Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformatio...

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-frac...

Let f be a locally injective analytic map of the unit disc D and let ff; zg be its Schwarzian derivative. Suppose jff; zgj 2p(jzj). We use the classical connection between Schwarzian derivative and second order linear equations to show that, for a particular class of functions p , the image f (D) is a quasidisc. The analysis centers on the diierential equation y 00 + py = 0 and a niteness condi...

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles which constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable discrete BKP equation and its Schwarz...

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles which constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable discrete BKP equation and its Schwarz...

S f t,,-,] \-fT,I f 2 t,,--fT,] The Schwarzian is important in many areas of complex analysis (see, for example, the recent book of O. Lehto ILl) but it occurs first and foremost through its connection with M6bius transformations. The basic facts are az+b (1.1) S(f) 0 if and only iff(z) ad bc :/: 0 cz+d’ and (1.2) S(fo h) S(h) if and only iff is M6bius. Equation (1.2) is a special case of a gen...

We study C2-structural stability of interval maps with negative Schwarzian. It turns out that for a dense set of maps critical points either have trajectories attracted to attracting periodic orbits or are persistently recurrent. It follows that for any structurally stable unimodal map the ω-limit set of the critical point is minimal.