Über die abstandsverträglichen Abbildungen auf dem Kreis und auf der reellen Geraden

Zusammenfassung. Eine Abbildung zwischen metrischen R?umen hei?t abstandsvertr?glich, wenn der Abstand der Bilder zweier Punkte nur vom Abstand der Punkte selbst abh?ngt. Wir zeigen, dass eine Abbildung genau dann abstandsvertr?glich ist, wenn der Cauchyschen Funktionalgleichung genügt, also ein Endomorphismus der Gruppe ist. Ein entsprechendes Resultat gilt auch für die abstandsvertr?glichen Abbildungen des Kreises (mit der Multiplikation komplexer Zahlen als Gruppenverknüpfung). Damit kann man sowohl alle messbaren abstandsvertr?glichen Abbildungen von bzw. in sich angeben, als auch einen Nachweis für die Existenz nichtmessbarer abstandsvertr?glicher Abbildungen auf und erbringen. Eingegangen am 20. Juni 2001 / Angenommen am 13. September 2001     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.     

Effects of in situ thermal annealing on the structural, optical, and electrical properties in Hg0.7Cd0.3Te epilayers grown on CdTe buffer layers

Scanning electron microscopy (SEM), Fourier transform infrared (FTIR) transmission, and Hall effect measurements were performed to investigate the structural, optical, and electrical properties of as-grown and in situ-annealed Hg_0.7Cd_0.3Te epilayers grown on CdTe buffer layers by using molecular beam epitaxy. After the Hg_0.7Cd_0.3Te epilayers had been annealed in a Hg-cell flux atmosphere, the SEM images showed that the surface morphologies of the Hg_0.7Cd_0.3Te thin films were mirror-like with no indication of pinholes or defects, and the FTIR spectra showed that the transmission intensities had increased in comparison to that of the as-grown Hg_0.7Cd_0.3Te epilayer. Hall-effect measurements showed that n-Hg_0.7Cd_0.3Te epilayers were converted to p-Hg_0.7Cd_0.3Te epilayers. These results indicate that the surface, optical, and electrical properties of the Hg_1 − xCd_xTe epilayers are improved by annealing and that as-grown n-Hg_1 − xCd_xTe epilayers can be converted to p-Hg_1 − xCd_xTe epilayers by in situ annealing.     

Optimality problem for infinite dimensional bilinear systems

For any finite dimensional control system with arbitrary cost, Pontryagin's Maximum Principle (PMP) [N. Bensalem, Localisation des courbes anormales et problème d'accessibilité sur un groupe de Lie hilbertien nilpotent de degré 2, Thèse de doctorat, Université de Savoie, 1998. [6]] gives necessary conditions for optimality of trajectories. In the infinite dimensional case, it is well known that these conditions are no more true in general. The purpose of this paper is to establish an “approached” version of PMP for infinite dimensional bilinear systems, with fixed final time and without constraints on the final state. Moreover, if the set of control is contained in a closed bounded convex subset with operators defining its dynamics are compact, or if it is contained in a finite dimensional space, we get an “exact” version of PMP. We also give two applications of these results. The first one deals with sub-Riemannian geometry on nilpotent Hilbertian Lie groups for which we can define a sub-Riemannian distance. The second one deals with heat equation for which we analyse the necessary conditions to give the optimal controls.     

On the structure of finite coverings of compact connected groups

Finite-sheeted covering mappings onto compact connected groups are studied. We show that for a covering mapping from a connected Hausdorff topological space onto a compact (in general, non-abelian) group there exists a topological group structure on the covering space such that the mapping becomes a homomorphism of groups. To prove this fact we construct an inverse system of covering mappings onto Lie groups which approximates the given covering mapping. As an application, it is shown that a covering mapping onto a compact connected abelian group G must be a homeomorphism provided that the character group of G admits division by degree of the mapping. We also get a criterion for triviality of coverings in terms of means and prove that each finite covering of G is equivalent to a polynomial covering.     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

On dimensional rigidity of bar-and-joint frameworks

Let V={1,2,…,n}. A mapping p:V→R^r, where p¹,…,pⁿ are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in R^r, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in R^r is said to be dimensionally rigid iff there does not exist a framework G(q) in R^s for s?r+1, such that ∥qⁱ-q^j∥²=∥pⁱ-p^j∥² for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p¹,…,pⁿ of the given r-configuration are in general position, is also investigated.     

The inverse scattering problem for Schrödinger and Klein–Gordon equations with a nonlocal nonlinearity

We study the inverse scattering problem for the nonlinear Schrödinger equation and for the nonlinear Klein–Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.     

A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.