Recognizing partitions - sets of strings.

(b) Let A be the set of words in the Oxford English Dictionary (OED). For each positive integer j, define Aj to be the set of all words with j letters in the OED. For example, the word "discrete" is an element of A8 because the word "discrete" has 8 letters. The longest word in the OED is "pneumonoultramicroscopicsilicovolcanoconiosis" which has 45 letters. You can assume that for any integer i in the range 1 through 45, there is at least one word in the OED that has i letters. Do the sets A1, …, A45 form a partition of the set of words in the OED?

(don't give explanation with reflexive, symmetric and transitive because we haven't learn it yet)

I know if f(x) is even the fourier series expansion will consists of consnx, for like f(x)=x^2sinx, or f(x)=2/(3+cosx). but if the f(x) is neither even or odd, would fourier expansion have both cosnx and sinnx? This is PDE.

Solve the partial differential equation by Laplace transformu_x (x,t)+u_t (x,t)=e^3t given that u(x,o)=0 , u(o,t)=e^3t

Let n greater than or equal to 2 and let k1,...,kn be positive integers. Recall that Ck1,..., Ckn denote the cyclic groups of order k1,...,kn. Prove by induction that their direct product Ck1×Ck2×....×Ckn is cyclic if and only if the ki's are pairwise coprime which means gcd(ki,kj)=1 for every i not equals j in {1,...n}.

find all eigenvalues and eigenvectors of the given matrix

A= [1 0 0

2 1 -2

3 2 1]

How many arrangements of six 0's, five 1's and four 2's are there in which

(a) the first 0 precedes the first 1?

(b) the first 0 precedes the first 1, which precedes the first 2?

The stockholders' equity section of the balance sheet appears as follows at January 1, 2014:

On March 1, 2014, the company repurchased 800 shares of its common stock at $12 per share but on April 6, 2014, it reissued 600 shares of the shares at $20 per share.

A) Prepare the journal entries to record for the March 1 and April 6 transactions.

B) How many shares of common stock are outstanding at March 31, 2014, and April 30, 2014, respectively?

March 31, 2014: $

April 30, 2014: $

Identify the parameter p in the following binomial distribution scenario. The probability of buying a movie ticket with a popcorn coupon is 0.772, and the probability of buying a movie ticket without a popcorn coupon is 0.228. If you buy 15 movie tickets, we want to know the probability that more than 10 of the tickets have popcorn coupons. (Consider tickets with popcorn coupons as successes in the binomial distribution.)

1. Create a scenario that can be used to model a counting question (permutation, combination or neither.)

2. Create a scenario that can be used to model a counting question (permutation, combination or neither.)

3. Create a scenario that can be used to model a counting question with multiple 'not', 'and', 'or' clauses

Question 1 [35 marks] A foundry that specializes in producing custom blended alloys has received an order for 1 000 kg of an alloy containing at least 5% chromium and not more than 50% iron. Four types of scrap which can be easily acquired can be blended to produce the order. The cost and metal characteristics of the four scrap types are given below: Scrap type Item 1 2 3 4 Chromium 5% 4% - 8% Iron 40% 80% 60% 32% Cost per kg R6 R5 R4 R7 The purchasing manager has formulated the following LP model: Minimise COST = 6M1 + 5M2 + 4M3 + 7M4 subject to 0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM) 0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON) M1 + M2 + M3 + M4 = 1000 (MASS) and all variables ≥ 0, where Mi = number of kg of scrap type i purchased, i=1,2,3,4. (a) Solve this model using LINDO or SOLVER. (b) Write down the foundry's optimal purchasing plan and cost. 4 PBA4804 OCTOBER/NOVEMBER 2019 PORTFOLIO EXAMINATION [TURN OVER] Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.) (c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order? (d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase? (e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this? (f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

Find the solution to the heat equation on 0 < x < l,
with u(0, t) = 0, u_x(l, t) = 0, and u(x, 0) = phi(x).
This is sometimes called a "mixed" boundary condition.

Given a commutative ring with identity R.

A. Prove that a unit is not a zero divisor.

B. Prove that a zero divisor cannot be a unit.

How to write a self-defined function to find the inverse of a matrix by using Gaussian Jordan method in MATLAB? Thanks.

Prove that there exists integers m and n such that 15m + 12n = 3

Please do not prove by assuming m=1 and n=-1, I'd like to prove by not assuming any actual numbers.

Show that orbit of the Thue-Morse sequence is dense in the shift space.