In the following I would like to present the main ideas of bifurcation theory along with some basic examples that illustrate the theory. For the sake of shortness let me formulate the fundamental question of bifurcation theory in an abstract way. Suppose you are given an equation \(F(x,\lambda)=0\) with the property that \(x=0\) is always a solution, the so-called trivial solution. Is it possible to find some \(\lambda^*\) such that a sequence of nontrivial solutions converges to \((0,\lambda^*)\)? In other words: Do nontrivial solutions bifurcate from the trivial solutions? In the following I will present three equations from analysis, linear algebra and ordinary differential equations showing that bifurcation theory is a topic worth studying! The reading requires some amount of advanced mathematics — do not hesitate to contact me if you need some additional explanations. By the way: Next semester I will give a lecture on that topic which is suited for master students or advanced bachelor students with a background in analysis, differential equations and possibly boundary value problems, see below for more information. Continue Reading →
About lasagne, splitting methods and your favourite film
The ‘Divide and Conquer’-principle was already used in the Roman foreign policy. The politicians and warlords of the ancient world noticed that a large group of people can be easier controlled or even conquered if you split them up into several smaller groups.
Luckily the ancient times are long gone, but the principle remains, in particular in computer science. Many algorithms are based on the idea of splitting a task into several easy-to-handle parts. After solving these simpler parts, the solution of the full problem needs to be reconstructed from the solutions of these subproblems. So, it is possible to ‘conquer’ a complex task.
A famous example is the well-known Quicksort algorithm to sort a list of numbers. You start by picking one number in the list (the pivot element) and compare it to every other number of the list. If the other number is smaller, we put it on the left side of our pivot element, if it is larger, we put it on the right side. Thus, we get two sublists, one with smaller and one with larger numbers than the pivot element. We continue to sort these sublists in the same way with new pivot numbers from the sublists. The complete list is sorted when all sublists are sorted.
You may wonder what this is all about and how this is related to our CRC?
Waveguides have been mentioned a few times in this blog (i. e. here and here), so it seems quite apt to explain (in laymans words) what a waveguide is. As any serious pundit, I will start by quoting the internets source of everlasting wisdom:
„A waveguide is a structure that guides waves, such as electromagnetic or sound waves.“
This, of course, does not really help, since the reader can guess this definition just from reading the word „waveguide“. But we learn, at least, that a waveguide is not a person but a structure, unlike – for example – a tour guide. Let’s try again to find out what a waveguide is by picking the word apart:
What’s a wave? Luckily, due to discovery of the wave-particle duality in the last century, the answer is quite clear: Almost everything is a wave. So, that’s done.
Dr. Ivan Fernandez-Corbaton from the Institute of Nanotechnology at the KIT presented his research in our CRC seminar April 21st. We are very happy to share his own short recap of his exciting talk on the question “How Electromagnetic Chiral is a Chiral Object?” with you:
Dr. Ivan Fernandez-Corbaton:
‘Take an object and place it in front of a mirror. If the “being in the other side of the mirror” handed you their version of the object, there are two possibilities: You can rotate it so that it is now identical to your original object, or you cannot. If you cannot, the object is called chiral. Try it at home with a wine opener and you will see that you will never be able to superimpose the two helices because they twist in opposite senses, as the two seashells in the picture.
Chirality is entrenched in nature: From the weak interactions between elementary particles to the empirical fact that most of the building blocks inside the human body are chiral (aminoacids, proteins, the DNA strands and many more).