Use the classification of groups with six elements to show that A(4) has no subgroup with 6 elements. [ Hint: check that the product of any two elements of A(4) of order 2 has order 2]

Let A∈R^{n× n} be a non-symmetric matrix.

Prove that |λ₁| is real, provided that |λ₁|>|λ₂|≥|λ₃|≥...≥|λ_n| where λ_i , i= 1,...,n are the eigenvalues of A, while others can be real or not real.

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is injective, but not surjective.

(b) Suppose g : R\{−1} → R\{a} is a function such that g(x) = x−1, where a ∈ R. Determine x+1

a, show that g is bijective and determine its inverse function.

Please review all of the course materials regarding Financial Markets: Savings and Investment Vehicles before you begin this assignment.  You will also need to reference the Tax Cuts and Jobs Act of 2017.

The questions below also rely on the following assumptions:

• You are 30 years old and your employer sponsors a 401(k) plan with a 4% employer match.  
• You earn $100,000 of gross wage income. This income is expected to stay constant over the next three years.
• At the start of every year you decide to invest 4% of your salary into your 401(k).
• Your expected return on your investments is 5% per year.
• You file your taxes as a single filer and you are in the 24% tax bracket.
• The long term capital gains tax is 15%.

A. Calculate the total amount of funds that you expect to be in your 401(k) at the end of three years. Explain your answer.

B. At the end of the third year, you decide to withdraw $15,000 from your 401(k) to pay for some home improvements. Calculate how much tax, if any, you will owe on this withdrawal. Explain your answer.

C. During this same three year period you also invested in a Roth IRA. At the end of this three year period, your Roth IRA had cumulative contributions of $15,000 and earnings or gains of $5,000.

Suppose you decided to fund your home improvements by withdrawing from your Roth IRA instead of your 401(k). Calculate how much tax, if any, would you owe in this withdrawal. Explain your answer.

Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

4e^-x2 sin(x) = x² − x + 1

The neighborhood of a vertex in a graph consists of the vertex itself, together with all vertices that are connected to it by an edge. Each graph has a variable xi associated with the i-th vertex, and the vertex has a known value that is equal to the sum of the variables for all neighborhood vertices. Start with a graph with 5 vertices forming a pentagon, with edges joining vertices 1 and 2, 2 and 3, 3 and 4, 4 and 5, and 5 and 1. Then draw an edge joining vertices 2 and 4, and an edge joining vertices 2 and 5. The known values at vertices 1 through 5 are, respectively, 2, 1, −1, 3, and 5.

(a) Find the augmented matrix for the system of equations satisfied by x₁, x₂, x₃, x₄, x_5.

Box-Jenkins : For each of the non-seasonal models presented below, indicate whether or not it complies with the Box-Jenkins approach, ie is it stationary and invertible? Justify your answers.
a) Yt = et − 0,67 Yt−1
b) Yt = et + 0,43 Yt−1 − 0,37 Yt−2
c) Yt = et + 0,25 et−1
d) Yt = et + 0,9 Yt−1 + 0,3 Yt−2
e) Yt = et + 0,9 Yt−1 + 0,3 et−1

MATLAB question!

4. (a) Modify the code, to compute and plot the quantity E = 1/2mv^2 + 1/2ky^2 as a function of time. What do you observe? Is the energy conserved in this case?

(b) Show analytically that dE/dt < 0 for c > 0 while dE/dt > 0 for c < 0.

(c) Modify the code to plot v vs y (phase plot). Comment on the behavior of the curve in the context of the motion of the spring. Does the graph ever get close to the origin? Why or why not?

given code
---------------------------------------------------------------

clear all;   
m = 4; % mass [kg]
k = 9; % spring constant [N/m]
c = 4; % friction coefficient [Ns/m]
omega0 = sqrt(k/m); p = c/(2*m);
y0 =-0.8; v0 = 0.3; % initial conditions
[t,Y] = ode45(@f,[0,15],[y0,v0],[],omega0, p); % solve for 0<t<15
y = Y(:,1); v = Y(:,2); % retrieve y, v from Y
figure(1); plot(t,y,'ro-',t,v,'b+-');% time series for y and v
grid on; axis tight;
%---------------------------------------------------
function dYdt = f(t,Y,omega0,p); % function defining the DE
y = Y(1); v = Y(2);
dYdt=[ v ; -omega0^2*y-2*p*v]; % fill-in dv/dt
end

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Differential Equations

Use a series solution to solve the IVP .

y′′−4y= 0

y(0) = 3

y′(0) = 5.

(Your answer will use one or more series.)

lambda calculus: what does the following expression evaluate to?

fst (snd (fst (pair (pair (pair 2 3) (pair 4 5)) (pair (pair 6 7) (pair 8 9))) ))

Discuss the importance of budgets in enhancing the financial and non-financial performances of an organisation. Explain how participative budgeting may assist the budgetary process in a decentralised organisation.

Your discussion should include the role of budgets:

◦           in communicating and coordinating across the organisation;

Solve the following differential equations.

i) y'''-6y''+10y'=0

ii) dy/dx= x²/(1+y²) with y(1)=3

iii) (x²-2y)y'+2x+2xy=0

iv) Use substitution to solve t²y'+2ty=y⁵ for t>0