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On the number of places of convergence for Newton’s method over number fields

Xander Faber, José Felipe Voloch (2011)

Journal de Théorie des Nombres de Bordeaux

Let f be a polynomial of degree at least 2 with coefficients in a number field K , let x 0 be a sufficiently general element of K , and let α be a root of f . We give precise conditions under which Newton iteration, started at the point x 0 , converges v -adically to the root α for infinitely many places v of K . As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v -adically to any given root of f for infinitely many places v . We also conjecture that...

On the optimal control of implicit systems

P. Petit (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding manifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold, defined by Rabier and Rheinboldt for determined implicit differential equations,...

On the Origin of Chaos in the Belousov-Zhabotinsky Reaction in Closed and Unstirred Reactors

M. A. Budroni, M. Rustici, E. Tiezzi (2010)

Mathematical Modelling of Natural Phenomena

We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations...

On the preservation of combinatorial types for maps on trees

Lluís Alsedà, David Juher, Pere Mumbrú (2005)

Annales de l'institut Fourier

We study the preservation of the periodic orbits of an A -monotone tree map f : T ⟶ T in the class of all tree maps g : S ⟶ S having a cycle with the same pattern as A . We prove that there is a period-preserving injective map from the set of (almost all) periodic orbits of f into the set of periodic orbits of each map in the class. Moreover, the relative positions of the corresponding orbits in the trees T and S (which need not be homeomorphic) are essentially preserved.

On the primary orbits of star maps (first part)

Lluis Alsedà, Jose Miguel Moreno (2002)

Applicationes Mathematicae

This paper is the first one of a series of two, in which we characterize a class of primary orbits of self maps of the 4-star with the branching point fixed. This class of orbits plays, for such maps, the same role as the directed primary orbits of self maps of the 3-star with the branching point fixed. Some of the primary orbits (namely, those having at most one coloured arrow) are characterized at once for the general case of n-star maps.

On the primary orbits of star maps (second part: spiral orbits)

Lluís Alsedà, José Miguel Moreno (2002)

Applicationes Mathematicae

This paper is the second part of [2] and is devoted to the study of the spiral orbits of self maps of the 4-star with the branching point fixed, completing the characterization of the strongly directed primary orbits for such maps.

On the ∗-product in kneading theory

Karen Brucks, R. Galeeva, P. Mumbrú, D. Rockmore, Charles Tresser (1997)

Fundamenta Mathematicae

We discuss a generalization of the *-product in kneading theory to maps with an arbitrary finite number of turning points. This is based on an investigation of the factorization of permutations into products of permutations with some special properties relevant for dynamics on the unit interval.

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