Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

Several results on the families of diffeomorphisms onS 1

In this paper, we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms onS ¹ to be stable and a necessary condition for the multi-parameter families to be stable; and, moreover, we prove that phase-locking is a generic property of the one-parameter families of diffeomorphisms onS ¹. We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms onS ¹ which strengthens a result in [2].     

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

An uncertainty relation for quantum systems with an arbitrarily large number of particles and the expansion rate of matter in the bulk

For any quantum system with an arbitrarily large number N of particles for which the lower end of the spectrum has a nonzero width and is bounded below by N, we rigorously derive an uncertainty relation for the product of N and the survival time T, a measure of the system resistance to change. We then use the derived inequality to investigate the highly nontrivial problem of the expansion rate of bulk matter as a function of the number N of electrons for large N. We approach the application to this problem by noting that resistance to the increase of the expansion rate can be quantum mechanically defined in terms of its survival time against such an increase. We show that a sufficient condition for matter to have a nonvanishing survival time against expansion rate increase is that the lower end of the spectrum is the lower end of an energy width for which applying the derived uncertainty relation becomes obvious. In turn, in particular, we show that if the expansion rate increases with its “size” of radius R, then the survival time decreases not faster than 1/R ³ for large R. For completeness and consistency of the analysis, we also consider the formal zero-width limit. Because the derived uncertainty relation is general, we expect it also to have other applications.     

Wiener’s test for super-Brownian motion and the Brownian snake

We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ ^d and x ∈ ℝ ^d , we provide a necessary and sufficientcondition for super-Brownian motion started at δ _x to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence in a domain D of a positivesolution of the equation Δ; u = u ² which explodes at a given point of ∂ D. Received: 5 January 1996 / In revised form: 30 October 1996     

The Closure of the Set of Tight Frame Wavelets

We study properties of the closure of the set of tight frame wavelets. We give a necessary condition and a sufficient condition for a function to be in this closure. In particular, we show that the collection of tight frame wavelets is not dense in L ²(? ⁿ ), which answers a question posed by D. Han and D. Larson (Preprint, 2008).     

Toeplitz operators hyponormal with the unilateral shift

For the unilateral shift operator U on the Hardy space H²(T), we describe conditions on operators T, acting on H²(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC     

Explosion of Jump-Type Symmetric Dirichlet Forms on ℝ d

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? ^d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? ^d .     

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

We consider the identification of a nonlinear corrosion profile from single voltage boundary data and show injectivity of the parameter-to-output map. We demonstrate that Tikhonov regularization can be applied in order to solve the inverse problem in a stable manner despite the presence of noisy data. In combination with a logarithmic stability estimate for the underlying Cauchy problem, rates for the convergence of the regularized solutions are proven using a source condition that does not involve the Fréchet derivative of the parameter-to-output map. We present sufficient conditions for the existence of a source function and illustrate our approach by means of numerical examples.     

To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by O(τ ²), where τ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left( {\tau ^2 \ln \frac{T} {\tau }} \right) $ , where T is the length of the time interval.     

A weak condition for the convexity of tensor-product Bézier and B-spline surfaces

A sufficient condition for a tensor-product Bézier surface to be convex is presented. The condition does not require that the control surface itself is convex, which is known to be a very restrictive property anyway. The convexity condition is generalised toC ¹ tensor-product B-spline surfaces.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

BV-Ellipticity and Lower Semicontinuity of Surface Energy of Caccioppoli Partitions of ℝ n

We give a new proof that BV-ellipticity is sufficient for lower semicontinuity of surface energy of Caccioppoli partitions of ? ⁿ , for any n≥2, with respect to convergence in volume. BV-ellipticity, introduced by L. Ambrosio and A. Braides two decades ago, is the only condition known to be necessary and sufficient for lower semicontinuity in the context of Caccioppoli partitions of ? ⁿ ; it is analogous, for this setting, to C.B. Morrey’s quasi-convexity. We also show, for the first time, that BV-ellipticity suffices for lower semicontinuity with respect to weaker notions of convergence involving the weak and flat topologies on integral currents.     

Boundaries of Holomorphic 1-Chains Within Holomorphic Line Bundles over $\mathbb{CP}^{1}$

We show that boundaries of holomorphic 1-chains within holomorphic line bundles of \mathbbCP¹\mathbb{CP}^{1} can be characterized using a single generating function of Wermer moments. In the case of negative line bundles, a rationality condition on the generating function plus the vanishing moment condition together form an equivalent condition for bounding. We provide some examples which reveal that the vanishing moment condition is not sufficient by itself. These examples also can be used to demonstrate one point of caution about the use of birational maps in this topic.     

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C^∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C^∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.     

CMV matrices with asymptotically constant coefficients. Szegö-Blaschke class, scattering theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrices acting in l², whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in l²) behaves asymptotically as the basis associated with the free system. (2) From the point of view of the scattering problem, the most natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely and (b) the associated Gelfand-Levitan-Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an A₂ kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions close to the optimal ones are given directly in terms of the scattering data.     

A Tridimensional Inverse Shaping Problem

We study the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Eilonal equation on the surface of the bubble which has self-contained interest. We answer the question for closed smooth surface which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a C¹ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. A counterexample proves that not all analytic perturbations of such surface are shapable.     

Homogenization and Enhancement of the G‐Equation in Random Environments

We study the homogenization of a G‐equation that is advected by a divergence free “small mean” stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G‐equation, and we give necessary and sufficient conditions for enhancement. Since the problem is not assumed to be coercive, it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition that is necessary to apply the subadditive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence that has both a long‐time averaged limit (due to the subadditive ergodic theorem) and stays asymptotically close to the minimal time. © 2013 Wiley Periodicals, Inc.     

Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L ^p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.     

An analytical study of a swapping operation in the money market

Investors in money markets have found a class of transaction in which return from investment can be increased without increasing the risk taken. We formally analyse one of these operations (usually referred to as a ‘swap’) and obtain the condition under which it is possible to obtain a swap profit.From this condition, an indicator is obtained to aid the decision of when to engage a swap. The indicator evidences the role played by the spread between yield rates, time to maturity and timing, in a swap.Finally the indicator is used to derive some mathematical properties of the swapping operation. We show that a trend toward widening spreads is neither a necessary nor a sufficient condition for a swap-profit to be obtained. We give examples using hard data to support our theoretical results.