The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001     

On the convergence of the DFP algorithm for unconstrained optimization when there are only two variables

Let the DFP algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets, where each line search finds the first local minimum. It is proved that the calculated gradients are not bounded away from zero if there are only two variables. The new feature of this work is that there is no need for the objective function to be convex. Received: June 16, 1999 / Accepted: December 24, 1999?Published online March 15, 2000     

Symmetries in the stochastic calculus of variations

Summary. Given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space-time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations. We prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals. We then prove a stochastic version of Noether theorem: to each family of transformations leaving the action invariant (or symmetries) we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints. Received: 29 June 1996 / In revised form: 19 July 1996     

Weak covering and the tree property

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if has the tree property then and is weakly compact in K. Received: 11 June 1997     

Foliations by complex curves and the geometry of real surfaces of finite type

We show that if the Levi form of a smooth CR manifold is de-generate in every conormal direction, then on a dense open set, the manifold is foliated by complex curves. As a consequence we show that every real analytic manifold of finite D'Angelo type can be stratified so that each stratum locally is contained in a Levi nondegenerate hypersurface. Received in final form: 11 June 2001 / Published online: 28 February 2002     

A stabilized finite element method for the incompressible magnetohydrodynamic equations

Summary. We propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large. This observation is confirmed by our numerical tests. Received August 31, 1998 / Revised version received June 16, 1999 / Published online June 21, 2000     

Positive Pinching, Volume and Second Betti Number

Our main theorem asserts that for all odd and 0 < , there exists a small constant, , such that if a simply connected n-manifold, M, with vanishing second Betti number admits a metric of sectional curvature, , then the injectivity radius of M is greater than . Submitted: June 1998, revised: December 1998, final version: June 1999.     

Riemannian submersions of open manifolds which are flat at infinity

We prove that a base of a Riemannian submersion is flat, if is flat at infinity and is compact. As a corollary we obtain a topological gap-phenomenon for open manifolds of nonnegative sectional curvature (Eschenburg-Schroeder-Strake conjecture). Received: October 21, 1997; revised: June 2, 1998.     

Complexes of Dirac operators in Clifford algebras

In this paper we study complexes of k Dirac operators (or variations of Dirac operators) in the real or complex Clifford algebras i.e. complexes in which the first map is induced by the matrix where is the Dirac operator with respect to the variable . In particular we prove that, if , the complex in the case of 3 operators can be described in terms of relations coming from the so called radial algebra. Moreover we show that if the dimension m is less than 2k-1, then the Fischer decomposition does not hold. Received: 2 February 2000; in final form: 20 June 2000 / Published online: 25 June 2001     

On the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy

We analyze the convergence of a sequential quadratic programming (SQP) method for nonlinear programming for the case in which the Jacobian of the active constraints is rank deficient at the solution and/or strict complementarity does not hold for some or any feasible Lagrange multipliers. We use a nondifferentiable exact penalty function, and we prove that the sequence generated by an SQP using a line search is locally R-linearly convergent if the matrix of the quadratic program is positive definite and constant over iterations, provided that the Mangasarian-Fromovitz constraint qualification and some second-order sufficiency conditions hold. Received: April 28, 1998 / Accepted: June 28, 2001?Published online April 12, 2002     

On projectively related Einstein metrics in Riemann-Finsler geometry

In this paper we study pointwise projectively related Einstein metrics (having the same geodesics as point sets). We show that pointwise projectively related Einstein metrics satisfy a simple equation along geodesics. In particular, we show that if two pointwise projectively related Einstein metrics are complete with negative Einstein constants, then one is a multiple of another. Received: 28 October 1999 / Revised: 10 January 2000 /Published online: 18 June 2001     

Binomial arithmetical rank of lattice ideals

 In this paper, we will show that a lattice ideal is a complete intersection if and only if its binomial arithmetical rank equals its height, if the characteristic of the base field k is zero. And we will give the condition that a binomial ideal equals a lattice ideal up to radical in the case of char k=0. Further, we will study the upper bound of the binomial arithmetical rank of lattice ideals and give a sharp bound for the lattice ideals of codimension two. Received: 12 June 2001 / Revised version: 22 July 2002     

The weak survival/strong survival phase transition for the contact process on a homogeneous tree

The contact process on a homogeneous tree of degree 3 or larger is known to have two survival phases: weak and strong. In the weak survival phase, the "Malthusian parameter" (the Hausdorff dimension of the set of ends of the tree in which the infection survives) is less than half the Hausdorff dimension of the entire boundary. It is shown that if the expected infection time of a vertex is bounded by a constant times the probability of infection, then the critical exponent for the Malthusian parameter is at least 1/2. Received: 30 June 2002     

The emergence of kinship behavior in structured populations of unrelated individuals

The paper provides an explanation for altruistic behavior based on the matching and learning technology in the population. In a infinite structured population, in which individuals meet and interact with their neighbors, individuals learn by imitating their more successful neighbors. We ask which strategies are robust against invasion of mutants: A strategy is unbeatable if when all play it and a finite group of identical mutants enters then the learning process eliminates the mutants with probability 1. We find that such an unbeatable strategy is necessarily one in which each individual behaves as if he is related to his neighbors and takes into account their welfare as well as his. The degree to which he cares depends on the radii of his neighborhoods. Received June 1996/Revised version October 1998     

s-expandable spaces

We introduce the concept of s-expandable spaces as a variation of expandable spaces. We show that an extremally disconnected semi-regular space is s-expandable if and only if it is expandable. This revised version was published online in June 2006 with corrections to the Cover Date.     

A partition relation for successors of Large Cardinals

We address partition problems of Erd?s and Hajnal by showing that for all , if and carries a -dense ideal. If is measurable we show that for where is a very large ordinal less than that is closed under all primitive recursive ordinal operations. Received: 27 June 2001 / Revised version: 5 December 2001 / Published online: 4 February 2003 The first author was partially supported by NSF grant DMS-0101155 and the Equipe d'Analyse Univ. of Paris 6. The second author was partially supported by NSF grants DMS-0072560 and DMS-9704477.     

Cohen–Macaulayness and squarefree modules

In this paper we consider the category of squarefree modules over the polynomial ring and an exact duality functor, which is an extension of the Alexander dual of a simplicial complex. We give a relationship between the squarefree components of local cohomology groups of a squarefree module and the Tor groups of its dual. With this result it is shown that a squarefree module is sequentially Cohen–Macaulay if and only if the dual is componentwise linear. Received: 7 June 1999 / Revised version: 6 September 2000     

Positive mass theorem for the Yamabe problem on spin manifolds

Let (M, g) be a compact connected spin manifold of dimension n ≥ 3 whose Yamabe invariant is positive. We assume that (M, g) is locally conformally flat or that n ∈ {3, 4, 5}. According to a positive mass theorem by Schoen and Yau the constant term in the asymptotic development of the Green’s function of the conformal Laplacian is positive if (M, g) is not conformally equivalent to the sphere. The proof was simplified by Witten with the help of spinors. In our article we will give a proof which is even considerably shorter. Our proof is a modification of Witten’s argument, but no analysis on asymptotically flat spaces is needed.Received: March 2004 Revised: June 2004 Accepted: June 2004