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Grab Bag
Ordinary Differential Equations
Model Fitting
Partial Differential Equations
Stochastic Methods
100
What is Gaussian elimination?
A method to solve a linear system of equations.
100
True or False: ODE methods can only be applied to equations of state where the forces depend only on position, such as the gravity.
False. Example: air resistance
100
You are trying to find the solution to a problem that has 10 unknown variables. What is the minimum number equations needed to determine the value of each variable?
10
100
What is the difference between ordinary differential equations and partial differential equations?
ODEs involve derivatives in only on variable (i.e. time) while PDEs have derivatives in multiple variables (i.e. time and space)
100
How would you code the algorithm, choose a random integer between 1 and 100?
x = floor(100*rand()) + 1
200
What are the three different types of second order partial differential equations?
Elliptic, hyperbolic, and parabolic.
200
Explain the Euler method for integrating position and velocity given an equation of state, a = f(x)
vi+1 = vi + a(xi) * dt
xi+1 = xi + vi * dt
200
I have 10 data points, which represent the temperature as function of length. I fit these points to a quadratic. How many degrees of freedom are there?
7
200
What method for solving the wave equation in one dimension introduces an artificial viscosity term to FTCS, but has the same order truncation error?
The Lax method.
200
If I flip a coin 6 times, what is the probability that I will get 3 heads?
6!/(3!*3!)*(1/2)^6 = 20/64 = 5/16
300
I throw a dart at a board that looks like a circle inscribed in a square, and it hits the square. What is the probability that it lands inside the circle?
pi/4 = 0.78
300
What do you call an ODE integrator that conserves energy?
Symplectic
300
You are searching for signals of frequency up to 100 GHz. What sampling rate do you need to detect them?
Minimum of 200 GHz.
300
How do you determine the maximum time step you should for stability?
The Courant-Friedrichs-Lewy (CFL) condition, (Courant condition): tau = h/c
300
How do you generate a exponentially distributed random value?
u = - lambda * log( 1 - rand() )
400
What does it mean if a matrix is singular?
The determinant is zero, or at least one of the rows is a linear combination of the others, or at least one of the columns is a linear combination of the others.
400
True or false: Runge-Kutta is a better algorithm than Euler because it conserves energy.
False
400
Find the mean and standard deviation of the set of numbers, [2,6,10,13,14].
mean = 9, std. dev = 5
400
What improvement does the Lax-Wendroff scheme make over the Lax method that makes it a better method for solving the wave equation?
It uses a higher order time derivative, or introduces an artificial viscosity.
400
Describe a method for generating random numbers over the interval [0,1) with the distribution P(x) dx = pi * x * sin( pi*x ) * dx
Choose a random uniform deviate x in the interval [0,1). Choose a second random number, y. If y < pi*x*sin(pi*x), then use than number, or else start over again with a new value of x.