Vortex reconnections in classical and quantum fluids

We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in $$\mathbb {T}^3$$ T 3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.

Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution.

An implicit function theorem for sprays and applications to Oka theory

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition [Formula: see text], and the equivariant localization principle is also given.

Generalized localization for spherical partial sums of multiple Fourier series

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

Devices for image-guided lung interventions: State-of-the-art review

Lung cancer is the leading cause of cancer-related death. According to the American Cancer Society, there were an estimated 222,500 new cases of lung cancer and 155,870 deaths from lung cancer in the United States in 2017. Accurate localization in lung interventions is one of the keys to reducing the death rate from lung cancer. In this study, a total of 217 publications from 2006 to 2017 about designs of medical devices for localization in lung interventions were screened, shortlisted, and categorized by localization principle and reviewed for functionality. Each study was analyzed for engineering characteristics and clinical significance. Research regarding interventional imaging equipment, navigation systems, and surgical devices was reviewed, and both research prototypes and commercial products were discussed. Finally, the future directions and existing challenges were summarized, including real-time intra-procedure guidance, accuracy of localization, clinical application, clinical adoptability, and clinical regulatory issues.