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In the present study, a set of physics-informed and data-driven approaches are examined towards the development of an accurate reduced-order model for a turbulent plane Couette flow. Based on the utilisation of the proper orthogonal decomposition (POD), a particular focus is given to the development of a reduced-order model where the number of POD modes are not large enough to cover the full dynamics of the given turbulent state, the situation directly relevant to the reduced-order modelling for turbulent flows. Starting from the conventional Galerkin projection approach ignoring the truncation error, three approaches enhanced by both physics and data are examined: (1) sparse regression of the POD-Galerkin dynamics; (2) Galerkin projection with an empirical eddy-viscosity model; (3) Galerkin projection with an optimal eddy viscosity obtained from a newly proposed sparse regression – an idea applying the sparse identification of nonlinear dynamics framework to an eddy-viscosity model. The sparse regression of the POD-Galerkin dynamics does not perform well, as the number of POD modes for the given chaotic dynamics appears to be too small. While the unsatisfactory performance of the Galerkin projection model with an empirical eddy viscosity is observed, the newly proposed approach, which combines the concept of an optimal eddy-viscosity closure with a sparse regression, more accurately approximates the chaotic dynamics than the other reduced-order models considered. This is demonstrated with the mean and time scales of the POD mode amplitudes as well as the first- and second-order turbulence statistics.

We show there exists a nontrivial $H^1_0$ solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. We also show the existence of a nontrivial solution to the steady Navier-Stokes equation in the same setting, whereas the analogous problem is open in the Euclidean case., Comment: 35 pages

Lymphomas of the central nervous system (CNS) are broadly classified into primary CNS lymphoma (PCNSL) and secondary CNS lymphoma (SCNSL). PCNSL refers to lymphoma restricted to the brain, leptomeninges, spinal cord, or eyes without evidence of it outside the CNS at primary diagnosis, while SCNSL refers to secondary CNS involvement by systemic lymphoma. The brain is the most common site of involvement and intracranial CNS lymphoma has a highly variable imaging appearance and varies according to the patient’s clinical profile and immunity. This pictorial essay aims to illustrate the different facets of intracranial CNS lymphomas on neuroimaging. This enables radiologists to be familiarized with their key diagnostic features and differentiate from their differential diagnoses, leading to early diagnosis, and treatment.

The problem for the stationary Navier–Stokes equation in 3D under finite Dirichlet norm is open. In this paper we answer the analogous question on the 3D hyperbolic space. We also address other dimensions and more general manifolds.

We consider the generalization of the Navier-Stokes equations from $\mathbb R^n$ to the Riemannian manifolds. There are inequivalent formulations of the Navier-Stokes equations on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds., Comment: 16 pages; published version. See version 1 for the details of the non-relativistic limit, which were omitted in the published version

We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to $0$ at infinity. We also address the decay rate for the vorticity and the behavior of the pressure.

Study design Retrospective study. Purpose To analyze the quantitative anatomy of C7 vertebra for insertion of lateral mass screws and pedicle screws in Southern Chinese patients. Overview of literature C7 lateral mass is smaller when compared to other subaxial cervical levels, which limits the length of lateral mass screws that can be used. Some studies have suggested pedicle screws for better fixation. But, this option is limited by the narrow pedicle width. Methods We have obtained computed tomography (CT) cervical spine data in 0.625 mm slices from our radiology department. The patients were adults. CTs were from May to August, 2015. The lateral mass screw length was measured using Margerl's technique and pedicle width and pedicle screw trajectory were determined in three-dimensional reformated images. Results CT scans of cervical spines of 94 patients were obtained and 188 lateral masses and pedicles of C7 vertebrae were measured. The mean lateral mass screw length was 13.2 mm (standard deviation [SD] 1.6 mm), mean outer pedicle width was 5.9 mm (SD 1.0 mm) and mean pedicle screw trajectory was 29.4 degrees (SD 3.6 degrees). Most (91.0%) of the pedicles had an outer diameter ≥4.5 mm. Conclusions The mean lateral mass screw length was longer when compared with other similar studies, while the mean outer pedicle width was narrower. Nearly 10% of the pedicles were unable to accommodate 3.5 mm screws. These findings favor the use of lateral mass screws to provide a safe and stable fixation for C7 vertebrae in Southern Chinese patients, while the final choice of fixation method should only be confirmed after careful preoperative planning with CT scan.

The Leray–Hopf solutions to the Navier–Stokes equation are known to be unique on R 2 . In our previous work, we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.

A gasket-type standard leak element with a microfluidic channel was developed and measured. The microfluidic channel with a diameter of several micrometers and a length of one hundred micrometers was fabricated by using femtosecond laser micromachining. The conductances (

This article is dedicated to the proof of \(C^\alpha \) regularization effects of Hamilton–Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent on the regularity of the Hamilton.

Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier-Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth., 10 pages, submitted

This article is dedicated to the regularity theory for solutions to a class of nonlinear integral variational problems. Those problems are involved in nonlocal image and signal processing.

We consider the fractional Burgers' equation on $\R^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur \cite{Driftdiffusion} and show existence of smooth solutions given any initial datum in $L^2(\R^N)$., Comment: 31 pages; corrected one of the references, and fixed some typos

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Holder continuous for large time.

This article is devoted to a Log improvement of Prodi-Serrin criterion for global regularity to solutions to Navier-Stokes equations in dimension 3. It is shown that the global regularity holds under the condition that |u|/(log(1+|u|)) is integrable in space time variables. keywords: Navier-Stokes, regularity criterion, a priori estimates MSC: 35B65, 76D03, 76D05

Purpose. To compare the tibial tuberosity–trochlear groove (TTTG) distance in Chinese patients with or without recurrent patellar dislocation. Methods. The TTTG distance of 14 knees in 5 men and 9 women aged 16 to 38 years with recurrent patellar dislocation were compared with that of 73 control knees using magnetic resonance imaging. Results. The mean TTTG distance was significantly greater in patients with recurrent patellar dislocation than in controls (16.4 mm vs. 10.1 mm, pConclusion. The TTTG distance is related to height and is shorter in Chinese than Caucasian patients; the threshold for tibial tuberosity transfer should be lower in Chinese patients.

Due to its large population, rapid economic growth and the fragility of its environment, China is environmentally more risky than Western countries. Over time we will see tougher and better-implemented environmental regulations in China. At the same time, sound environmental management is being viewed by investors as an essential part of good corporate governance. Hong Kong listed companies have failed to provide sufficient information for investors to assess their positions with respect to environmental management. This increases their vulnerability to possible accusations of poor environmental performance. The lack of environmental activity and accountability identified amongst the companies leads us to conclude that they are taking substantial yet avoidable risks. In the short term, Hong Kong security market regulators should expedite better voluntary environmental reporting. In the long term, a fundamental reform of security regulations in partnership between regulators, professionals and listed companies should be pursued. Copyright © 2005 John Wiley & Sons, Ltd and ERP Environment.

We consider the Navier-Stokes equation on $\mathbb{H}^{2}(-a^{2})$, the two dimensional hyperbolic space with constant sectional curvature $-a^{2}$. We prove an ill-posedness result in the sense that the uniqueness of the Leray-Hopf weak solutions to the Navier-Stokes equation breaks down on $\mathbb{H}^{2}(-a^{2})$. We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions., Comment: 30 pages