4. Suppose there are 99 lockers numbered 1 through 99, and 99 students numbered
1 through 99. Initially, all lockers are closed. Now each odd-numbered student
1, 3, 5, 7, · · · , 99, in numerical order from 1 through 99, will open/close all the lockers
that are numbered to be a multiple of the number of the student. For example, student
1 will open/close all the lockers, and student 3 will open/close all the lockers numbered
by 3, 6, 9, 12, · · · , 99, etc. Now find all the lockers that will be open when all students
are done.
Note: An answer containing a computer search only is worth 10 marks. But you can
start a computer search first and see if the results inspire you.

Solve by variation of parameters:

A. y" + 6y' + 9y = e^(-3t)/(1 + t^2)

B. y" − y = e^t − e^-t.

Suppose that k is a field which is not algebraically closed. a. Show that if I ⊂ k[x1, . . . , xn ] is maximal, then V(I) is either empty or a point in kn . Hint: Examine the proof of Theorem 11. b. Show that there exists a maximal ideal I in k[x1, . . . , xn ] for which V(I) = ∅. Hint: See the previous exercise. c. Conclude that if k is not algebraically closed, there is always a maximal ideal of k[x1, . . . , xn ] which is not of the form <x1 − a1, . . . , xn − an >

Problem 2: Let f and g be two differentiable functions defined on an interval (a,b).

Assume that g(x) dne 0 for all x ∈ (a, b). Prove that f/g is differentiable and (f/g)'(x) = (f'(x)g(x)-f(x)g'(x))/(g^2(x))

for all x ∈ (a, b)

Let R be the following relation on the set of all alive people in the world:

x R y if and only if x and y have the same pair of biological parents.

Prove that R is an equivalence relation.

a. Let f be a real function. Prove that f is convex iff −f is concave.

b. Let f and g be real functions. Prove that if f and g are convex, then f + g is convex

Prove the set identities...

Theorem 2.2.19 (Set identities I). Let A be a set with universal set U.

Identity laws: A ∩ U = A; A ∪ ∅ = A

Domination laws: A ∪ U = U; A ∩ ∅ = ∅

Idempotent laws: A ∪ A = A; A ∩ A = A

Complementation law: A = A

Complement laws: A ∪ A = U; A ∩ A = ∅

Please help me with this question.. Thanks in advance,

Sincerely,

Kind Regards..

Find the charge on the capacitor in an LRC-series circuit at

t = 0.05 s

when

L = 0.05 h,

R = 1 Ω,

C = 0.04 f,

E(t) = 0 V,

q(0) = 3 C,

and

i(0) = 0 A.

(Round your answer to four decimal places.)

Determine the first time at which the charge on the capacitor is equal to zero. (Round your answer to four decimal places.)

THIS IS THE ENTIRE PROBLEM GIVEN -

National signing day for rugby recruiting season 2020 has been completed – now, as the recruiting coordinator for the San Diego State University Aztec rugby team, it is time to analyze the results and plan for 2021.

You’ve developed complex analytics and data collection processes and applied them for the past few recruiting seasons to help you develop a plan for 2021. Basically, you have divided the area in which you actively recruit rugby players into 8 different regions. Each region has a per target cost, a “star rating” (average recruit ‘star’ ranking, from 0 to 5, similar to what Rivals uses for football), a yield or acceptance rate percentage (the percentage of targeted recruits who come to SDSU), and a visibility measure, which represents a measure of how much publicity SDSU gets for recruiting in that region, measured per target (increased visibility will enhance future recruiting efforts).

Your goal is to create a LINEAR mathematical model that determines the number of target recruits you should pursue in each region in order to have an estimated yield (expected number ) of at least 25 rugby recruits for next year while minimizing cost. (Region 1 with yield of 40% - if we target 10 people, the expected number that will come is .4*10 = 4.)

In determining the optimal number of targets in each region (which, not surprisingly, should be integer values), you must also satisfy the following conditions.

1) No more than 20% of the total targets (not the expected number of recruits) should be from any one region.

2) Each region should have at least 4% of the total targets (again, not the expected number of recruits, but the number of targets).

3) The average star rating of the targets must be at least equal to 3.3.

4) The average visibility value of the targets must be at least equal to 3.5.

Off on the recruiting trail you go!! (IN EXCEL)

a) Define a cyclic group

b) Give an example of a cyclic group with exactly four distinct generators

c) Give an example of a cyclic group with infinitely many cyclic subgroups

d) Prove that no cyclic group can have exactly three distinct generators

3.1.24 Prove that <f,g> = the integral f(x)*g(x)dx from 0 to 1 does not define an inner product on the vector space C0[-1,1]. Explain why this does not contradict the fact that it defines an inner product on the vector space C0[0,1]. Does it define an inner product on the subspace P(n) C C0[-1,1] consisting of all polynomial functions?

Use the golden Section search method to find the minimum of f(x)= x/5−sin(x) . Start with the range of 0 to 3, i.e., xl=0, xu=3 . Show two iterations of the Golden Section Search Method by populating the following Table. Again please do all calculations in MATLAB and make sure you have included it in your submission.

i xl f(xl) x2 f(x2) x1 f(x1) xu f(xu) d

5a

Please code in language Ocaml

In each of the three parts in this problem, you will get full credit if you use foldT and define at most one helper function.

(a) Define an OCaml function leafCount : ’a binTree -> int which returns an integer representing the total number of leaves in a tree.

Starter Code:

type 'a binTree =
| Leaf
| Node of 'a * ('a binTree) * ('a binTree)

let leafCount (t : 'a binTree) : int = (* your work here *) 0

prove by using induction. Prove by using induction. If r is a real number with r not equal to 1, then for all n that are integers with n greater than or equal to one, r + r^2 + ....+ r^n = r(1-r^n)/(1-r)

1) Given f(x) = x^2 + x + 1 and g(x) = x^3 + x, compute f(x) * g(x) in GF(2^4) with the irreducible polynomial m(x) = x^4 + x + 1.

1.5) Give the set of polynomials for finite field of the form GF(2 ⁴ ).