An inversion in σ ∈ S_n is a pair (i, j) such that i < j and σ(i) > σ(j). Take σ, τ ∈ S_n and take i < j.

- Suppose (i, j) is not an inversion in τ. Show that (i, j) is an inversion in στ if and only if (τ(i), τ(j)) is an inversion in σ.

- Suppose (i, j) is an inversion in τ. Show that (i, j) is an inversion in στ if and only if (τ(j), τ(i)) is not an inversion in σ.

For each collection of three groups of addition problems below, choose one strategy from the following list that may be useful in solving each collection of three problems. (“One more than and two more than”, “doubles”, “combinations of 10”, “making 10”, “using 5 as an anchor”, “near-doubles”).

a. 5+2, 1+6, 2+7

b. 6+8, 5+4, 6+5

c. 7+4, 7+9, 9+3

A loan of $310,000 is amortized over 30 years with payments at the end of each month and an interest rate of 6.9%, compounded monthly.

Answer the following, rounding to the nearest penny.

a) Find the amount of each payment. $
b) Find the total amount of interest paid during the first 15 payments. $

c) Find the total amount of interest paid over the life of the loan. $

d) Find the total of all payments made over 30 years. $


Suppose that payment number 6 is skipped and the interest owed for month 6 is added to the balance. Payments then resume as usual for the remainder of the 30 years.
e) Find the balance owing at the end of month 6. $
f) Find the balance remaining after the 360th payment. $

Find the general solution of the following differential equations:

a. y′′ − 2y′ + 6y = 0

b. y′′ + 6y′ + 13y = 0

13. Approximately how high would a stack of 4 million $1 bills be? (Assume there are 233 new $1 bills per inch. Round your answer to the nearest yard.)

14. A school in Oakland, California, spent $200,000 in changing its mascot sign. If the school had used this amount of money for chalk, estimate the length of the chalk laid end-to-end. (Assume a box of chalk contains 12 four inch pieces and a box costs $1. Round your answer to the nearest mile.)

16. Imagine you have written down the numbers from 1 to 10,000,000. What is the total number of zeros you have recorded?

2. Write each of the numbers in scientific notation and in floating-point notation (as on a calculator).

(a)    googol

Please show work! I have the answers to these parts, but I am having trouble working through the problem to find the answer.

Find the solution of the IVP:

d.) 5y" - y' = 0, y(0) = -1, y'(0) = -1 (answer: y(t)=4 - 5 e^t/5)

e.) 3y" - y = 0, y(0) = 3, y(t) is bounded for 0 <= t < infinity (answer: y = 3 exp(-(1/3)sqrt(3)t) )

g.) y" + 2y' + y = 0, y(0) = -1, y'(0) = 0 (answer: y(t) = - e^-t - e^-t t )

h.) y" + 9y = 0, y(0) = -1, y'(0) = -1 (answer: y(t) = -1/3 sin(3t) - cos(3t) )

i.) y" + 2y' + 5y = 0, y(0) = -1, y'(0) = -1 (answer: y(t) = -e^-t sin(2t) - e^-t cos(2t) )

Say as much as possible about each of the following system’s solution set (recall that there are 3 possible solution set sizes to linear systems. Can you use process of elimination to rule some out?) State the theorems you use to reach your conclusions.

(a) A consistent system of 5 equations in 8 variables.

(b) A system with 11 equations in 12 variables.

(c) A system with 16 equations and 13 variables. The reduced row-echelon form of the augmented matrix of the system has 14 pivot columns

Another common second-order R-K method is Ralston’s method. For this method, a2=2/3. Derive expressions for a1, p1, and q11.

Example #2: Write the following set of four linear equations with 4 unknowns x1, x2, x3, and x4 in the matrix form. Solve the equations using MATLAB.

0.1 x1+ 2.3 x2 + 3x3 + 4x4 =1

x1+ 3x2 -7x3 +5x4 =2

3x1+2x2+7x3 =3

x1 +2x2 +x3 +10x4=0

(b)Roots of Polynomials:

In order to obtain the roots of a polynomial with the coefficients a1,a2,a3 ,... (where a1 is the coefficient of the highest power, and so on in a descending order) using MATLAB, organize the coefficients of the equation using the following format: p=[a1, a2, a3, a4]

Then the roots (all the roots - real and complex) can be obtain by the following command: r=roots(p)

1. What was Benoit Mandelbrot and Ian Stewart’s arguments about simple formulas? What connections do these arguments have with the Mandelbrot Set?

Find the multiplicative inverse of x^4 + 1 using the extended euclidean algorithm with GF(2^8), modulo = 2

The number of basketballs that can be produced: Min 30,000 – Max 60,000

The number of footballs that can be produced: Min 20,000 – Max 40,000

Time to manufacture a Basketball: 0.5 hrs.

Time to manufacture a Football: 0.3 hrs.

Cost of labour -- 1 machine hour: $6.00

Cost of material-- 1 Basketball: $2.00

Cost of material-- 1 Football:  $1.25

Purpose of the Report

Description of the Problem

Methodology (which would include the model formulation)

Findings or Results

Recommendations or Conclusions

Be sure to address all relevant points, discuss any assumptions you are making, and highlight the following items in your report:

A recommendation for the number of basketballs and footballs to manufacture that maximizes net profit after taxes given the existing constraints.

A discussion of which constraints are binding and the amount of slack or surplus in the remaining constraints.

A list of recommendations as to what actions the company may take in the future to increase profitability, and how much extra profit the company might expect if the action is taken.  Note that these values can be used by the company to determine whether the expected gain in net profit will offset any capital investment required to implement your recommendations.

Remember that you are writing the report from the point of view of a consultant with senior management of ABCD, Ltd. as the intended audience.
Hints

You need to assume, or guess, an initial number of production units for each product and proceed with using Excel to calculate your Net Revenue for manufacturing. It is ideal to set up a separate section on your spreadsheet that presents the information to be used in the analysis. This information should be organized under the headings “Changing Cells,” “Constants,” “Calculations,” and “Income Statement.”

Once your spreadsheet model is designed, you can  proceed with setting Excel SOLVER to carry the calculation.  Excel SOLVER is an add-in for MS Excel that can be used for optimization and other linear programming models. Appendix 7.1 on page of 298 of your textbook provides an overview of how to formulate a model and use Solver to extract the required information.

Please also note that your tax will be applied to your Net profit [TR – TC], and if your total cost [TC] is greater than your total revenue [TR], you will have a loss that will be exempted from tax. So, in calculating your Tax you need to use an “IF Statement”, i.e., IF (profit <=0, then put Tax=0, otherwise calculate Tax).

Let A and B be sets. Prove the following please show in as much detail as possible

i. A ⊆ B is and only if A U B = B

ii. A ⊆ B is and only if A ∩ B = A

iii. A ⊆ B is and only if A \ B = empty set