1. Calculate the Transversality Conditions

2. and then solve the ff:

a.)  Ux +Uy =1-U

U (X, X+X^2) = SIN X

b) XUx + YUy= 4U

with initial condition u=1 on the unit circle x^2 +y^2 =1

( you will need to parameterize the circle in terms of parameter r).

What is Monte Carlo simulation? What principles underlie its use, and what steps are followed in applying it? Be detailed and give an example.

1. From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the
   1. average number of cars inline.
   2. average time a car waits before it is washed.
   3. average time a car spends in the service system.
   4. utilization rate of the car wash.
   5. probability that no cars are in the system.

Please be detailed and show your work.

If we rename y=x^2, f(x)=x^2, could we rename the inverse f^-1(x)? Explain your answer

The Discounted Cash Flow methodology. The difference between interest rates and different compounding methods. The concepts of present value and future value as well as discounting. How to determine a firm's cash flow in the context of required rates of return and compensation for risk.Please give us one example from your research, work, or personal life of an application of the above concepts.

Two events E1 and E2 are said to be independent if p(E1∩E2) =p(E1)p(E2). Two (fair, standard) dice are rolled, and their sum is found.

(a) Are the events “the sum is even” and “the sum is greater than 7” independent?

(b) Are the events “the sum is a multiple of 3” and “the sum is greater than 7”independent?

Suppose we assume the logic has just 5 statement letters (i.e. Boolean variables): A, B, C, D, and E.

E1. Show a model that is a counterexample to (A ^ B) |= (B --> (not C))?

E2. Expand the definition of entailment in the following two statements, explaining without using "entails" what they mean:

a. P |= (Q v R)

b. either (P |= Q) or (P |= R) or both Here, P, Q, and R are arbitrary formulas.

E3. Prove that 2a and 2b are not necessarily the same by finding formulas P, Q, and R for which one is true and the other is false.

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1 distinct degrees (up to isomorphism). The other answers on the website are incorrect.

Let G be a group of order mn where gcd(m,n)=1

Let a and b be elements in G such that o(a)=m and 0(b)=n

Prove that G is cyclic if and only if ab=ba

4. Gradient descent. Gradient descent is one of the most popular algorithms in data science and by far the most common way to optimise neural networks. A function is minimised by iteratively moving a little bit in the direction of negative gradient. For the two-dimensional case, the step of iteration is given by the formula xn+1 , yn+1 = xn, yn − ε ∇f(xn, yn). In general, ε does not have to be a constant, but in this question, for demonstrative purposes, we set ε = 0.1. Let f(x, y) = 3.5x 2 − 4xy + 6.5y 2 and x0 and y0 be any real numbers. (a) For all x, y ∈ R compute ∇f(x, y) and find a matrix A such that [3] A x y = x y − ε ∇f(x, y). Write an expression for xn yn in terms of x0 and y0 and powers of A. (b) Find the eigenvalues of A. [1] (c) Find one eigenvector corresponding to each eigenvalue. [2] (d) Find matrices P and D such that D is diagonal and A = P DP −1 . [1] (e) Find matrices Dn , P −1 and An . Find formulas for xn and yn. [4] (f) Suppose x0 = y0 = 1. Find the smallest N ∈ N such that xN yN ≤ 0.05. [3] (g) Sketch the region R consisting of those (x0, y0) such that xN ≥ 0, yN ≥ 0 and [4] xN yN ≤ 0.05, xN−1 yN−1 > 0.05, where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not?

1) When a mass of 3 kilograms is attached to a spring whose constant is 48 N/m, it comes to rest in the equilibrium position. Starting at

t = 0, a force equal to f(t) = 180e^−4t cos(4t) is applied to the system. Find the equation of motion in the absence of damping.

x(t) =

2) Solve the given initial-value problem. d^(2)x/dt^2 + 9x = 5 sin(3t), x(0) = 6,  x'(0) = 0

x(t) =

Use Stokes' Theorem to find the circulation of F⃗ =2yi⃗ +2zj⃗ +4xk⃗  around the triangle obtained by tracing out the path (3,0,0) to (3,0,4), to (3,4,4) back to (3,0,0).

Circulation = ∫CF⃗ ⋅dr⃗  =

Second time I've asked this question because chegg cant solve this problem correct. 24sqrt(2) is the wrong answer