Obstruction à la transversalité de lagrangiens

In this Note, we give a necessary and sufficient condition for Lagrangians in a symplectic vector bundle to be deformed stably into transversal Lagrangians. In the case of three Lagrangians, we show that the associated Grothendieck group can be identified with a Hermitian K-theory group. To cite this article: M. Karoubi, M.L. Lapa de Souza, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point.We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Stró?yna–?o?ladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations.     

High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces

Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic growth in the velocities or in the momenta. Such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions. In this note we show how these results can be extended to the classical setting of Tonelli Lagrangians (Lagrangians which are C²-convex and superlinear in the velocities), or to Hamiltonians which are superlinear in the momenta and have a coercive action integrand.     

The local regularity for strong solutions of the Hessian quotient equation

By means of the Reilly formula and the Alexandrov maximum principle, we obtain the local C1,1 estimates of the W2,p strong solutions to the Hessian quotient equations for p sufficiently large, and then prove that these solutions are smooth. There are counterexamples to show that the integral exponent p is optimal in some cases. We modify partially the known result in the Hessian case, and extend the regularity result in the special Lagrangian case to the Hessian quotient case.     

Convergence Properties of Dikin’s Affine Scaling Algorithm for Nonconvex Quadratic Minimization

We study convergence properties of Dikin’s affine scaling algorithm applied to nonconvex quadratic minimization. First, we show that the objective function value either diverges or converges Q-linearly to a limit. Using this result, we show that, in the case of box constraints, the iterates converge to a unique point satisfying first-order and weak second-order optimality conditions, assuming the objective function Hessian Q is rank dominant with respect to the principal submatrices that are maximally positive semidefinite. Such Q include matrices that are positive semidefinite or negative semidefinite or nondegenerate or have negative diagonals. Preliminary numerical experience is reported.     

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality (0<p≤k) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k<p≤k^∗, where k^∗ is the critical Sobolev embedding index of k-Hessian type. We also prove that k^∗ is optimal by one-dimensional Hardy’s inequality.     

Large deviations and linear statistics for potential theoretic ensembles associated with regular closed sets

A two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature β is investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equilibrium measure for K.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Black Box Low Tensor-Rank Approximation Using Fiber-Crosses

In this article we introduce a black box type algorithm for the approximation of tensors A in high dimension d. The algorithm adaptively determines the positions of entries of the tensor that have to be computed or read, and using these (few) entries it constructs a low rank tensor approximation X that minimizes the ? ₂-distance between A and X at the chosen positions. The full tensor A is not required, only the evaluation of A at a few positions. The minimization problem is solved by Newton’s method, which requires the computation and evaluation of the Hessian. For efficiency reasons the positions are located on fiber-crosses of the tensor so that the Hessian can be assembled and evaluated in a data-sparse form requiring a complexity of $\mathcal{O}(Pd)$ , where P is the number of fiber-crosses and d the order of the tensor.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

The Aubry set for periodic Lagrangians on the circle

For periodic convex Lagrangians on \(\mathbb{S}^1\), we show that, generically, in the sense of Mañé, there exists a dense open set of cohomology classes such that the Aubry set of these Lagrangians is a hyperbolic periodic orbit. This allows us to prove Mañé’s conjecture on \(\mathbb{S}^1\).     

Bilinear Relative Equilibria of Identical Point Vortices

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n?m+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z ²+η ², where η is real, there is a unique p(z) of degree n, and a unique value of η ²=A _n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $A_{n} \approx\frac{2}{3}n + \frac{1}{2}$ , as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.     

Convergence properties of the inexact Lin-Fukushima relaxation method for mathematical programs with complementarity constraints

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, form a difficult class of optimization problems. The feasible set of MPECs is described by standard equality and inequality constraints as well as additional complementarity constraints that are used to model equilibrium conditions in different applications. But these complementarity constraints imply that MPECs violate most of the standard constraint qualifications. Therefore, more specialized algorithms are typically applied to MPECs that take into account the particular structure of the complementarity constraints. One popular class of these specialized algorithms are the relaxation (or regularization) methods. They replace the MPEC by a sequence of nonlinear programs NLP(t) depending on a parameter t, then compute a KKT-point of each NLP(t), and try to get a suitable stationary point of the original MPEC in the limit t→0. For most relaxation methods, one can show that a C-stationary point is obtained in this way, a few others even get M-stationary points, which is a stronger property. So far, however, these results have been obtained under the assumption that one is able to compute exact KKT-points of each NLP(t). But this assumption is not implementable, hence a natural question is: What kind of stationarity do we get if we only compute approximate KKT-points? It turns out that most relaxation methods only get a weakly stationary point under this assumption, while in this paper, we show that the smooth relaxation method by Lin and Fukushima (Ann. Oper. Res. 133:63–84, 2005) still yields a C-stationary point, i.e. the inexact version of this relaxation scheme has the same convergence properties as the exact counterpart.     

Lagrangian mean curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in ${{\mathbb R}^{2n}}$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t?=?0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L ^∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.     

Inference for earthquake models: A self-correcting model

Questions of asymptotic inference are discussed for a point process model in which the conditional intensity function increases monotonically between events and drops by determined (nonrandom) amounts after each event. Parameter estimates are shown to be consistent and, except under the null hypothesis of a Poisson process, normally distributed. Under the null hypothesis, however, the Hessian matrix is not asymptotically constant, and the limiting distribution of the likelihood ratio statistics is not χ², but has a form related to that of the Cramer-von Mises ω² statistic for the test of goodness of fit.     

Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems

We introduce a new class of curvature PDOs describing relevant properties of real hypersurfaces of . In our setting, the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second-order fully nonlinear PDOs not elliptic at any point. However, when computed on generalized s-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures.     

Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.     

On the mutual embeddability of (2k, k, k − 1) and (2k − 1, k, k) quasi-residual designs

The complements of blocks containing a given point in a (2k ? 1, k, k) design, enlarged by this point, and the blocks not containing it, form a (2k ? 1, k, k) design. Likewise, the complements of blocks containing a given point in a (2k, k, k ? 1) design and the blocks not containing it, form a (2k ? 1, k, k) design. In this paper we show that if a quasi-residual (2k ? 1, k, k) design is obtained from an embeddable (2k ? 1, k, k) or (2k, k, k ? 1) design, then it is also embeddable, and describe an example of non-embeddable (12, 6, 5) design such that all (11, 6, 6) designs obtained from it are embeddable.     

Representations and the Jacobian Conjecture

In this paper we apply the representation theory of the Lie algebra sl₂(C) to the problem of describing Hessian nilpotent polynomials, which are important in the theory of the Jacobian Conjecture. In the two variable case we describe them as the maximal and minimal weight vectors of the irreducible representations of sl₂(C). For the first time this gives a characterization of the Hessian nilpotent polynomials in terms of linear differential operators.