Exact Controllability of a Piezoelectric Body. Theory and Numerical Simulation

We study the exact controllability of a three-dimensional body made of a material whose constitutive law introduces an elasticity-electricity coupling. We show that a coupled elastic-electric control acting on the whole boundary of the body drives the system to rest after time large enough. Two-dimensional numerical experiments suggest that controllability can still be achieved by relaxing this restrictive condition using either both controls on a reduced support or elastic control alone. B. Miara supported by the European project “INTAS, Research Project for South Caucasian Republics 06-1000017-8886”. A. Münch partially supported by grants ANR-05-JC-0182-01 and ANR-07-JC-1832-84.     

Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e., patterns of disks appearing in the definition of the weighted Delaunay decomposition associated with a set of disjoint disks, possibly with cone singularities at the centers of those disks. Hyperideal circle patterns are associated with hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]), however, the proof is completely different (and more intricate) since Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]) used a shortcut which is not available here. The author would like to thank the RIP program at Oberwolfach, where part of the research presented here was conducted. Partially supported by the “ACI Jeunes Chercheurs” Métriques privilégiés sur les variétés à bord, 2003-06, and the ANR program Representations of surface groups, 2007-09.     

Links of surface singularities and CR space forms

We classify 3-dimensional compact locally homogeneous non-degenerate CR-manifolds (“CR space-forms”). Most of them are links of normal complex surface singularities, and we classify these singularities. In memory of Peter Scherk Research partially supported by the NSF.     

Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality

We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov–Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality. Dedicated to the memory of Silvano Bispuri. The work of A. L. Knutsen is partially supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The work of A. F. Lopez is partially supported by the MIUR national project “Geometria delle varietà algebriche” COFIN 2002--2004.     

Copulas: Tales and facts

“But he does not wear any clothes” said the little child in Hans Christian Andersen’s “The Emperor’s New Clothes.”Mikosch’s research is partially supported by the Danish Research Council (SNF) GrantNo 21-04-0400. This is a discussion paper which was initiated at the 4th InternationalConference on Extreme Value Analysis in Gothenburg, 15–19 August, 2005; see.     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action     

Patchworking singular algebraic curves I

In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation. The second author was also partially supported by the EC-network ‘Algebraic Lie Representations” contract no. ERB-FMRX-CT97-0100.     

On eigenfunctions of the structures described by the “shallow-water” model on the plane

We propose a new method for solving the “shallow-water” equations. We show that from the equations of “shallow water” one obtains nonlinear Liouville-type equations, Helmholtz equations, etc. This allows one to construct eigenfunctions of various structures that appear in the flow in the two-dimensional case. We obtain exact and asymptotic solutions in an elliptic domain with singularities. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 17–32, 2006.     

A Tutorial on Computational Cluster Analysis with Applications to Pattern Discovery in Microarray Data

Microarrays offer unprecedented possibilities for the so-called omic, e.g., genomic and proteomic, research. However, they are also quite challenging data to analyze. The aim of this paper is to provide a short tutorial on the most common approaches used for pattern discovery and cluster analysis as they are currently used for microarrays, in the hope to bring the attention of the Algorithmic Community on novel aspects of classification and data analysis that deserve attention and have potential for high reward. R. Giancarlo is partially supported by Italian MIUR grants PRIN “Metodi Combinatori ed Algoritmici per la Scoperta di Patterns in Biosequenze” and FIRB “Bioinformatica per la Genomica e la Proteomica” and Italy-Israel FIRB Project “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”. D. Scaturro is supported by a MIUR Fellowship in the Italy-Israel FIRB Project “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”.     

Justification of a construction of the solution of the problem on a point source ofSH waves in a layered medium with vertical boundary

The solution is determined by the method of separation of variables. The method is justified by using a suitable space of generalized functions. Boundary conditions “are hidden” in this space. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 56–66. Original article This work was supported by the Russian Foundation of Fundamental Research (Grant No. 93-011-16148). Translated by T. N. Surkova.     

Sparse second moment analysis for elliptic problems in stochastic domains

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain. This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.     

Hilbert–Kunz multiplicity, McKay correspondence and good ideals¶in two-dimensional rational singularities

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. Received: 25 October 2000     

On Hardy spaces associated with Bessel operators

In this paper, we study Hardy spaces associated with two Bessel operators. Two different kind of Hardy spaces appear. These differences are transparent in the corresponding atomic decompositions. The first author was partially supported by MTM2004/05878. The second author was supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by Polish funds for science in years 2005–2008 (research project 1P03A03029).     

Remarks on the nef cone on symmetric products of curves

Let C be a very general curve of genus g and let C ⁽²⁾ be its second symmetric product. This paper concerns the problem of describing the convex cone of all numerically effective -divisors classes in the Néron–Severi space . In a recent work, Julius Ross improved the bounds on in the case of genus five. By using his techniques and by studying the gonality of the curves lying on C ⁽²⁾, we give new bounds on the nef cone of C ⁽²⁾ when C is a very general curve of genus 5 ≤ g ≤ 8. This work has been partially supported by (1) PRIN 2007 “Spazi di moduli e teorie di Lie”; (2) Indam (GNSAGA); (3) FAR 2008 (PV) “Varietá algebriche, calcolo algebrico, grafi orientati e topologici”.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

On symplectic half-flat manifolds

We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups. We prove that the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures. Moreover, we provide an example of a compact 6-dimensional symplectic half-flat manifold whose real part of the complex volume form is d-exact. Finally we discuss the 4-dimensional case. This work was supported by the Projects M.I.U.R. “Geometric Properties of Real and Complex Manifolds”, “Riemannian Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.