Real Birkhoff normal forms and complex coordinates

When dealing with Birkhoff normal forms, one can often simplify notations and computations by interpreting pairs of real coordinates as complex coordinates. In the present paper we try to establish an adequate frame for this old and useful trick: Starting with an embedding of a real vector space in a complex one, we construct the respective embedding of the vector fields and the analogue of the homological operator. The construction offers a systematic technique of introducing complex coordinates; in the last section we show how it applies to some well-known examples.

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Zusammenfassung Man kann sich die Arbeit mit Birkhoff-Normalformen häufig wesentlich vereinfachen, indem man Paare reeller Koordinaten als je eine komplexe Koordinate auffaßt. Diesem alten und nützlichen Trick versuchen wir in der vorliegenden Arbeit einen angemessenen Rahmen zu geben: Ausgehend von der Einbettung eines reellen Vektorraumes in einen komplexen, konstruieren wir die entsprechende Einbettung der Vektorfelder und das Analogon des homologischen Operators. Wir stellen damit einen Algorithmus zur Einführung komplexer Koordinaten bei Birkhoff-Normalformen bereit; im letzten Abschnitt erproben wir ihn an einigen wohlbekannten Beispielen.

     

Endomorphisms of the plane sending linear coordinates to coordinates

Let be a field of characteristic zero. We show that an endomorphism of which sends each linear coordinate to a coordinate is an automorphism of .

     

Collective stochastic versions of playable games as metaphors for complex biosystems: Team connect four

In an effort to better understand complex biological systems, the game “Connect Four” is generalized to be a stochastic contest between two teams. Members of each team typically possess “sensors” that provide some information on the nearby deployment of the pieces. Sensing something interesting increases the probability that a given team member will move. Simulations show the relative strengths of various sensor weightings and thereby cast some light on the use of sensors in more general complex autonomous systems. © 2003 Wiley Periodicals, Inc.     

Restricted homogeneous coordinates for the Cayley projective plane

I. Porteous has shown that the Cayley projective plane can be coordinatized in a way resembling homogeneous coordinates. We will show how to construct line coordinates in a similar way. As an illustration, we give an explicit example to show that the Cayley projective plane is not Desarguean.     

On residual coordinates and stable coordinates of R [3]

Let R be a polynomial ring over ${\mathbb{C}}$ . In this paper we show that a polynomial in R ^[3] is a residual coordinate if and only if it is a stable coordinate.     

On the dislocation mechanism of polymer deformation: Defects of spiral molecules

Different types of point dislocations for polymer spirals are introduced. The energy and width of a torsional dislocation in polyoxymethylene are calculated as a numerical example. It is suggested that point dislocations largely determine the development of large deformations in polymers.     

Asymptotics and analytic modes for the wave equation in similarity coordinates

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long-time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self-similar solution χ _T of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long-time behaviour (in similarity coordinates) of linear perturbations around χ _T is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of χ _T with the sharp decay rate for the perturbations.      

Branching Approximation for the Collective Epidemic Model

A new approach is developed that allows us to establish and analyze a branching-type approximation for the collective epidemic model. Firstly, a necessary and sufficient condition is obtained for the vague convergence of the final size of the epidemic to the total progeny in an appropriate branching model, as the initial number of susceptibles tends to infinity. Then, an upper bound for the L ₁ distance between the statistics under study is derived, showing inter alia that the approximation may hold even when the initial number of infectives is arbitrarily large. The results are illustrated with several particular models of special interest.     

On the regularity of oricyclic coordinates

Suppose there is defined in the plane a complete metric W^?, whose curvature K satisfies the inequality ?k ₂ ² ≤ K≤ ?k ₁ ² (k₁ and k₂ are positive constants) and some regularity hypothesis. Then in the entire domain of definition of the metric W^? one can construct regular oricyclic coordinates (x, y), in which the line element has the form ds² = dx²+ B² (x, y) · dy².     

Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

Uniform coordinates and spectral asymptotics for solutions of the scattering problem for the Schrödinger equation

Uniform asymptotic formulas at large distances and for a large value of the spectral parameter are obtained for solutions of the scattering problem. The singularities of the scattering amplitude in the angular variables and its asymptotic expression in the spectral parameter are described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 171–199, 1977.     

Discretization orders and efficient computation of cartesian coordinates for distance geometry

Distance geometry is a class of problems where the position of points in space is to be identified by using information about some relative distances between these points. Although continuous approaches are usually employed, problems belonging to this class can be discretized when some particular assumptions are satisfied. These assumptions strongly depend on the order in which the points to be positioned are considered. We discuss new discretization assumptions that are weaker than previously proposed ones, and present a greedy algorithm for an automatic identification of discretization orders. The use of these weaker assumptions motivates the development of a new method for computing point coordinates. Computational experiments show the effectiveness and efficiency of the proposed approaches when applied to protein instances.     

Threshold for the volume spanned by random points with independent coordinates

Let μ be an even compactly supported Borel probability measure on the real line. For every N > n consider N independent random vectors X ₁, ..., X _N in ℝ ⁿ , with independent coordinates having distribution μ. We establish a sharp threshold for the volume of the random polytope K _N ≔ conv{X ₁, ..., X _N }, provided that the Legendre transform λ of the cumulant generating function of μ satisfies the condition