Notes on very ample vector bundles on 3-folds

Let be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of is provided when is nef but not big, and when a suitable positive multiple of defines a morphism X → B with connected fibers onto a smooth projective curve B, where K_X is the canonical bundle of X. As an application, the case where the genus of B is positive and has a global section whose zero locus is a smooth hyperelliptic curve of genus ≧ 2 is investigated, and our previous result is improved for threefolds. Received: 27 January 2005; revised: 26 March 2005     

Dirichlet forms and diffusion processes for fermion random point fields

We construct the Dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the Euclidean space , for which certain fermion random point fields are invariant.     

A Global Wellordering of Norms Defined via Blackwell Games

We define a global order of norms using strongly optimal strategies in Blackwell games and prove that it is a prewellordering under the assumption of the Axiom of Blackwell determinacy.     

Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory

A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where and R(x, y) is a covariance function.The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation as 0.$$" align="middle" border="0"> The domain D can be an interval or a domain in Rⁿ, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, D_x)R(x, y) = P(x, D_x)δ(x − y), where Q(x, D_x) and Px, D_x) are elliptic differential operators.     

Image of the Spectral Measure of a Jacobi Field and the Corresponding Operators

By definition, a Jacobi field is a family of commuting selfadjoint three-diagonal operators in the Fock space The operators J(ϕ) are indexed by the vectors of a real Hilbert space H₊. The spectral measure ρ of the field J is defined on the space H₋ of functionals over H₊. The image of the measure ρ under a mapping is a probability measure ρ_K on T₋. We obtain a family J_K of operators whose spectral measure is equal to ρ_K. We also obtain the chaotic decomposition for the space L²(T₋, d_{ρ K}).     

On the coupling of MIXED-FEM and BEM for an exterior Helmholtz problem in the plane

Summary This paper deals with an elliptic boundary value problem posed in the plane, with variable coefficients, but whose restriction to the exterior of a bounded domain reduces to a Helmholtz equation. We consider a mixed variational formulation in a bounded domain that contains the heterogeneous medium, coupled with a boundary integral method applied to the Helmholtz equation in . We utilize suitable auxiliary problems, duality arguments, and Fredholm alternative to show that the resulting formulation of the problem is well posed. Then, we define a corresponding Galerkin scheme by using rotated Raviart-Thomas subspaces and spectral elements (on the interface). We show that the discrete problem is uniquely solvable and convergent and prove optimal error estimates. Finally we illustrate our analysis with some results from computational experiments.     

Estimates on path delocalization for copolymers at selective interfaces

Starting from the simple symmetric random walk {S_n}_n, we introduce a new process whose path measure is weighted by a factor exp with α,h≥0, {W_n}_n a typical realization of an IID process and N a positive integer. We are looking for results in the large N limit. This factor favors S_n>0 if W_n+h>0 and S_n<0 if W_n+h<0. The process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents. It has been shown [6] that this model undergoes a (de)localization transition: more precisely there exists a continuous increasing function λ↦h_c(λ) such that if h<h_c(λ) then the model is localized while it is delocalized if h≥h_c(λ). However, localization and delocalization were not given in terms of path properties, but in a free energy sense. Later on it has been shown that free energy localization does indeed correspond to a (strong) form of path localization [3]. On the other hand, only weak results on the delocalized regime have been known so far. We present a method, based on concentration bounds on suitably restricted partition functions, that yields much stronger results on the path behavior in the interior of the delocalized region, that is for h>h_c(λ). In particular we prove that, in a suitable sense, one cannot expect more than O( log N) visits of the walk to the lower half plane. The previously known bound was o(N). Stronger O(1)–type results are obtained deep inside the delocalized region. The same approach is also helpful for a different type of question: we prove in fact that the limit as α tends to zero of h_c(λ)/λ exists and it is independent of the law of ω₁, at least when the random variable ω₁ is bounded or it is Gaussian. This is achieved by interpolating between this class of variables and the particular case of ω₁ taking values ±1 with probability 1/2, treated in [6].     

Weak convergence of random <Emphasis Type="Bold">p</Emphasis>-mappings and the exploration process of inhomogeneous continuum random trees

We study the asymptotics of the p-mapping model of random mappings on [n] as n gets large, under a large class of asymptotic regimes for the underlying distribution p. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2004) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of “attracting points” to emerge.Research supported by NSF Grant DMS-0203062.Research supported by NSF Grant DMS-0071468.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove L^p estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.     

A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue

We prove a Brunn-Minkowski-type inequality for the eigenvalue Λ of the Monge-Ampère operator: Λ^-1/2n is concave in the class of domains in Rⁿ endowed with Minkowski addition. The equality case is explicitly described too. The main device of the proof is a notion of addition for convex functions, called infimal convolution, which corresponds to the Minkowski addition of the graphs of the involved functions.