Suppose X1, X2, ..., Xn iid∼ Multinomial (p). a) What is the MLE for the situation where p = (p1, p2, p3, p4), such that p1 = p2 and p3 = p4? b) What can you interpret from these estimates in terms of the number of observations of type k (k = 1, 2, 3, 4) compared the total number of trials n?

Let G, H be groups and define the relation ∼= where G ∼= H if there is an isomorphism ϕ : G → H.

(i) Show that the relation ∼= is an equivalence relation on the set of all groups.

(ii) Give an example of two different groups that are related.

Given a directed graph, prove that there exists an Eulerian cycle that is also a hamiltonian cycle if and only if the graph is a single cycle.

Let G be a group and let N ≤ G be a normal subgroup.

(i) Define the factor group G/N and show that G/N is a group.

(ii) Let G = S4, N = K4 = h(1, 2)(3, 4),(1, 3)(2, 4)i ≤ S4. Show that N is a normal subgroup of G and write out the set of cosets G/N.

Prove that every product of three glide reflection is a reflection or a glide reflection.

Puzzle #1 A kindergarten teacher knows that exactly one of four students, Kathy, Andy, Jose, or Dana, took Amy’s cookie. When asked about who did it, Kathy said: “Andy did it.”, Andy said: “Dana did it.”, Jose said: “I didn’t do it.”, and Dana said: “Andy lied when he said I did it.” 1. If the teacher knows that exactly one of the students is telling the truth, who did it? Explain.

2. If the teacher knows that exactly one of the students is lying, who did it? Explain. Puzzle #2 There are three neighbors living in a row. Each house has a different color and a different animal and person living in/at the house. Each person has a different profession. The horse lives in the first house. The doctor is Jamal’s neighbor. Jamal does not have a ferret as a neighbor. Carlos does not live in the blue house and Jamal does not live in the green house. Ann is a lawyer and lives in the 3rd house. The professor has the horse as a neighbor. The mouse does not live in the red house. The lawyer does not live in the blue house. 1. Determine the color of the house the horse lives in. This project will be scored out of 10 points in the following way:

**NUMBER THEORY**

Without calculating the products or using the calculator, find the remainders of the division. Demonstrate the process, the solution has been shown already.

a) 5²⁸⁵⁷⁴ divided by 17.

Solution : 15.

b) 35³⁴⁶ divided by 41.

Solution : 2.

c) 34 × 17 divided by 29.

Sol : 27

d) 19 × 14 divided by 23.

Sol : 13.

Determine if the following statements are true or false:

1. There is exactly one two-digit number (in the range from 10 through 99} that is congruent to 1 modulo 3, to 3 modulo 4, and to 3 modulo 5.

2. A minimum element for a partial order R on a set X is an element a such that for every element b, R(a, b) is true. If X is a nonempty finite set, then there must be exactly one minimum element for R.

3. If a and b are any two positive naturals, then the set {a + bi: i is a natural} contains infinitely many prime numbers.

4. There are exactly 32 numbers in the range from 0 through 95 that have inverses modulo 96.

Let x and y be integers such that 17 | (3x +5y). Prove that 17 | (8x + 19y).

Let arcman(y) = arcsin(y) - arctan(y). Find range and domain of arcman(y). Prove that y=man(z) exists when z=arcman(y). And find third order Taylor polynomial for arcman(y) and man(z).

1. Let M denote set of m&ms in a bag. This consist of red, yellow, green, blue and brown candies.

a. Device on equivalence relation on M.

b. Define relation R on M by: aRb if and only if either: a is green or b is blue, or, a is yellow abd b is not yellow. Is R symmetric, reflexive, transitive or anti-symmetric? Explain.

2. On set Zx(Z not including {0}) define relation R (a,b) a,b e Z, b#0 as follows:(a,b) R (c,d) if and only if ad=bc. Prove that R is an equivalence relation.

Producers' Surplus

The demand function for a certain brand of CD is given by

p = −0.01x² − 0.2x + 11

where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. The supply function is given by

p = 0.01x² + 0.6x + 1

where p is the unit price in dollars and x stands for the quantity that will be made available in the market by the supplier, measured in units of a thousand. Determine the producers' surplus if the market price is set at the equilibrium price. (Round your answer to the nearest dollar.)

Can someone make an example problem from these instructions?

This is Linear Algebra and this pertains to matrices.

Find the inverses and transposes of elementary and permutation matrices and their products.

Use your own numbers to create a problem for this or post a similar problem that describes this. I need this knowledge for a quiz.