1.) How many relations are there from a set of size n to a set of size m?

2.) Determine the number of entries in the following sequences:

a.) {13, 19, 25, . . . , 601}

b. {7, 11, 19, 35, 67, . . . , 131075}

Let G be a nonabelian group of order 253=23(11), let P<G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup.

a. What are the orders of P and Q. (Explain and include any theorems used).

b. How many distinct conjugates of P and Q are there in G? n23? n11? (Explain, include any theorems used).

c. Prove that G is isomorphic to the semidirect product of P and Q.

(a) Find the equilibrium solution, or critical point, of the given system.

(b) Use a computer to draw a direction field and phase portrait centered at the critical point.

(c) Describe how solutions of the system behave in the vicinity of the critical point.

x′ =−0.25x−0.75y+8, y′ =0.5x+y−11.5

(d) Let x= x_c+u and y= y_c+v, where x_c and y_c give the critical point you found in (a). Plug these into the system and show that you obtain a homogeneous system u′ = Au for u = (u v)^T .

(e) Solve the resulting homogeneous system for u and v, and show that the solutions you obtain match the phase portrait that you generated in (b).

Q1. The following table shows the quantity supplied and quantity demanded of a commodity at certain unit prices.

Unit Price
$1.50
1150
340

Quantity Demanded
$2.75
712.5
652.5

Quantity Supplied
$3.25
537.5
777.5

a. From the information given in the table above, describe in your own words, how would you go about showing that the quantity demanded and supplied are linear functions of price, without having to plot a graph.

b. By demonstrating what you have describe above, show that the quantity demanded and supplied are linear functions of price and that price is equal to $2.85 when quantity demanded is equal to quantity supplied.

Q4. A fish processing company in Charlotteville processes three types of products: Type A, Type B and Type C. Its operations involve three basic activities: cleaning, cutting and packaging. Type A products require 4 minutes of cleaning 2 minutes of cutting and 2 minutes of packaging time. Type B products require 6 minutes of cleaning 4 minutes of cutting and 2 minutes of packaging time. Type C products require 8 minutes of cleaning 2 minutes of cutting and 4 minutes of packaging time.

Given that the total time available for cleaning, cutting and packaging is 3.5hours, 2.5 hours and 1.5 hours respectively.

a. Clearly defining your variables, show the model which represents the information given in this problem. State any restrictions on the variables in the problem.

Ten students shall be sorted into the four houses (namely Gryffindor, Slytherin, Ravenclaw, and Hufflepuff) in Hogwarts. If at least one student shall go to each house, how many ways can the students be sorted?

in how many ways can you distribute 13 identical pieces of candy to 5 children, if two of the children are twins and must get an equal number of pieces and every child must receive at least one piece?

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]_p, [2]_p, . . . , [p-1]_p}. Prove that for y ≠ 0, L_y restricts to a bijective map L_y|_s : S → S.

11.5 Prove Fermat's Little Theorem

solve the initial value problem

Y" + 2Y' - Y = 0, Y(0)=0,Y'(0) = 2sqrt2

With regards to elliptic curves, what is the point at infinity? Consider cases where x and y vary over real numbers.

In Elliptic Curves, when computing A⨁B = C , we take the line through A and B and find the point it intersects the curve. We then reflect through the x-axis. Why do we reflect at the x-axis?

use laplace to answer y"-3y'+2y=1+cost+e^-t,y(0)=1,y'(0)=0

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the 19th, 24th, 60th, and 70th percentiles.

If needed, round your answers to two decimal digits.

Percentile 19% 24% 60% 70%

Value for percentile?