We prove that for impulsive exposure patterns there is no uniform exponential energy law in wall-clock time t, which explains why past t-based unifications of continuous damping with impulses fail. We therefore replace t by a measure-valued clock, sigma, that aggregates absolutely continuous exposure and atomic doses within a single Lyapunov ledger. On this ledger we prove an observability-dissipation principle in the sense of the Hilbert Uniqueness Method (HUM): there exists a structural constant c_sigma > 0 such that the energy decays at least at a product-exponential rate with respect to sigma. When sigma = t, the statement reduces to classical exponential stabilization with the same constant. For the damped wave under the Geometric Control Condition (GCC), the constant is calibrated by the usual observability and geometric factors. The framework yields a monotonicity principle ("more sigma-mass implies faster decay") and unifies intermittent regimes where quiescent intervals are punctuated by impulses. As robustness, secondary to the main contribution, the same decay law persists under structure-compatible discretizations and along compact variational limits; a stochastic extension supplies expectation and pathwise envelopes via the compensator. The contribution is a qualitative dynamics backbone: observability implies sigma-exponential decay with sharp constants.
Particular adaptive Morley finite element schemes for the approximation to a regular root of the incompressible stream function vorticity formulation of the stationary 2D Navier-Stokes equations guarantee optimal convergence rates. A smoother in the quadratic nonlinearity enables reliable and efficient explicit residual-based a posteriori error estimators in a standard adaptive loop and allows for the first rate-optimality result for the Navier-Stokes equations. A numerical comparison verifies the efficiency of the proposed error estimators and the optimal convergence rates in the discrete energy and in weaker piecewise Sobolev norms.
We prove a T1 theorem for fractional vector Riesz transforms mapping one weighted Sobolev space to another, where the weights are doubling measures on Euclidean space. Boundedness is characterized by the classical A_2 condition and two dual testing conditions on indicators of cubes. We also show the equivalence of various weighted Sobolev norms when the measure is doubling, something that fails in general.
This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journé, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight restricted weak type inequality. Assuming σ and ω are locally finite positive Borel measures, with one of them an A infinity weight, we show that the two weight restricted weak type inequality for an α-fractional Calderón-Zygmund singular integral T holds if and (provided T is elliptic) only if the classical fractional Muckenhoupt condition holds. In the case α = 0, boundedness of T on unweighted L2 is assumed as well. Applications are then given to a Tp theorem for CZO's with doubling weights when one is A infinity, and then to optimal cancellation conditions for CZO's in terms of polynomial testing in similar situations.
We begin an investigation into extending the T1 theorem of David and Journé, and the corresponding cancellation conditions of Stein, to more general pairs of distinct doubling weights. For example, assuming the measures satisfy a fractional A infinity condition and are comparable in the sense of Coifman and Fefferman, we characterize the two weight norm inequality for a strongly elliptic fractional Calderón-Zygmund singular integral, in terms of the one-tailed fractional Muckenhoupt conditions, and the usual cube testing conditions. We then apply this result to give a version, in the setting of two comparable fractional A infinity weights, of Stein's characterization of cancellation conditions on a kernel K in order that there exists a bounded operator T that is associated with K. More generally we prove a T1 theorem involving a bilinear indicator/cube testing inequality in place of the weak boundedness property of David and Journeé - where we must test over all bounded functions instead of just Holder continuous functions. We use a proof strategy based on an adaptation of the `pivotal' argument of Nazarov, Treil and Volberg to the weighted Alpert wavelets of Rahm, Sawyer and Wick using a Parallel Corona decomposition of Lacey, Sawyer, Shen and Uriarte-Tuero.
We introduce the notion of a pre-sequence of matrix orthogonal polynomials to mean a sequence {F_n} of matrix orthogonal functions with respect to a weight function W, satisfying a three term recursion relation and such that det(F_0) is not zero almost everywhere. By now there is a uniform construction of such sequences from irreducible spherical functions of some fixed K-types associated to compact symmetric pairs (G,K) of rank one. Our main result is that {Q_n=F_nF_0^{-1}} is a sequence of matrix orthogonal polynomials with respect to the weight function F_0WF_0*, see Theorem 2.1.
Randal L. Croshaw, Megan L. Marshall, Tesha L. Williams
et al.
Breast-conserving therapy (BCT) for sporadic breast cancer has been widely accepted by surgeons and patients alike. While BCT is associated with a higher risk of ipsilateral breast tumor recurrence (IBTR), it has not been shown to decrease overall survival (OS) in comparison with mastectomy. Many women with aBRCA1/2mutation opt for mastectomy instead of breast-conserving measures at the time of a breast cancer diagnosis. In some cases, this is due to fear of aggressive disease, but to date, there have been no studies offering strong evidence that breast conservation should not be offered to these women.BRCA1/2-associated breast cancer has not been found to be more aggressive or resistant to treatment than comparable sporadic tumors, and no study has shown an actual survival advantage for mastectomy in appropriately treated affected mutation carriers. This paper reviews the available literature for breast conservation and surgical decision making inBRCA1/2mutation carriers.