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## mathematics

### 4. Series 21. Year - 1. bees and geometry

When you look at honeycomb you can admire its periodic structure. In a cut the walls cell form regular hexagons and fill whole plane.

Why do the bees make cells as hexagons? Why not for example rectangle of pentagon?

Zadal Honza Prachař inspirován knihou Matematika kolem nás.

### 4. Series 21. Year - P. project 5

Suggest a shape of the most fairness cube of 5-sides. We mean to find such 5-sided object, where the probability of stopping on each side is same for all sides.

Vymysleli Aleš Podolník a Marek Scholz.

### 4. Series 21. Year - S. quantum harmonic oscillator

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called coherent stat of harmonic oscillator and wave packet should oscillate around origin with angular frequency √( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

### 3. Series 21. Year - S. wandering of a sailor, pi-circuit and epidemic in Prague

<h3>Integral</h3>

Integrate using Monte Carlo method function e^{$-x}$ on interval [ $-100,100]$. Try numerically find value of this integration interval from −∞ till +∞.

Hint: Function is symmetrical in origin, therefore it is sufficient to integrate on interval [ 0, +∞ ) . Make substitution $x=1⁄t-1$, where you change limits of integration from 0 to 1.

<h3>Wandering of sailor</h3>

Drunken sailor stepped out onto pier of length 50 steps and wide 20 steps. He goes to land. At each step forward looses balance and makes one step left or right. Find, what is probability of reaching land and what is probability of falling off the pier into the sea.

Sailor was lucky and survived. However the second night he goes (again drunken) from ship to land. This time there is strong wind of speed of 3 m\cdot s^{−1}, which causes change of probability of stepping to the left to 0.8 and 0.2 to the right. Again, find the probability, that he reaches the other side or will fall into the sea.

Third night the situation repeats again. The wind is blowing randomly, following normal distribution with mean value 0 m\cdot s^{−1} and dispersion 2 m\cdot s^{−1}. Find the probability of sailor reaching land. You can assume, that sailor walks slowly and inertia of wind is negligible, therefore wind is uncorrelated between individual steps.

<h3>Pi-circuit</h3>

Having 50 resistor of resistance 50 Ω we want to create a circuit with the resistance in Ohms closest to number π. Solve it using simulated annealing.

For this task you can adapt our program, which can be found on our web pages.

If you do not feel like solving this problem, try to solve problem of „traveling salesman“ with introducing curved Earth surface into a model and find solution for concrete set of towns, e.g. capitals of European countries.

<h3>Epidemic in Prague</h3>

Investigate evolution of epidemic in Prague. Assume 1 million inhabitants. Intensity of infection $β$ is 0$,4⁄1000000$ per day, cure $γ$ is ( four days )^{$-1}$. At the beginning there is 100 infected people. Compare the evolution with case of vaccinated population of 20% of population. Also compare with vaccination during the epidemic, where 0.5% population is vaccinated per day. The end of epidemic is, when less than 20 people are ill.

There is a lot of data you can get from computer simulation. Apart from the mean value also plot a graph, where you will show five random simulations. You can also observe fluctuations. Compare your results with deterministic model which does not assume randomness of process of infection. The number of points, which we will give out will reflect how many interesting data you will process.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.

### 1. Series 19. Year - S. probability

• Three cards are randomly selected from 36 cards. Calculate probability of possibility that (i) just one ace was selected, (ii) at least one ace is selected and (iii) no ace is selected.
• $N$ identical particles is in a container. Calculate probability of case, that in the left part of container is $m$ particles more than in the right half. Draw a graph of dependence for $N=10^{10}$. Range of $m$ select so that the probability on the sides will be one tenth of the probability in the middle. How the width depends on $N?$ (Width is difference $m_{2}-m_{1}$, where $m_{2}>0$ a $m_{1}<0$ are values of $m$, for which the probability is half compared to the maximum).
• Estimate size of ln$n!$ (without using Stirling formula).

Autor seriálu Matouš.