Self-adjoint elliptic operators and manifold decompositions Part II: Spectral flow and Maslov index

This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous “static” results in obtaining results on the decomposition of spectral flows. Some of these “dynamic” results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of “jumping Lagrangians.” In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce “canonical perturbations” of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant. © 1996 John Wiley & Sons, Inc.     

Actions of the dipper-donkin quantization GL 2 on the clifford algebra C(l,3)

Following the method already developed for studying the actions of GL_q (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL ₂ constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases.

We also prove that each irreducible finite dimensional algebra representation of the quantum GL ₂ q^m ≠1, is one dimensional.

By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

L 2-topological invariants of 3-manifolds

Summary We give results on theL ²-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, along with some analytic results on geometric 3-manifolds, are used to compute theL ²-Betti numbers of compact 3-manifolds which satisfy a weak form of the geometrization conjecture, and to compute or estimate their Novikov-Shubin invariants.Oblatum 6-V-1993 & 20-VI-1994Partially supported by NSF-grant DMS 9101920Partially supported by NSF-grant DMS 9103327     

L (α)-Conjugates on Parabolic Bergman Spaces

The parabolic Bergman space is the set of all L ^p -solutions of the parabolic operator L ^(α). In this paper, we define L ^(α)-conjugates by using fractional derivatives, which are the extension of harmonic conjugates. We study several properties of L ^(α)-conjugates on parabolic Bergman spaces.     

The Best Multipoint Padé Approximant

Abstract

This paper is dealing with the problem of finding the “best” multipoint Padé approximant of an analytic function when data in some neighborhoods of sampling points are more important than others. More exactly, we obtain a multipoint Padé approximants as limits of best rational L^p-approximations on union of disks, when the measure of them tends to zero with different speeds. As such, this technique provides useful qualitative and analytic information concerning the approximants, which is difficult to obtain from a strictly numerical treatment.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Integral self-affine sets with positive Lebesgue measures

A self-affine region is an integral self-affine set with positive Lebesgue measure. In this note we give two criteria for integral self-affine sets being self-affine regions. As their applications we study the L ¹-solutions of refinement equations, which play an important role in constructing wavelets, and we give several interesting examples. Received: 8 May 2007     

Symplectic modules

A symplectic module is a finite group with a regular antisymmetric form. The paper determines sufficient conditions for the invariants of the maximal isotropic subgroups (Lagrangians), and asymptotic values for a lower bound of a group which contains Lagrangians of all symplectic modules of a fixed orderp ⁿ. These results have application to the splitting fields of universal division algebras.     

Boundary value problem for second-order differential operators with integral conditions

In this article, we study a second-order differential operator with mixed nonlocal boundary conditions combined weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions and on the coefficients in the boundary conditions, called regular and nonregular cases, we prove that the resolvent decreases with respect to the spectral parameter in L ^p ?(0,?1), but there is no maximal decreasing at infinity for p?>?1. Furthermore, the studied operator generates in L ^p ?(0,?1) an analytic semigroup for p?=?1 in regular case, and an analytic semigroup with singularities for p?>?1 in both cases, and for p?=?1 in the nonregular case only. The obtained results are then used to show the correct solvability of a mixed problem for a parabolic partial differential equation with nonregular boundary conditions.     

“Zero” temperature stochastic 3D ising model and dimer covering fluctuations: A first step towards interface mean curvature motion

We consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “−” spins disappears within a time τ₊, which is at most L²(log L)^c and at least L²/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between “+” and “−” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ₊ ≃ const × L², conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ₊ can be shown to be of order L² without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like for L large, as conjectured in [2]. © 2011 Wiley Periodicals, Inc.     

Limiting Absorption Principle for Stratified Operators with Thresholds: A General Method

Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z ∈ I. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).     

Some characterizations of the analytic Radon-Nikodým property in Banach spaces

The analytic Radon-Nikodým property for a complex Banach space X is characterized in several ways analogous to the common characterizations of the Radon-Nikodým property. For example, the Banach space X has the analytic Radon-Nikodým property if and only if any operator from L¹ to X that factors through L¹/H₀¹ is representable. The characterizations are used to study complemented subspaces of L¹/H₀¹ and to define the analytic Radon-Nikodým property for real Banach spaces.     

Analyticity on L 1 of the heat semigroup with Wentzell boundary conditions

We show that the semigroup generated by the realization of the Laplace operator with Wentzell boundary conditions in a bounded smooth domain is analytic on ${L^1(\Omega) \oplus L^1(\partial \Omega)}We show that the semigroup generated by the realization of the Laplace operator with Wentzell boundary conditions in a bounded smooth domain is analytic on L¹(W) ?L¹(?W){L^1(\Omega) \oplus L^1(\partial \Omega)} .     

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L²(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star‐shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star‐combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second‐kind integral operator for which convergence of the Galerkin method in L²(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star‐combined operator implies frequency‐explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high‐frequency case. The proof of coercivity of the star‐combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.     

Dense Subsets of -Solutions to Linear Elliptic Partial Differential Equations

Let Ω ^N (N2) be an unbounded domain, and L_m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing ¹-solutions to L_m (in Ω) approximate all ¹-solutions to L_m (in Ω), provided there exist real numbers R_j→∞, 0, and a sequence {y^j} such that B(y^j, )∩Ω= and where |·| means the volume and

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for z ^N, R>0 and D ^N. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to L_m may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in .