Publications

2006

• The canonical pencils on Hokirawa surfaces
  
  • Auroux Denis
  , 2006.
• The trace problem for Sobolev spaces over the Heisenberg group
  
  • Vigneron François
  , 2006.
• The canonical pencils on Horikawa surfaces
  
  • Auroux Denis
  , 2006.
• On the controllability of anomalous diffusions generated by the fractional Laplacian
  
  • Miller Luc
  Mathematics of Control, Signals, and Systems, Springer Verlag, 2006, 18 (3), pp.260-271. This paper introduces a "spectral observability condition" for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one dimensional output systems from Müntz-Szasz theorem on the closed span of sets of power functions. (10.1007/s00498-006-0003-3)
  DOI : 10.1007/s00498-006-0003-3
• The odd-dimensional Goldberg Conjecture
  
  • Apostolov Vestislav
  • Draghici Tedi
  • Moroianu Andrei
  Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2006, 279, pp.948-952. An odd-dimensional version of the Goldberg conjecture was formulated and proved by Boyer and Galicki, using an orbifold analogue of Sekigawa's formulas, and an approximation argument of K-contact structures with quasi-regular ones. We provide here another proof of this result and give some applications. (10.1002/mana.200410404)
  DOI : 10.1002/mana.200410404
• Hamiltonian 2-forms in Kähler gerometry I : general theory
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  Journal of Differential Geometry, International Press, 2006, 73 (3), pp.359-412.
• On exponential observability estimates for the heat semigroup with explicit rates
  
  • Miller Luc
  Rendiconti Lincei. Matematica e Applicazioni, European Mathematical Society, 2006, 17 (4), pp.351--366. This note concerns the final time observability inequality from an interior region for the heat semigroup, which is equivalent to the null-controllability of the heat equation by a square integrable source supported in this region. It focuses on exponential estimates in short times of the observability cost, also known as the control cost and the minimal energy function. It proves that this final time observability inequality implies four variants (an integrated inequality with singular weights, an integrated inequality in infinite times, a sharper inequality and a Sobolev inequality) with roughly the same exponential rate everywhere and some control cost estimates with explicit exponential rates concerning null-controllability, null-reachability and approximate controllability. A conjecture and open problems about the optimal rate are stated. This note also contains a brief review of recent or to be published papers related to exponential observability estimates: boundary observability, Schrödinger group, anomalous diffusion, thermoelastic plates, plates with square root damping and other elastic systems with structural damping.
• An anomaly formula for L2-analytic torsions on manifolds with boundary
  
  • Ma Xiaonan
  • Zhang Weiping
  , 2006, pp.247-274.
• An anomaly formula for Ray-Singer metrics on manifolds with boundary
  
  • Ma Xiaonan
  • Bruning J.
  Geometric And Functional Analysis, Springer Verlag, 2006, 16 n°4, pp.767-837.
• Mapping class group factorizations and symplectic 4-manifolds: some open problems
  
  • Auroux Denis
  , 2006, pp.123--132.
• Killing Forms on Symmetric Spaces
  
  • Belgun Florin
  • Moroianu Andrei
  • Semmelmann Uwe
  Differential Geometry and its Applications, Elsevier, 2006, 24, pp.215-222. Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere and $k>p$. (10.1016/j.difgeo.2005.09.007)
  DOI : 10.1016/j.difgeo.2005.09.007
• On the asymptotic expansion of Bergman kernel
  
  • Ma Xiaonan
  • Liu Kefeng
  • Dai Xianzhe
  Journal of Differential Geometry, International Press, 2006, 72, pp.1-41.
• Non-structural controllability of linear elastic systems with structural damping
  
  • Miller Luc
  Journal of Functional Analysis, Elsevier, 2006, 236 (2), pp.592-608. This paper proves that any initial condition in the energy space for the plate equation with square root damping z''- r Delta z' + Delta^2 z' = u on a smooth bounded domain, with hinged boundary conditions z=Delta z=0, can be steered to zero by a square integrable input function u supported in arbitrarily small time interval [0,T] and subdomain. As T tends to zero, for initial states with unit energy norm, the norm of this u grows at most like exp(C_p /T^p) for any real p>1 and some C_p>0. Indeed, this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non-structural controls satisfying a spectral observability condition. Moreover, under some geometric optics condition on the subdomain allowing to apply the control transmutation method, this estimate is improved into p=1 and the dependence of C_p on the subdomain is made explicit. These results are analogous to the optimal ones known for the heat flow. (10.1016/j.jfa.2006.03.001)
  DOI : 10.1016/j.jfa.2006.03.001
• Determining a magnetic Schrodinger operator from partial Cauchy data
  
  • Sjostrand Johannes
  • dos Santos Ferreira David
  • Kenig Carlos
  • Uhlmann Gunther
  , 2006.
• New Asymptotic profiles of nonstationnary solutions of the Navier-Stokes systems
  
  • Vigneron François
  , 2006.
• Groupes et Symetries: groupes finis et algèbres de Lie, représentations, deuxième édition révisée
  
  • Kosmann-Schwarzbach Yvette
  , 2006, pp.193.
• The control transmutation method and the cost of fast controls
  
  • Miller Luc
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2006, 45 (2), pp.762-772. In this paper, the null controllability in any positive time T of the first-order equation (1) x'(t)=e^{i\theta}Ax(t)+Bu(t) (|\theta|<\pi/2 fixed) is deduced from the null controllability in some positive time L of the second-order equation (2) z''(t)=Az(t)+Bv(t). The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method explicits the input function u of (1) in terms of the input function v of (2): u(t,x)=\int k(t,s)v(s)ds, where the compactly supported kernel k depends on T and L only. It proves that the norm of a u steering the system (1) from an initial state x_{0} to zero grows at most like ||x_{0}||\exp(\alpha_{*}L^{2}/T) as the control time T tends to zero. (The rate \alpha_{*} is characterized independently by a one-dimensional controllability problem.) In the applications to the cost of fast controls for the heat equation, L is the length of the longest ray of geometric optics which does not intersect the control region. (10.1137/S0363012904440654)
  DOI : 10.1137/S0363012904440654
• Les théorèmes de Noether: Invariance et lois de conservation au XXe siecle
  
  • Kosmann-Schwarzbach Yvette
  , 2006, pp.173.
• Symplectic 4-manifolds, singular plane curves, and isotopy problems
  
  • Auroux Denis
  , 2006, pp.263--276.
• On the non-randomness of modular arithmetic progressions: a solution to a problem by V. I. Arnold
  
  • Cesaratto Eda
  • Plagne Alain
  • Vallée Brigitte
  Discrete Mathematics and Theoretical Computer Science, DMTCS, 2006, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.271-288. We solve a problem by V. I. Arnold dealing with "how random" modular arithmetic progressions can be. After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory. In conclusion, this study shows that modular arithmetic progressions are far from behaving like purely random sequences, according to Arnold's definition. (10.46298/dmtcs.3510)
  DOI : 10.46298/dmtcs.3510