Let E be an extension field of a finite field F, where F has q elements. Let a in E be an element which is algebraic over F with degree n. Show that F(a) has q^n elements. Please provide an unique answer and motivate all steps carefully. I also prefer that the solution is provided as written notes.

Let f : R → S and g : S → T be ring homomorphisms.

(a) Prove that g ◦ f : R → T is also a ring homomorphism.

(b) If f and g are isomorphisms, prove that g ◦ f is also an isomorphism.

Problem 10.

a. Construct of partition of N with exactly 4 elements and describe the equivalence relation defined by your partition. Remember the elements of a partition are sets.

b. Construct of partition of N with infinitely many elements and describe the equivalence relation defined by your partition.

c. Construct a partion of the plane with exactly 4 elements and describe the equivalence relation defined by your partition.

d. Construct a partion of the plane with infinitely many elements and describe the equivalence relation defined by your partition.

T/F

1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear function, where x = (x1,...,xn).

2) The function f(x1,x2,x3,x4) = (x2,x1,x4,x3) is linear.

3) For a given matrix A and vector b, equation Ax = b always has a solution if A is wide

a) ln(x² + y²)

calculate the Laplacian ∇² of the scalar field using both cartesian and cylindrical coordinate systems

b) (x² + y² + z²)^-1/2

calculate the Laplacian ∇² of the scalar field using both cartesian and spherical coordinate systems

PROVE THAT COS Z,SIN Z,cosh Z and sinh z are entire function

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F.

(a) Prove that

ev(0,0) : F[x, y] → F

p(x, y) → p(0, 0)

is a surjective ring homomorphism.

(b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]}

(c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y] is a maximal ideal.

(d) Find an ideal I ⊆ F[x, y] such that I is prime but not maximal. [HINT: Find a surjective homomorphism F[x, y] → F[x].]

(e) Find an ideal J ⊆ F[x, y] such that J is not prime

ASSIGNMENT #1

McGovern is a car manufacturing company. It builds 2 types of cars: a sports car and a sports utility vehicle (SUV). Its vehicles are very popular among its customers. Recently, increased demand for both vehicles has caused the company to revisit its total number of cars to produce and unit costs for those vehicles. Each sports car generates 10 kilowatt hours of energy to be produced and each SUV requires 20 kilowatt hour of energy to be produced. Each kilowatt hour costs .25. The following chart breaks down McGovern’s expenses for producing the companies.  Presume the production of the sports car and SUV’s are split equally between the two vehicles.

McGovern normally produces 2000 sports cars and 1500 SUV’s. The increased demand has the company estimating production needing to increase to 3500 sports cars and 3000 SUV’s. However, McGovern has the capacity to produce 5000 sports cars and 4000 SUV’s.

For this assignment, please do the following:

1.      Develop a graphical analysis of the operating costs in relation to the units produced for the sports car and SUV. Following the development of the graphs, provide an explanation of your graphs discussing the relationship of the costs with respect to the number of units produced.

2.      Determine the behavior per unit costs in relation to the fixed costs and variable costs and explain what takes place when you increase and decrease the number of units of the sports car and the SUV. (It may be best to create a table for each vehicle.)

3.      Determine the fixed cost per unit for each vehicle if it is at normal production, production due to increased demand, and if McGovern were to produce the vehicles at maximum capacity production. After calculating the fixed cost per unit for each vehicle, provide an explanation as to what happens to the fixed costs as the number of units increase with each production increase. Please be sure to show the work done to reach your conclusions.

4.      Finally, based on the cost information provided and the calculation you have performed, determine whether the company should maintain production, increase production based on demand, or produce at maximum capacity and provide an explanation as to why your selected option is the best option.

Problem 4.9.4 (10) In Section 2.10 we proved that every
partial order is the “path-below” relation of a graph called
a Hasse diagram. How does the Hasse diagram relate to
the graph of the partial order itself? Present the proof of
the Hasse Diagram Theorem using mathematical induction.

1. For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence.
   1. 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ...
   2. 7,11,15,19,23,27,31,35,39,43,...
   3. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, . . .
   4. 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, ...
   5. 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, ...
   6. 2, 4, 16, 256, 65536, 4294967296, ...

state and prove spectral mapping theorem

How many integer solutions are there to x₁+x₂+x₃+x₄= 100 with all of the following constraints:

10 ≤ x_{1 ,} 0≤ x₂ < 20 , 0 ≤ x₃ < 40 , 10 ≤ x₄< ₅₀

_{Please solve using the Principle of Inclusion/Exclusion.}

Suppose you are deciding whether to buy an electric or a natural gas hot water heater. Assume you live in California.

Note: 1 therm of natural gas equals ~30 kWh of electricity

Natural Gas Price in CA = $13.35 per Thousand Cubic Feet

Electric price in CA = .16 cents per kilowatt-hour (kWh)

Gas Water Heater Uses 258 therms per year

Water heater numbers are based on 40-gallon models with a 0.63 EF rating using 40,000 Btu/hr and based on the U.S. Department of Energy’s (DOE) national average of 64 gallons of hot water use daily.

Electric water heater Uses 4,881.25 kWh per year

An average water heater runs three hours daily. A 50-gallon, 5,500-watt water heater with a .90 EF and an electricity rate of $.16 per kilowatt hour.

For each sub question, show your assumptions and calculations.

1a) Which would you pick to minimize your operating costs?

1b) If the electric water heater cost $1250 and the natural gas water heater cost $1600, how much would this change your calculation for 1a?

1c) Which would you pick to minimize your total energy footprint, and why?

1d) Which would you pick to minimize your environmental footprint, and why?