Resolvent estimates and local energy decay for hyperbolic equations

Abstract We examine the cut-off resolvent R_χ(λ) = χ (–Δ_D – λ²)^–1χ, where Δ_D is the Laplacian with Dirichlet boundary condition and equal to 1 in a neighborhood of the obstacle K. We show that if R_χ(λ) has no poles for , then This estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray. Keywords: Trapping obstacles, Resonances, Local energy decay, Cut-off resolvent     

A Translation and Rotation Invariant Gauss–Newton Like Scheme for Image Registration

A Gauss–Newton like method is considered to obtain a d–dimensional displacement vector field , which minimizes a suitable distance measure D between two images. The key to find a minimizer is to substitute the Hessian of D with the Sobolev-H²(Ω)^d norm for . Since the kernel of the associated semi-norm consists only of the affine linear functions we can show in this way, that the solution of each Newton step is a linear combination of an affine linear transformation and an affine-free nonlinear deformation. Our approach is based on the solution of a sequence of quadratic subproblems with linear constraints. We show that the resulting Karush–Kuhn–Tucker system, with a 3×3 block structure, can be solved uniquely and the Gauss–Newton like scheme can be separated into two separated iterations. Finally, we report on synthetic as well as on real-life data test runs. AMS subject classification (2000) 65F20, 68U10     

Results on Local Times of a Class of Multiparameter Gaussian Processes

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.     

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations

Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Z_t serves as a physical model for diffusions in a crack. If τ_D(Z) is the first exit time of this processes from a domain D⊂ℝⁿ, started at z∈D, then P_z[τ_D(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of which gives exponential integrability of for parabola-shaped domains of the form P_α={(x,Y)∈ℝ×ℝⁿ⁻¹:x>0, |Y|<Ax^α}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ² as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated Brownian motion in a parabola we find that for z∈℘

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

On a generalization of compound Newton-Cotes quadrature formulas

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse _,n in the estimate

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to , M → 0, though producing unphysical results. A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M^–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments. AMS subject classification (2000) 35L65, 35C20, 76G25     

On a semigroup generated by a convex combination of two Feller generators

Let be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in with related Feller processes {X _A (t), t ≥ 0} and {X _B (t), t ≥ 0} and let α and β be two non-negative continuous functions on with α + β = 1. Assume that the closure C of C ₀ = αA + βB with generates a Feller semigroup {T _C (t), t ≥ 0} in . It is natural to think of a related Feller process {X _C (t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point , with probability α(p) the process behaves like {X _A (t), t ≥ 0} and with probability β(p) it behaves like {X _B (t), t ≥ 0}. We provide an approximation of {T _C (t), t ≥ 0} via a sequence of semigroups acting in that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].     

On the geometry of metric measure spaces

We introduce and analyze lower (Ricci) curvature bounds  ⩾ K for metric measure spaces . Our definition is based on convexity properties of the relative entropy regarded as a function on the L ₂-Wasserstein space of probability measures on the metric space . Among others, we show that  ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,  ⩾ K if and only if  ⩾ K for all . The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.     

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if is a Gabor frame for with frame bounds A and B, then the following two inequalities hold: and . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form , where Δ _k and Λ _k are arbitrary sequences of points in and , 1 ≤ k ≤ r. Corresponding author for second author Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ²⁺¹→S ² of the form where u is the polar angle on the sphere, is the ground state harmonic map, λ(t)=t ^-1-ν, and is a radiative error with local energy going to zero as t→0. The number can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis. Mathematics Subject Classification (1991) 35L05, 35Q75, 35P25     

Characterization of Gaussian semigroups on separable Banach spaces

Let E be a separable real Banach space and denote by BUC (E) the space of bounded and uniformly continuous functions on E. For a C ₀-semigroup acting on BUC(E), we obtain necessary and sufficient conditions ensuring that is Gaussian.     

On the Cauchy problem for D t 2–D x a(t,x) n D x

Abstract The well posedness of the Cauchy problem for the operator P=D_t²–D_xa(t,x)ⁿD_x, with data on t=0 is studied assuming a ∈ C^N( (R)), s₀>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ^(s), for , n≥ n₀. Keywords: Partial differential equations, Cauchy problem, Well posedness     

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.     

The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel μ（t） =∑ω=1^∞ exp（-itEω^1/2）, where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the （x^1, x^2, x^3）-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ （t） on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j （j = 1,……, n）, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^＊ （i = 1 ＋ kj-1,……, kj） of the bounding surfaces S j are considered, such that S j = Ui-1＋kj-1^kj Si^＊, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω （e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature） are determined from the asymptotic expansion ofexpansion of μ（t） for small │t│.