Define a sequence from R as follows. Fix r > 1. Let a1 = 1 and define recursively, an+1 = (1/r) (an + r + 1). Show, by induction, that (an) is increasing and bounded above by (r+1)/(r−1) . Does the sequence converge?

Please solve the following:

u_t=u_xx, 0<x<1, t>0

u(0,t)=0, u(1,t)=A, t>0

u(x,0)=cosx, 0<x<1

if s₁ and s₂ are two simple functions then prove that the max and minimum of then are also simple function.

If |G| = p1* p2* p3 (distinct primes)
Then G has a normal subgroup.

Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING POLYNOMIAL exist and its unique.

Provide the definition of an Empirical Model, and provide two examples.

*** PLEASE EXPLAIN ANSWER ***

Dear Students:

I have recently been employed by HMS Nautical Inc to work on their submarine program. I have only some basic data to work with and no idea how to use it to get the information I need.

Here is what I know. First, I know that our Subs have a maximum running depth of 500 feet below sea level. I also know that a sub functioning at acceptable levels should be able to reach maximum depth in 10 minutes. Finally I know that I need to multiply the decent by a factor of 5 to achieve an accurate model. I also have a chart that lists times and depths for the sub.

Finally, I know that the sub follows a quadratic model when it descends and then ascends.

I have been told that you will be able to take this data and make sense of it. I would like a model for the path of the submarine as it descends to its running depth and then returns to the surface. I want to be able to use this model to predict where the sub will be at any time during its decent/ascent cycle. I would also like to know after how many minutes I should expect the sub to breech the surface of the water again.

Please explain clearly how you came up with the model so that I can repeat the process for new additions to our fleet of submarines. I appreciate any help you can give me in this matter. I would like your response returned to me either as a business letter !!!

Sincerely,

Nemo Hook

HMS Nautical In

Prove the converse of Theorem 3.3.4 by showing that if a set K ⊆ R is closed and bounded, then it is compact.

Theorem 3.3.4 A set K ⊆ R is compact if and only if it is closed and bounded.

Solve the given differential equation by undetermined coefficients. y'' − 8y' + 20y = 200x2 − 65xex

1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|.

(a) Prove that (R2,ρ) is a metric space.

(b) In (R2,ρ), sketch the open ball with center (0,0) and radius 1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if xn → a and xn → b for some a,b ∈ X, then a = b.

3. (Optional) Let (C[a,b],ρ) be the metric space discussed in example 10.6 on page 344 of Wade. If {fn} and f are in C[a,b], prove that fn → f with respect to ρ if and only if fn → f uniformly (in the sense of Chapter 7).

Prove the following.

Let T denote the integers divisible by three. Find a bijection f : Z→T (Z denotes all integers).

8. (20 pts)

a. RSA encryption. Let n = pq = (7)(17) = 119 and e = 5 define a (very modest) RSA public key encryption. Since 25 < 119 < 2525, we can only encode one letter (two digit representation) at a time. Use the function ? = ? mod ? to encode the word MATHY into a series of five numbers that are less than n.

b. To decrypt an RSA encrypted message, we need to find d, the multiplicative inverse of e modulo (p-1)(q-1). Use Euclidian algorithm and two-pass method to determine the Bezout coefficient of e for the RSA in Part a. above. Then write down the decryption function.

A 0 B 1 C 2 D 3 E 4 F 5 G 6 H 7 I 8 J 9 K 10 L 11 M 12 N 13 O 14 P 15 Q 16 R 17 S 18 T 19 U 20 V 21 W 22 X 23 Y 24 Z 25

TMA manufactures 37-in. high definition LCD televisions in two separate locations, Locations I and II. The output at Location I is at most 6000 televisions/month, whereas the output at Location II is at most 5000 televisions/month. TMA is the main supplier of televisions to the Pulsar Corporation, its holding company, which has priority in having all its requirements met. In a certain month, Pulsar placed orders for 3000 and 4000 televisions to be shipped to two of its factories located in City A and City B, respectively. The shipping costs (in dollars) per television from the two TMA plants to the two Pulsar factories are as follows.

TMA will ship x televisions from Location I to city A and y televisions from Location I to city B. Find a shipping schedule that meets the requirements of both companies while keeping costs to a minimum.