line, a circle, or neither.

b) State the error ε that results in the approximation of the largest eigenvalue
of a symmetric matrix An×n by the power method. Assume the x not equals 0 is a given
real vector to be used in the computation.

4. Sampling Without Replacement

We have n−2 beer bottles b1,…,bn−2 and 2 cider bottles c1 and c2 . Consider a uniformly random permutation π1,…,πn of these n bottles (so that each of the n! permutations is equally likely).

1. Let X be the index of the first beer bottle in the permutation. That is, {π1,…,πX−1}⊆{c1,c2} and πX∈{b1,…,bn} . What is E[X] ?
2. Let Y be the index of the first cider bottle in the permutation. That is {π1,…,πY−1}⊆{b1,…,bn} and πX∈{c1,c2} . What is E[Y] ?