Academic literature on the topic 'Log-Lipschitz continuity'

Let $f$ be a function from $\mathbf{R}^p$ to $\mathbf{R}^q$ and let $\Lambda$ be a finite set of pairs $(\theta, \eta) \in \mathbf{R}^p \times \mathbf{R}^q$. Assume that the real-valued function $\langle\eta, f(x)\rangle$ is Lipschitz continuous in the direction $\theta$ for every $(\theta, \eta) \in \Lambda$. Necessary and sufficient conditions on $\Lambda$ are given for this assumption to imply each of the following: (1) that $f$ is Lipschitz continuous, and (2) that $f$ is continuous with modulus of continuity $\le C\epsilon |{\log \epsilon}|$.

Abstract By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law.

AbstractWe prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form\mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dxwith non-autonomous integrand {F(x,\xi)} which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map {x\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class.

AbstractWe study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) mapf: ℙ2ℂ → ℙ2ℂ, where ℙ2ℂ stands for the complex projective space of dimension 2. Letδs(x)denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote byδs(x)the Hausdorff dimension of∩ Λ, whereris some fixed positive number andis the local stable manifold atxof sizer;δs(x)is calledthe stable dimension at x. Mihailescu and Urba ńnski introduced a notion of inverse topological pressure, denoted by P−, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates off, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each pointxfrom Λ has the same numberd′of preimages in Λ, then we show thatδs(x)is independent of x; in factδs(x)is shown to be equal in this case with the unique zero of the mapt → P(tϕs−log d′). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

Abstract Magnetic resonance (MR) images are frequently corrupted by Rician noise during image acquisition and transmission. And it is very challenging to restore MR data because Rician noise is signal-dependent. By exploring the nonlocal self-similarity of natural images and further using the low-rank prior of the matrices formed by nonlocal similar patches for 2-D data or cubes for 3-D data, we propose in this paper a new nonlocal low-rank regularization (NLRR) method including an optimization model and an efficient iterative algorithm to remove Rician noise. The proposed mathematical model consists of a data fidelity term derived from a maximum a posteriori (MAP) estimation and a nonlocal low-rank regularization term using the log-det function. The resulting model in terms of approximated patch/cube matrices is non-convex and non-smooth. To solve this model, we propose an alternating reweighted minimization (ARM) algorithm using the Lipschitz-continuity of the gradient of the fidelity term and the concavity of the logarithmic function in the log-det function. The subproblems of the ARM algorithm have closed-form solutions and its limit points are first-order critical points of the problem. The ARM algorithm is further integrated with a two-stage scheme to enhance the denoising performance of the proposed NLRR method. Experimental results tested on 2-D and 3-D MR data, including simulated and real data, show that the NLRR method outperforms existing state-of-the-art methods for removing Rician noise.

The present thesis is devoted to the study both of strictly hyperbolic operators with low regularity coefficients and of the density-dependent incompressible Euler system. On the one hand, we show a priori estimates for a second order strictly hyperbolic operator whose highest order coefficients satisfy a log-Zygmund continuity condition in time and a log-Lipschitz continuity condition with respect to space. Such an estimate involves a time increasing loss of derivatives...