Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

We consider front solutions of the Swift–Hohenberg equation ∂ _t u= -(1+ ∂ _x ²)² u + ɛ² u -u ³. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using renormalization techniques and a decomposition into Bloch waves, we show the non-linear stability of these solutions. It turns out that this problem is closely related to the question of stability of the trivial solution for the model problem ∂ _t u(x,t) = ∂ _x ² u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t) ^p with p>3. In particular, we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front. Received: 23 February 2001 / Accepted: 27 August 2001     

Zur Sch?nbein'sehen Reaction auf Blaus?ure

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Scaling of Mandelbrot sets generated by critical point preperiodicity

Letzf (z) be a complex holomorphic function depending holomorphically on the complex parameter . If, for =0, a critical point off ₀ falls after a finite number of steps onto an unstable fixed point off ₀, then, in the parameter space, near 0, an infinity of more and more accurate copies of the Mandelbrot set appears. We compute their scaling properties.On leave from the University of Geneva     

On the computation of the Galois group over the quotient field of ℂ[λ]

Summary We give an algorithm for the computation of the Galois group of the splitting field of polynomials in two variables with integer coefficients over the quotient field (), (the rational functions in ). The algorithm uses a constructive version of the Newton polygon method and analytic continuations.Supported in part by the Fonds National Suisse     

Extensive Properties of the Complex Ginzburg–Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in R^d, d <3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q_L of side L. We cover this set by a (minimal) number N_QL(l) of balls of radius l in $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, H_\epsilon =\lim_{L\to\infty} L^{-d} \logtwo N_{Q_L}(\epsilon)< /FORMULA > exists. In particular, we bound < FORMULA FORM=ÏNLINE» exists. In particular, we bound H_\epsilon< /FORMULA > by < FORMULA FORM=ÏNLINE» by \OO\bigl(\logtwo(1/\epsilon )\bigr)< /FORMULA > , which shows that the attracting set is < SMALL > smaller < /SMALL > than the set of bounded analytic functions in a strip. We finally give a positive lower bound: < FORMULA FORM=ÏNLINE», which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H_\epsilon>\OO\bigl (\logtwo(1/\epsilon)\bigr)$.     

Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon über die Komposition quadratischer Formen

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Dynamics of Triangulations

We study a few problems related to Markov processes of flipping triangulations of the sphere. We show that these processes are ergodic and mixing, but find a natural example which does not satisfy detailed balance. In this example, the expected distribution of the degrees of the nodes seems to follow the power law d ⁻⁴.