Wedge removability of metrically thin sets and application to the CR-meromorphic extension

We give a wedge removability theorem for metrically thin sets of two codimensional Hausdorff null measure. Following [22], this removability theorem combined with the wedge removability theorem of [21] for closed subsets of two codimensional manifolds, gives a CR-meromorphic extension theorem in the greater codimensional case. Received: 28 August 1999; in final form: 10 April 2000 / Published online: 17 May 2001     

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

Summary. We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Lévy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Lévy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space. Received: 8 August 1996 / In revised form: 8 January 1997     

Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

THE CODIMENSION FORMULA ON QUASI-INVARIANT SUBSPACES OF THE FOCK SPACE

Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space L_a~2 in the case of n > 2.     

Regular submanifolds, graphs and area formula in Heisenberg groups

We describe intrinsically regular submanifolds in Heisenberg groups Hn. Low dimensional and low codimensional submanifolds turn out to be of a very different nature. The first ones are Legendrian surfaces, while low codimensional ones are more general objects, possibly non-Euclidean rectifiable. Nevertheless we prove that they are graphs in a natural group way, as well as that an area formula holds for the intrinsic Hausdorff measure. Finally, they can be seen as Federer-Fleming currents given a natural complex of differential forms on Hn.     

Moments of the number of upcrossings of an absolutely continuous process at points of density of a sample path derivative associated set

For a separable process on the unit interval with a.s. absolutely continuous sample paths a kth factorial moment formula is found for the number of sample path upcrossings of zero which occur at points of density (in a weak sense) of the set where the sample path derivative exceeds a fixed value. In the case where the sample path derivative is continuous on the closed unit interval the moment formula reduces to a simple variation of the Cramér-Leadbetter formula for the corresponding kth factorial moment of the number of unconstrained upcrossings.     

Spectral distribution and L<Superscript>2</Superscript>-isoperimetric profile of Laplace operators on groups

We give a formula relating the L ²-isoperimetric profile to the spectral distribution of a Laplace operator on a finitely generated group Γ. We prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner’s function of the group. This leads to upper bounds on spectral distributions of some non-solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth.     

Ricci Curvature and Bochner Formulas for Martingales

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two‐sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature.© 2018 Wiley Periodicals, Inc.     

Spectral flow as winding number and integral formulas

A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.

     

The Green’s function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve

We consider a regular Riemann surface of finite genus and “generalized spectral data,” a special set of distinguished points on it. From them we construct a discrete analog of the Baker-Akhiezer function with a discrete operator that annihilates it. Under some extra conditions on the generalized spectral data, the operator takes the form of the discrete Cauchy-Riemann operator, and its restriction to the even lattice is annihilated by the corresponding Schrödinger operator. In this article we construct an explicit formula for the Green’s function of the indicated operator. The formula expresses the Green’s function in terms of the integral along a special contour of a differential constructed from the wave function and the extra spectral data. In result, the Green’s function with known asymptotics at infinity can be obtained at almost every point of the spectral curve.     

Prime orbit theorems with multi-dimensional constraints for axiom a flows

In this paper we give a new proof of an asymptotic formula for the number of closed orbits of an Axiom A flow subject to certain constraints due to S. P. Lalley. We consider finite group extensions and, for transitive Anosov flows, give an application to homology. We also discuss asymptotics for closed orbits in a fixed homology class, extending a result of Katsuda and Sunada.Supported by S.E.R.C. grant no. 88001623.     

频谱曲线计算与模拟曲线DFT理论

一、实函数曲线抽样与其光滑模拟曲线的DFT设震动波实函数x(t)之CFT为X(f),x(t)?X(f),     

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreĭn's formula is established. Some examples of trace compatible affine spaces of operators are given.

     

Difference equations with a point interaction

In this paper, we consider an impulsive second‐order difference equation on the whole axis. We determine eigenvalues, spectral singularities, continuous spectrum corresponding to this difference equation with an impulsive condition by using the asymptotic properties of Jost functions, and uniqueness theorems of analytic functions. Finally, we demonstrate that the impulsive difference equation has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.     

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction. We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.     

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

     

Ray-Singer type theorem for the refined analytic torsion

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.     

The loop-product spectral sequence

The purpose of this paper is to prove that the shriek map associated to a finite codimensional sub-fiberwise embedding between Hilbert manifolds behaves properly in regard of the associated Serre Spectral sequences. We apply this result to evaluate the Chas–Sullivan loop product of the total space of a fibration. Then, we compute up to extension issues the loop homology of sphere bundle of spheres.