A list of six positive integers, p, q, r, s, t, u satisfies p < q < r < s < t < u. There are exactly 15 pairs of numbers that can be formed by choosing two different numbers from this list. The sums of these 15 pairs of numbers are: 25, 30, 38, 41, 49, 52, 54, 63, 68, 76, 79, 90, 95, 103, 117. Which sum equals r + s?

A 2nd order homogeneous linear differential equation has odd-even parity. Prove that if one of its solutions is an even function and the other can be constructed as an odd function.

'Find all conjugacy classes and their sizes in Z3XA4'. The posted solutions to this question are vague and lacking details and I cannot follow what was done. There is no full explanation how to do this problem and I do not understand how to even start. I do not want the solution. I want to understand how to do this problem.

Please start by defining Z3 and A4. Then please go step by step through this problem as if explaining to a simpleton.  

Thank you.

1. You have an order to give Penicillin 150,000 Units (I.M.). You have a vial of Penicillin containing 3,000,000 Units in 10ml. How many ml. will you give?______ 2. The doctor orders Demerol 25 mg. (I.M.) q4h prn. You have Demerol 100mg. in 2ml. How many ml will you give? ________ 3. Your patient is to receive Digoxin 0.125 mg. On hand you have Digoxin 500 mcg in 2 ml. How many ml will you give? ______ _ 4. The drug order reads ASA gr.X q4h prn. for headache. You have ASA 325 mg tablets. How many tablets will you give? ________ 5. The doctor's order reads: Phenobarbital gr. ss (1/2) q6h. You have Phenobarbital 15 mg tablets. How many tablets will you give? ____ 6. In question #5, how many tablets will you give in 24 hours? ____ 7. The doctor orders Atropine gr. 1/200 I.M. stat. You have a vial that reads: Atropine gr. 1/150 in 2ml. How many ml will you give? ______ 8. Your patient is to receive Atropine 0.3 mg. You have Atropine 1/150 gr. in 1 ml. How many ml will you give?______ 9. The order reads: Valium 15 mg po T.I.D. The labe! reads: Valium gr. ½. How many tablets would you give in a 24-hour period? ______ 10. The doctor orders Streptomycin 400mg. The vial label reads Streptomycin 1g in 2.5ml. How many ml will you give? _

Apply Newton’s method and the modified method to the equation f(x) = 1−cos(x−5) to approximate the a double root 5. Compare the results and demonstrate the superiority of the modified method. Numerically identify the rates of convergence of both the methods.

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0.

1. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS.

   (ii) Show that S has a least upper bound if and only if cS has a least upper bound, and that (when these sets have least upper bounds), they are related by

   l.u.b.(cS) = c · l.u.b.(S).

(a) Consider the function f(x)=(e^x −1)/x.

Use l’Hˆopital’s rule to show that lim f(x) = 1 when x approaches 0

(b) Check this result empirically by writing a program to compute f(x) for x = 10−k, k = 1,...,15. Do your results agree with theoretical expectations? Explain why.

(c) Perform the experiment in part b again, this time using the mathematically equivalent formulation, f(x)=(e^x −1)/log(e^x), evaluated as indicated, with no simplification. If this works any better, can you explain why?

Please ignore the programming part. I am interested in knowing why this happens.

Suppose Alice is using a block cipher to send the message
THE ORDER IS KARL, ANDY, FRED AND IAN. IAN AND ANDY HAVE LEFT.

to Bob. Assume that

• the block cipher is used in ECB mode,

• the English is divided into plaintext blocks of 2 letters (ignore spaces and punctuation)

• the ciphertext blocks are denoted C1,C2,...,C23
  
  (a) Write down the 23 plaintext blocks.
  (b) Will any of the ciphertext blocks be repeated? If so, which ones?
  (c) Suppose an attacker manipulates the ciphertext so that instead of receiving C1,C2,...,C23.
  
  Bob receives

C1, C2, C3, C4, C5, C10, C11, C6, C7, C8, C9, C12, C13, C14, C15, C16, C17, C6, C7, C20, C21, C22, C23

What does Bob think the message Alice sent is? (add spaces and punctuation)

(d) If Bob receives

C10, C8, C6, C17, C6, C7, C20, C21, C22, C23
what does he think the message is? (add spaces and punctuation)

(e) Find another sequence of ciphertext blocks that decrypts to a plausible plaintext sentence of at least five words (add spaces and punctuation to your sentence).

Using a system of equations, prove that a 2x2 matrix A has a right inverse if and only if it has a left inverse.

Find the straight line distance from Glassboro, nj to New Delhi, India. Use the assumptions given in Problem 75 of Section 12.7 (p. 684). You must use spherical coordinates. You will need to look up the coordinates of the two cities.

Glassboro, NJ Coordinates: (39.7029° N, 75.1118° W)

New Delhi, India Coordinates: (28.6139° N, 77.2090° E)

Problem 75 of section 12.7 states:

Consider a rectangular coordinate system with its origin at the center of the earth, z-axis through the north pole, and x-axis through the prime meridian. Find the rectangular coordinates of Sydney Australia (34 S, 151 E), and Bogota Columbia (4 32' N, 74 15 ' W). A minute is 1/60 degrees. Assume that the earth is a sphere of radius R = 6370 km.

You can verify that the differential equation: −7x2y′′−14x(x−1)y′+14(x−1)y=0−7x2y″−14x(x−1)y′+14(x−1)y=0, x>0x>0 has solutions y1=3xy1=3x and y2=5xexp(−2x)y2=5xexp⁡(−2x).

1. Compute the Wronskian WW between y1y1 and y
2. 
   The solutions y1y1 and y2y2 form a fundamental set of solutions because there is a point x0x0 where W(x0)≠0W(x0)≠0

• Let G be an abelian group.

  (a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G. (b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G? (Give a proof or counterexample.)

Solve each system. Express the solution set using vectors. Identify a particular solution and the solution set of the homogeneous system.

(a) 3x + 6y = 18

x + 2y = 6

(b) x + y = 1

x − y = −1

(c) x₁ + x₃= 4

x₁ − x₂ + 2x₃ = 5

4x₁ − x₂ + 5x₃ = 17

(d) 2a + b − c = 2

2a + c = 3

a − b = 0

(e) x + 2y − z = 3

2x + y + w = 4

x − y + z + w = 1

(f) x + z + w = 4

2x + y − w = 2

3x + y + z = 7

Reduce the following argument into an argument form, and then perform the truth table test for validity. Create and label the entire truth table.
If your program contains a syntax error or a semantic error, then your program will fail to compile.
Your program contains a syntax error and a semantic error.
Therefore, your program will not fail to compile.