Serial of year 31

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.

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.

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.

1. Find all (three) real roots of the function $\exp (x)-5x^2$. Choose an appropriate method yourself and comment on the reasons behind your choice.
2. Newton’s method works even for functions of complex variable. Your task is to render so called Newton fractals, i.e. areas in complex plane in which choosing an initial guess for Newton’s method leads to converging on a specific root. Render the fractals for the functions $z^3-1$ and $z^6+z^3-1$, where $z$ is a complex number. The derivations of these functions are $3z^2$, and $6z^5+3z^2$ respectively. For calculations and rendering you may utilize the Python code attached to this task.
   Note: Complex derivation, if it exists, can be calculated the same way as normal derivation..
   Bonus: Find as beautiful or interesting Newton fractal as you can.
3. Simulate on computer (or calculate by hand) an elementary cellular automaton abiding by the rule 54 on a grid with size 20 and periodical conditions for at least 10 time steps (more certainly can’t hurt). At the beginning, one arbitrary cell has the value 1, the rest 0. Plot the result on a spacetime diagram.
4. Simulate the changes in roughness $W$ of a 1D surface using a model of random deposition. The roughness $W$ is given by the equation \[\begin{equation*} W(t,L) = \sqrt {\frac {1}{L^d}\sum _i h_i^2-\(\frac {1}{L^d}\sum _i h_i\)^2} \, . \end {equation*}\] Where $d$ is the dimension, $L = 100$ is the length of the surface and $h_i$ is the height of the i-th column. Initially, the surface is perfectly flat. Plot the roughness as a function of time for at least $10^8$ steps (one step $=$ one new particle), discuss the results.
   Note: Random deposition simply means that in each step of the simulation, the height of one randomly selected column will increase by one.

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.

1. Simulate the dynamics of a predator-prey system using Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} &= r\_x x - D\_x xy ,\\ \frac{\d y}{\d t} &= r\_y xy - D\_y y . \end {align*}\] where $x$ and $y$ are the population sizes of prey and predator respectively, the parameters $r\_x$ and $r\_y$ represent the populations’ growth and the parameters $D\_x$ and $D\_y$ represent the shrinking of the populations. Set the parameters to be $r\_x = 0{.}8$, $D\_x0= 1{.}0$, $r\_y = 0{.}75$, $D\_y = 1{.}5$. Run the simulations for several different value pairs for initial population sizes $x = 0{.}5$ and $y = 2{.}0$; $x = 1{.}5$ and $y = 0{.}5$; $x = 1{.}95$ and $y = 0{.}75$. Plot the predator population size as a function of the prey population size. Discuss the results.
   Bonus: Find the solutions for the same situations analytically (by integrating the differential equations).
2. Using the competitive Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} = r\_x x \(1 - \(\frac {x + I\_{xy} y}{k\_x}\)\)  ,
   \frac{\d y}{\d t} = r\_y y \(1 - \(\frac {y + I\_{yx} x}{k\_y}\)\) . \end {align*}\] simulate the dynamics of two competing populations (e.g. hawks and eagles) for the following values of parameters: $r\_h = 0{.}8$, $I\_{he} = 0{.}2$, $k\_h = 2{.}0$, $r\_e = 0{.}6$, $I\_{eh} = 0{.}3$, $k\_e = 1{.}0$. Set the initial population sizes to be $h = 0{.}01$, $e = 1{.}0$. Then, simulate the same situation, but change the interaction coefficients to $I\_{he} = 1{.}5$ a $I\_{eh} = 0{.}6$. Plot the results in one graph - the sizes of populations vs time. Discuss the results.
3. Verify the importance of pivoting.
   Solve the system of linear equations \[\begin{equation*} \begin{pmatrix} 10^{-20} & 1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \end {equation*}\] at first exactly (on paper), then using LU factorization with partial pivoting (you may utilize some Python module, e.g. scipy.linalg.lu()), and finally, solve the system using LU factorization without pivoting. Compare the resultant  $\vect {x}$ obtained from the three methods and the results of matrix multiplication $L^{-1}\cdot U$ ($P\cdot L^{-1}\cdot U$ in the case of pivoting).
4. Consider an infinite parallel-plate capacitor. The gap between plates has a thickness $L=10 \mathrm{cm}$ and the voltage between the plates is $U=5 \mathrm{V}$. Between the plates of the capacitor grounded electrode in the shape of an infinitely long prism with square base of side length $a=2 \mathrm{cm}$, whose center lies $l=6{,}5 \mathrm{cm}$ away from the grounded plane of the original capacitor. The prism is oriented such that one of its short sides is perpendicular to the capacitor plates. Find the distribution of electric potential in the condensator. Since the problem has a translational symmetry in the direction of the infinite side of the prism, it is sufficient to solve it only in the plane parallel to the plates, i.e. it is a 2D problem. Render the potential distribution in this plane. You may utilize the code attached to this task.
   Bonus: Calculate and render the distribution of the electric field strength $\vect {E}$.