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mathematics

5. Series 31. Year - 5. sneaky dribblet

Let's take a rounded drop of radius $ r_0 $ made of water of density $ \rho \_v $ which coincidentally falls in the mist in the homogeneous gravity field $g$. Consider a suitable mist with special assumptions. It consists of air of density $\rho \_{vzd}$ and water droplets with an average density of $ rho\_r $ and we consider that the droplets are dispersed evenly. If a drop falls through some volume of such mist, it collects all the water that is in that volume. Only air is left in this place. What is the dependence of the mass of the drop on the distance traveled in such a fog?

Bonus: Solve the motion equations.

Karal wanted to assign something with changing mass.

5. Series 31. Year - S. Differential equations are growing well

  1. Solve the two-body problem using the Verlet algorithm and the fourth-order Runge-Kutta method (RK4) over several (many) periods. Use a step size large enough for the numerical errors to become significant. Observe the way the errors manifest themselves on the shape of the trajectories.
  2. Solve for the time-dependent position equation of a damped linear harmonic oscillator described by the equation $\ddot {x}+2\delta \omega \dot {x}+\omega ^2 x=0$, where $\omega $ is the angular velocity and $\delta $ is the damping ratio. Change the parameters around and observe the changes in the oscillator’s motion. For which values of the parameters is damping the fastest?
  3. Model sedimentation using the method of ballistic deposition \[\begin{equation*} h_i(t+1) = max(\(h_{i-1}(t), h_i(t)+1, h_{i+1}(t)\)) \, , \end {equation*}\] where $h_i$ is the height of i-th column. And study the development of the roughness of the surface $W(t,L)$ (see this year’s series 4, problem S). Initially (for small values of $t$) the roughness is proportional to some power of $t$: $W(t,L) \sim t^{\beta }$. For large values of $t$, however, it is proportional to some (possibly different) power of the grid length $L$. $W(t,L) \sim L^{\alpha }$. Find the powers $\alpha $ and $\beta $. Choose an appropriate step size so that you could study both modes of sedimentation. The length of the surface should be at least $L = 256$. (Warning: the simulations may take several hours.)
  4. Simulate on a square grid the growth of a tumor using the Eden growth model with the following variation: when a healthy and an infected cell come into contact, the probability of the healthy one being infected is $p_1$ and the probability of the infected one being healed is $p_2$. Initially, try out $p_1 \gg p_2$, the proceed with $p_1 > p_2$ and then with $p_1 < p_2$. At the beginning, let only 5 cells (arranged into the shape of a cross) be infected.
    Describe qualitatively what you observe.
  5. Rewrite the attached code for the growth of a fractal (diffusion limited aggregation model) on a hexagonal grid to the growth of a fractal on a square grid and calculate the dimension of the resultant fractal.

Note: Using the codes attached to this task is not mandatory, but it is recommended.

Mirek and Lukáš have already grown their algebra, now they have different seeds.

4. Series 31. Year - 3. weirdly shaped glass

We have a cylindrical glass with a small hole at the bottom of the glass. The surface area of the hole is $S$. The glass is filled with water and the water flows into a second glass by itself. The second glass has no holes. What shape should the second glass have so that the water level grows linearly inside it? The glass is supposed to have cylindrical symmetry.

Bonus: The bottom of both glasses is at the same high and the glasses are connected by the hole.

Karel was watching how the glass is being filled.

4. Series 31. Year - 4. solve it yourself

We have a black box with three outputs (A, B, and C). We know that it consists of $n$ resistors with the same resistance but we don't know the circuit diagram. So we measure the resistance between each pair of outputs $R\_{AB} = 3 \mathrm{\Omega }$, $R\_{BC} = 5 \mathrm{\Omega }$ a $R\_{CA} = 6 \mathrm{\Omega }$. Your task is to find the minimum possible $n$ and calculate the corresponding resistance of one resistor.

Matěj solved it quickly.

3. Series 31. Year - 1. slowed down

Let's suppose a camera with a frame rate of 24 frames per second (consider evenly spaced and perfectly sharp shots). We record a flight of a helicopter with the rotor rotation velocity of $2 900 \mathrm {cycles/min}$. Then the record is played. What is the apparent rotational velocity of the rotor in the record?

3. Series 31. Year - S. a walk with integrals

  1. Propose three different examples of Markov chains, at least one of which is related to physics. Is a random walk without backtracking (a step cannot be time reversed previous step) an example of Markov chain? What about a random walk without a crossing (it can lead to each point at most once)?
  2. Consider a 2D random walk without backtracking on a square grid beginning at the point $(x,y) = (0,0)$. It is constrained by absorbing states $b_1\colon y = -5$, $b_2\colon y = 10$. Find the probability of the walk ending in $b_1$ rather than in $b_2$.
  3. Simulate the motion of a brownian particle in 2D and plot the mean distance from the origin as a function of time. Assume a discrete time and a constant step size. (One step takes $\Delta t = \textrm{const} $, and the step size is $\Delta l = \textrm{const} $). A step in any arbitrary direction is possible, i.e. every step is described by it’s length and an angle $\theta \in [0,2\pi )$, while all directions are equally probable. Focus especially on the asymptotic behavior, i.e. the mean distance for $t \gg \Delta t$.
  4. Error function is defined as \[\begin{equation*} {erf}(x)=\frac {2}{\sqrt {\pi }}\int _0^x \eu ^{-t^2} \d t . \end {equation*}\] Calculate the integral for many different values of $x$ and plot it’s value as a function of $x$. What do you get by numerically deriving this function?
  5. Look up the definition of Maxwell-Boltzmann probability distribution $f(v)$, i.e. the probability distribution of speeds of particles in an idealized gas. Utilizing MC integration calculate the mean value of speed defined as \[\begin{equation*} \langle v\rangle = \int _0^{\infty } vf(v) \d v , \end {equation*}\] Use the Metropolis-Hastings algorithm for sampling the Maxwell-Boltzmann distribution. Compare the values of particular parameters with the values from literature.

Mirek and Lukáš random-walk to school.

2. Series 31. Year - 5. raining glass

A worker brought a bag of marbles to a skyscraper construction, to show off in front of his colleagues. But, what an unlucky accident – the marbles pour out and start falling through the scaffolding towards the ground. The scaffolding consists of different levels separated by height $h$. The floor of each level is made out of identical metal grid in which the holes constitute $k  \%$ out of the whole grid area. Consider a simplified model of marbles falling through the scaffolding, in which if marble lands in the hole of the grid it goes through unobstructed and if it lands on the solid part of the grid its velocity drops to $0$ and starts to fall down again immediately (i.e. the size of the marbles is insignificant with respect to the size of the holes in the scaffolding and the marbles don't bounce upon landing, instead they stop and immediately roll down into a hole and continue with their fall). Ignore any potential collisions between marbles themselves. If we assume the marbles pour out of the bag with a constant mass flow of $Q$, what is the force on each level of the scaffolding, when the situation comes to a steady state?

Mirek wanted to transfer Ohm's law into mechanics.

2. Series 31. Year - P. ooh Oganesson

What properties does the $118^{\rm th}$ element in the Periodic table have? Alternatively, what sort of properties would it have, had it been stable? Discuss at least three physical qualities.

Karel wanted to have something on extrapolation.

2. Series 31. Year - S. derivatives and Monte Carlo integration

 

  1. Plot the error as a function of step size for the method \[\begin{equation*} f'(x)\approx \frac {-f(x+2h)+f(x-2h)+8f(x+h)-8f(x-h)}{12h} \end {equation*}\] derived using Richardson extrapolation. What are the optimal step size and minimum error? Compare with forward and central differences. Use $\exp (\sin (x))$ at $x=1$ as the function you are differentiating.
    Bonus: Use error estimate to determine the theoretical optimal step size.
  2. There is a file with experimentally determined $t$, $x$ and $y$ coordinates of a point mass on the website. Using numerical differentiation, find the time dependence of components of speed and acceleration and plot both functions. What is the most likely physical process behind this movement? Choose your own numerical method but justify your choice.
    Bonus: Is there a better method for obtaining velocity and acceleration, then direct application of numerical differentiation?
  3. We have an integral $\int _0^{\pi } \sin ^2 x \d x$.
    1. Find the value of the integral from a geometrical construction using Pythagoras theorem.
    2. Find the value of the integral using a Monte Carlo simulation. Determine the standard deviation.
      Bonus: Solve the Buffon's needle problem (an estimate of the value of $\pi $) using MC simulation.
  4. Find the formula for the volume of a six-dimensional sphere using Monte Carlo method.
    Hint: You can use the Pythagoras theorem to measure distances even in higher dimensions.

Mirek and Lukáš read the Python documentation.

1. Series 31. Year - S. Taking Off

  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.

Mirek and Lukas dusted off some old textbooks.

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