Fibonacci and no end? / Irregular functions and functional equations

Prof. Dr. W. Lex
Fibonacci and no end?

The Fibonacci sequence -
_Fn= n for n = 0, 1 and _Fn+1 = _Fn + _Fn-1 for n from the set of natural numbers - should be recalled and, from the wealth of existing examples, manifold references to combinatorics (Pascal's triangle), theory of recurrent sequences (characteristic polynomial), number theory (divisibility, continued fractions), geometry (golden ratio), algebra (monoids, lattices), complexity theory (Euclidean algorithm) and game theory (Nim) should also be touched on. Much of the material can be directly and easily integrated into modern mathematics lessons, from elementary school to A-levels, depending on ability and knowledge.

Prof. Dr. H.-H. Kairies
Irregular functions and functional equations

Real functions are discussed, some of which are far from being differentiable. Examples:
a) Continuous functions that cannot be differentiated anywhere, associated with the names Weierstrass, Tagaki, Knopp, Wunderlich,
b) Riemann's famous function that is differentiable exactly at certain rational points,
c) Singular functions (i.e. continuous, monotonically increasing functions that have the derivative zero almost everywhere), associated with the names Cantor, Minkowski, de Rham.

The selection of these functions is motivated by two facts:
1) Their irregularity, which is difficult to visualize, has fascinated mathematicians for more than a hundred years and has given rise to many works on the fine structure of real functions. 2) They all satisfy a system of simple functional equations and on the other hand can be characterized as a unique solution of this system.

Program