A person buying a personal computer system is offered a choice of three models of the basic unit, four models of keyboard, and three models of printer. How many distinct systems can be purchased?

  systems

1) Show that if A is an open set in R and k ∈ R \ {0}, then the set kA = {ka | a ∈ A} is open.

1. Find the 100th and the nth term for each of the following sequences.

a. 50,90,130,...

b. 1,4,16,...

c. 7, 7⁴,7⁷,7¹⁰,...

d. 197+7x3²⁷, 197+8x3²⁷, 197+9x3²⁷,...

2. Find the first five terms in sequences with the following nth terms.

a. 10ⁿ-5

b.2n-1

3. How many terms are there in each of the following sequences?

a. 11,15,19,23,...331

b. 59,60,61,62,...459

find all primes p such that 6 is a square mod p?

please with clear hand writing

Can you explain how kernels related with uniqueness, while images related with consistent?

Problem 4-23 (Algorithmic)

EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 15,000 windows and ended the month with 9,500 windows in inventory. EZ-Windows’ management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month is possible.

The company’s cost accounting department estimates that increasing production by one window from one month to the next will increase total costs by $1.00 for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $0.65 for each unit decrease in the production level. Ignoring production and inventory carrying costs, formulate and solve a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts. If required, round your answers to two decimal places. If an amount is zero, enter "0".

Let:

F = number of windows manufactured in February

M = number of windows manufactured in March

A = number of windows manufactured in April

I_m = increase in production level necessary during month m

D_m = decrease in production level necessary during month m

s_m = ending inventory in month m

If required, round your answers to the nearest dollar.

Cost: $  

Differential Geometry

3. Evaluate the 1-form f = x² dx - y² dz on the vector fields V = xU1 + yU2 + zU3, W = xy (U1 - U3) + yz (U1 - U2), and (1/x)V + (1/y)W.

Let Q(t)=x^2. Find a formula for the slope of the secant line over the interval [9,t] and use it to estimate the slope of the tangent line at t=9t=9. Repeate for the interval [6,t] and for the slope of the tangent line at t=6

The slope of tangent line at t=6 is approximately
The slope of tangent line at t=9 is approximately

Prove: There are infinitely many primes of the form 6n − 1 (n is an integer).

1. Develop a well-structured MATLAB function to compute the Maclaurin series expansion

   for the cosine function and name the function cosMac. The function should have the following features:

   1. Iterate until the relative error (variable name “relErr” ) falls below a stopping criterion OR exceeds a maximum number of iterations (variable name“maxIter”) for a given value of x.

   2. Include a default value of relErr = 1e-6 if the user does not enter the value (use nargin function).

   3. Include a default value for maxIter = 100 if the user does not enter the value (use nargin function).

   4. Return the estimate of cos(x), the approximate relative error, the number of iterations, and the true relative error (that you can calculate based on the built- in cosine function).

Use the Newton’s method to find the root for e^x+1 = 2 + x, over [−2, 2]. Can you find a way to numerically determine whether the convergence is roughly quadratic using error produced at each iteration? Include your answers as Matlab code comments

Describe a setting where you could use exponential functions to make investment decisions.

What kind of information could exponential functions tell you that would be valuable?

show that for any n the matrix ring M_n(F) is simple over a field F.

show your work. Do not use quotient rings!

A number of guests gather around a round table for a dinner. Between every adjacent pair of guests, there is a plate for tips. When everyone has finished eating, each person places half their tip in the plate to their left and half in the plate to their right. Suppose you can only see the amount of tips in each plate after everyone has left. Can you deduce the amount that each individual tipped?

(a) Suppose six guests sit around a table and there are six plates of tips. If we know the amount of tip in each plate, P1 to P6, can we determine each individual’s tip amount, G1 to G6 (G1+G6 = P1, G1+G2 = P2, G2+G3 = P3, ... , G5+G6 = P6)? If yes, explain why by examining the relationship between the plate values, P1 to P6, and guest tips, G1 to G6. If not, give two different assignments of G1 to G6 that will result in the same P1 to P6.

(b) Now lets consider five guests at the table, G1 to G5, and we can see the amount of tips in the five plates, P1 to P5 ((G1+G5 = P1, G1+G2 = P2, G2+G3 = P3, ... , G4+G5 = P5)). In this new setting can you figure out each guests tip values, G1 to G5?

(c) If n is the total number of guests sitting around a table, for which values of n can you figure out everyone’s tip? You do not have to rigorously prove your answer. (Hint: consider what is different about parts a and b.)

Suppose that we have access to an unlimited number of 5 and 11 cent stamps. Prove, using simple induction, that we can use these stamps to make any amount of postage that is at least 40 cents.