An element a in a ring R is called nilpotent if there exists an n such that aⁿ = 0.

(a) Find a non-zero nilpotent element in M2(Z).

(b) Let R be a ring and assume a, b ∈ R have a^t = 0 and b^m = 0 for some positive integers t and m. Find an n so that (a + b)ⁿ = 0. (You just need to find any n that will work, not the smallest!)

(c) Show that the set of nilpotent elements in a commutative ring R forms a subring of R.

(d) Does this subring from the previous question contain a unity?

(e) Are the Gaussian integers from an earlier question an integral domain? Explain your answer.

Each time draw 6 different numbers from 1 ~ 45 randomly

Let M denote the times you have to draw such that all the number from 1~45 has been drawn
What's the expected number of M.

please explain thoroughly. Thanks.

Find a basis and the dimension of the subspace:

V = {(x₁, x₂, x₃, x₄)| 2x₁ = x₂ + x₃, x₂ − 2x₄ = 0}

7.

(Please, if you are not willing to answer the question completely, please leave the question to someone who is!)

a. At the time of his daughter's birth, a man deposited $ 1,000 in an account that pays 6%; this amount is set every birthday. When he turned 12, he increased his appropriations to $ 1,500. Calculate the amount that will be available to her at age 18.

b. José earned $ 4,000,000 from the Puerto Rican lotus and will receive a check for $ 200,000 now and a similar one every year for 19 years. To guarantee these payments, the Electronic Lottery bought an anticipated annuity at the 10% interest rate compounded monthly. How much did the electronic Lottery cost the annuity?

Use MATLAB to figure out the following problem, if you do not know how to use MATLAB then please do not answer. Coding is required for the exercise.

For f(x) = arctan(x), find its zeros by implimenting Newtons method and the Secant method in Matlab. (Hint: Use Newtons method to calculate x1 for Secant method)

Comment all code please since I would like to learn how to do this correctly in MATLAB. Thank you.

Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no two vectors are parallel, and ?3 ⃗ is not in the plane spanned by ?1 ⃗ and ?2 ⃗ . Prove that {?1 ⃗ , ?2 ⃗ , ?3 ⃗ } forms a basis for ℝ3

Formulate the following problem using the following steps.

a. Define the decision variables.

b. Specify the objective function.

c. Specify the constraints and simplify them so that the left hand side of each constraint only contains terms involving the decision variables.

ChemLabs uses raw materials I and II to produce two domestic cleaning solutions A and B. The daily availabilities of raw materials I and II are 150 and 145 units respectively. One unit of solution A consumes .5 unit of raw material I and .6 unit of raw material II. One unit of solution uses .5 unit of raw material I and .4 unit of raw material II. The profits per unit of solutions A and B are $8 and $10 respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Formulate and solve graphically the optimal daily production amounts of A and B

Nepal and Tibet can both produce butter [B] or tea [T].

        If Nepal allocates all resources to butter, it can produce a maximum of 500 units a year. If all its

        resources are allocated to tea, it can produce a maximum of 2500 units.

If Tibet allocates all resources to butter, it can produce a maximum of 1500 units a year. If all its resources are allocated to tea, it can produce a maximum of 3000 units of tea.

[a] Political considerations initially mean that trade is not possible between Nepal and Tibet.

        If Nepal produces 1250 units of tea for itself, how many units of butter does it produce for itself?

               

        If Tibet also produced 1250 units of tea for itself, how many units of butter does it produce for itself?

        What is the combined amount of tea produced by the countries when trade is not possible? The combined amount of butter?

[b] If political considerations change and trade becomes possible, which good will Nepal trade to

        Tibet? Prove your answer.

[c]   If Nepal and Tibet pursue complete specialization in the production of their comparative advantage products when initiating trade, what is the combined amount of tea produced by the countries?

        Under complete specialization in comparative advantage, what is the combined amount of butter?

        Comparing combined production prior to the possibility of trade [see [a]] with combined production available under specialization, what are the potential gains from trade as measured by additional butter and/or tea?  

[d] Assuming complete specialization by both countries, identify a specific trade [i.e. an amount of tea traded for an amount of butter] that will leave both countries better off when compared to their positions in [a].

Solve the following initial value problem: tdy/dt+5y=5t with y(1)=8.

Put the problem in standard form.

Then find the integrating factor, ρ(t)=

and finally find y(t)=

A) Solve the initial value problem:

8x−4y√(x^2+1) * dy/dx=0

y(0)=−8

y(x)=

B)  Find the function y=y(x) (for x>0 ) which satisfies the separable differential equation

dy/dx=(10+16x)/xy^2 ; x>0

with the initial condition y(1)=2
y=

C) Find the solution to the differential equation

dy/dt=0.2(y−150)

if y=30 when t=0

y=

Q: Find the power series solution centered at the ordinary point x = 0 for each equation.
((x^2) + 1) y''-6y=0

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 400 eggs and 750 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha. How many quarts of each flavor should you make to earn the largest profit? HINT [See Example 2.] (If an answer does not exist, enter DNE.)

For each n ∈ N, let f_n(x) = a_nx + b_n where a_n,b_n are sequences of real numbers. Prove that if f_n → f on [0,1] where f is a function on [0,1], then the convergence is necessarily uniform.

These questions are about math cryptography

1) Encrypt the plaintext "this is a secret message" using the affine function
f(x) = 5x + 7 mod 26.

2) Determine the number of divisors of 2n, where n is a positive integer.

The simplex algorithm is to continue in this manner, always performing basis exchanges which improve the objective function, until no more exchanges are possible. We conclude with an example: Buzz Buzz Buzz Coffee has on hand 1 kg of coffee grounds, 1 gallon of milk and 10 cups of sugar. They can use these to make espressos, containing 8 grams of grounds and no milk or sugar; lattes, containing 15 grams of grounds, 0.0625 gallons of milk and 0.125 cups of sugar; or caf´e cubano, containing 7.5 grams of grounds, no milk and 0.125 cups of sugar. They will be able to sell all they produce, which they will sell at prices of $2 for espressos, $4 for lattes and $5 for a caf´e cubano.

Question 10. (5 points) Let e, l and c be the number of espressos, lattes and caf´es cubanos manufactured, and let g, m and s be the amounts of grounds, milk and sugar left over when they are done. Let p be the amount of money they take in. Record the linear equations relating e, l, c, g, m, s and p.

Question 11. (15 points) Start at the point where no drinks are made (so e = l = c = 0). Exchange one of these variables, in order to increase p. Repeat the process of exchanging a basis variable to increase p until there are no exchanges which will make p larger. How many of each drink should be made?