Series 1, Year 31

When is it most efficient to pour cold milk into a hot coffee for the coffee to be drinkable as soon as possible? We expect a detailed description of the cooling process rather than a precise calculation.

Consider an optical switch (transfer speed $10 \mathrm{Gb s^{-1}}$), whose output (after any necessary amplification) is used to illuminate the Moon. Thanks to the mirrors left behind by the Apollo mission, the signal comes back and can be used (after any necessary amplification) as an input to the switch. If we make sure the switch works reliably the transmitted data will circle in the system indefinitely. Thus we acquire a memory. What is its maximum capacity? Ignore any delays caused by the processing of the signal and any headers of the data.

We have a thin homogeneous pipe in the shape of an L with segments of lengths $b,c$. It is hung freely by one of the ends in a rail car so that its bend points in the direction of travel. Find the acceleration of the rail car required for the bottom side of the L to stay parallel to the direction of travel? Ignore relativistic effects.

Bonus: Consider relativistic effects.

What happens when we close an almost empty 1.5 liter PET bottle in a well-heated office, let's say at  $t\_k = 26 \mathrm{\C }$, and then walk down the stairs with it? The bottle starts to make cracking noises. What has a bigger effect on this phenomenon, the fact the atmospheric pressure is changing as we walk down 10 floors of the building, or the fact that the temperature on the stairs is a bit lower, let's say, $t\_s = 15 \mathrm{\C }$?

You have probably already thought about whether there are alien civilizations in the universe. As a general rule, the bigger a star the higher is its radiation power and the shorter its life. Let's focus on two stars, where one has twice the power output than the other. If the band where life is possible is given by the steady state temperature of a black body orbiting there, which has a lower and upper limit (that are the same for each and every system), around which star is this life-supporting band wider? How many times larger it will be in comparison to the other star?

How would a plane behave in microgravity (in other words ignore the effects of gravity)? Describe what effects would the ailerons, the rudder, the elevators and thrust vectoring have of the engines? Which acrobatic maneuvers would be possible? (E.g. a flat spin probably wouldn't be).

Measure the bending of a wooden skewer stick freely supported at its ends as a function of the force applied in its middle (see picture).

1. Modify the expression $\sqrt {x+1}-\sqrt {x}$, so that it isn't so prone to the problems of cancellation, ordering and smearing. Which of these problems would have originally caused a trouble with the expression and why? What is the difference between the original and the corrected expression when we evaluate it using double precision with $x=1{,}0 \cdot 10^{10}$?
2. Describe the effects of the following code. What is the difference between the functions \texttt {a()} and \texttt {b()}? With which values of \texttt {x} can they be evaluated? Don't be afraid to run the code and play with different values of the variable \texttt {x}. What is the asymptotic computation time complexity as a function of the variable x.

   def a(n):
     if n == 0:
       return 1
     else:
       return n*a(n-1)
   def b(n):
     if n == 0:
       return 1.0
     else:
       return n*b(n-1)
   x=10
   print("{} {} {}".format(x, a(x), b(x)))

3. Let's designate $o_k$ and $O_k$ as the circumference of a regular $k$sided polygon inscribed and circumscribed respectively in a circle. The following recurrent relationships then apply \[\begin{equation*} O_{2k}=\frac {2o_k O_k}{o_k + O_k} ,\; o_{2k}=\sqrt {o_k O_{2k}} . \end {equation*}\] Write a program that can calculate the value of $\pi $ using these relations. Start with an inscribed and a circumscribed square. How accurately can you approximate $\pi $ using this method? (A similar method has been originally used by Archimedes for this purpose.)
   
   
4. Lukas and Mirek play a game. They toss a fair coin: when it's tails (reverse) Mirek gives Lukas one Fykos t-shirt when it's heads (obverse) Lukas gives one to Mirek. Together they have $t$ t-shirts of which $l$ belongs to Lukas and $m$ to Mirek. When one of them runs out of t-shirts the game ends.
   1. Let $m = 3$ and Lukas's supply be infinite. Determine the most probable length of the game, i.e. the number of tosses after which the game ends (because Mirek runs out of t-shirts).
   2. Let $m = 10$, $l = 20$. Simulate the game using pseudorandom number generator and find the probability of Mirek winning all of Lukas's t-shirts. Use at least 100 games (more games means more precise answer).
   3. How will the result of the previous task change in case Mirek „improves“ the coin and heads now occur with the probability of $5/9$?
      Bonus: Calculate the probability analytically and compare the result with the simulation.
5. Consider a linear congruential generator with parameters $a = 65 539$, $m = 2^{31}$, $c = 0$.
   1. Generate at least $1 000$ numbers and determine their mean and variance. Compare it to the mean and variance of a uniform distribution over the same interval.
   2. Find the relationship that gives the next number in the generated sequence as a linear combination of the two preceding numbers. I.e. find the coefficients $A$, $B$ in the recurrence relationship $x_{k+2} = Ax_{k+1} + Bx_k$. If we consider each three sequential numbers as the coordinates of a point in 3D, how does the recurrence relationship influence the spatial distribution of these points?
      Bonus: Generate a sequence of at least $10 000$ numbers and plot the points on a 3D graph that will illustrate the significance of the given recurrence relationship.