Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to general targets

Extending our previous work [5] on this subject we show global existence of smooth solutions to the Cauchy problem for wave maps from the (1 + 2)-dimensional Minkowski space to an arbitrary smooth, compact Riemannian manifold without boundary, for arbitrary smooth, radially symmetric data. Received: 14 December 2001 / Accepted: 1 March 2002 / Published online: 6 August 2002     

Lower Bounds for the Height and Size of the Ideal Class Group in CM-Fields

 We prove that, under the assumption of the Generalized Riemann Hypothesis, the exponent of the ideal class group of a CM-field goes to infinity with its absolute discriminant. This gives a positive answer to a question raised by Louboutin and Okazaki [4]. Received September 10, 2001; in revised form April 5, 2002     

Localization transition of <Emphasis Type="Italic">d</Emphasis>-friendly walkers

 Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S ^k _j } _j≥0 , k=1,2,…,d by introducing ``non-crossing condition': and ``reward for collisions' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (β _m )≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d−1)-dimensional space for the parameters . Received: 13 June 2002 / Revised version: 24 August 2002 Published online: 28 March 2003 Mathematics Subject Classification (2000): 82B41, 82B26, 82D60, 60G50 Key words or phrases: Random walks – Random surfaces – Lattice animals – Phase transitions – Polymers – Random walks     

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.     

Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type

 Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact K?hler manifold, then its Euler number χ(M) satisfies (−1) ^m χ(M)>0. Received: 25 September 2001 / Published Online: 16 October 2002     

Ideals and Subalgebras in BCI-Algebras

We prove that there exists a maximal ideal of a BCI-algebra X in which each subalgebra is an ideal.AMS Subject Classification (1991): 06F35, 03G25.The second author was supported by the LG Yonam Foundation (1995).     

An additive problem in the Fourier coefficients of cusp forms

 We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms. Received: 2 October 2001 / Revised version: 9 September 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): Primary 11F30, 11F37; Secondary 11M41.