## Lectures in English

### Primary schools

**Tantrix: a geometric puzzle,****Nuno Rafael Bastos/14.12.2021, hours: 16-17, from 7 years old (lecture) /****Zoom**Meeting ID: 876 5482 1181

Tantrix is the brand name for a set of hexagonal Bakelite tiles and was invented by Mike McManaway of New Zealand in 1991. Since then, it has won major awards around the world. Each tile has three lines, of different colours, which go from one side to another. Some sets of tiles can be used as puzzles, in which you have to arrange the tiles so that the edges match in colour, thus forming coloured lines or even loops. The tiles can also be used for playing a game with two or more people.

In this talk, where the main focus will be on the puzzles, participants will learn Tantrix rules, how to use tiles to solve some puzzles and work with some mathematical concepts (e.g. rotations, symmetries, angles, polygons, etc) . The presented puzzles are best solved by applying logic and deduction, although trial and error can sometimes work too.

To fully participate, please go to https://bit.ly/nbplmdays21 and then cut each one of the 14 hexagonal tiles.

### First years of university and High schools/Secondary schools

*Complex numbers and fractals,***Zbigniew Bartosiewicz/14.12.2021, hours: 12-13 / MSTeams**

I will show how to pass from the sad one-dimensional world of real numbers to imaginary two-dimensional reality of complex numbers, where one can find square roots of negative numbers and fabulously colorful fractal dragons. Along the way we will learn how to compute powers of complex numbers and see where we will arrive, when we keep computing the powers to infinity.**Mathematicians in our neighbourhood****, Krzysztof Ciesielski/****12.2021, hours: 13-14 / MSTeams**

When we wander around various cities, we often pass monuments to famous people people. Various people are commemorated in this way, including (though definitely too rare) excellent scholars. Are there mathematicians among them? If yes, why were they awarded in this way? This is what it will be on speech lecture. The most attention will be paid to a bench with figures of Stefan Banach and Otton Nikodym, unveiled in 1916 in Krakow.

**MATLAB Toolbox for fractional-variable-order closed loop systems,****Piotr Oziabło/14.12.2021, hours: 16-17 / MSTeams**

One of the most common control approaches in modern industry is a closed loop PID control. PID controllers as an input take the control error which is a difference between expected and actual (measured) value of the control variable. The control error is in this case multiplied by the constant value, integrated and differentiated which corresponds to the proportional (P), integral (I) and derivative (D) terms of PID controller. As a result of the mentioned operations the control signal is generated which is then send to the controlled object/process. A generalization of PID is a fractional-order PID (FOPID) controller where the integral and derivative terms utilize fractional-order calculus to calculate the output control signal. In the other words FOPID controllers instead of calculating first order integral/difference of the control error, perform the operation of fractional order integration and differentiation. FOPID controllers proved their usefulness in many applications outperforming standard PID approach. Further generalization of FOPID control are fractional-variable-order controllers (FVOPID). In this case the integral/difference terms of the controller are not only fractional but are also functions which values change during the control process. This could potentially result in more robust, efficient, and flexible control processes in comparison to the systems which only perform constant order operations. Since FVOPID concept is relatively new there are not so many tools which support engineers and scientists in the study of such systems. In this work a MATLAB Toolbox is presented that allows to simulate and tune closed-loop systems with fractional-variable-order digital PID (FVOPID) controllers. The toolbox provides MATLAB Simulink blocks of FVOPID based on two different implementations. An additional part of the toolbox is an application with a GUI interface which main purpose is to simulate and find the optimal parameters of FVOPID for a given closed-loop system. The tuning process, in this case, is performed by different optimization methods which are set to minimize the preselected objective function.

**About surface visualization in SURFER, Małgorzata Wyrwas/15.12.2021, hours: 13-14 / MSTeams**

The participants will be able to see that mathematical equations are not boring at all, and with their help one can visualize surfaces with interesting shapes that can be an inspiration for designers, computer graphic designers, architects. During the classes, examples of surfaces obtained with the Surfer program will be presented, including the equations of a lemon, a hat and other interesting surfaces.