Entropy solutions to the Dirichlet problem for nonlinear diffusion equations with conservative noise

Motivated by porous medium equations with randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness and $L_{1}$-stability of non-negative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov's doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations.Comment: 33 page

Quantum multipartite maskers vs quantum error-correcting codes

Since masking of quantum information was introduced by Modi et al. in [PRL 120, 230501 (2018)], many discussions on this topic have been published. In this paper, we consider relationship between quantum multipartite maskers (QMMs) and quantum error-correcting codes (QECCs). We say that a subset $Q$ of pure states of a system $K$ can be masked by an operator $S$ into a multipartite system \H^{(n)} if all of the image states $S|\psi\>$ of states $|\psi\>$ in $Q$ have the same marginal states on each subsystem. We call such an $S$ a QMM of $Q$. By establishing an expression of a QMM, we obtain a relationship between QMMs and QECCs, which reads that an isometry is a QMM of all pure states of a system if and only if its range is a QECC of any one-erasure channel. As an application, we prove that there is no an isometric universal masker from \C^2 into \C^2\otimes\C^2\otimes\C^2 and then the states of \C^3 can not be masked isometrically into \C^2\otimes\C^2\otimes\C^2. This gives a consummation to a main result and leads to a negative answer to an open question in [PRA 98, 062306 (2018)]. Another application is that arbitrary quantum states of \C^d can be completely hidden in correlations between any two subsystems of the tripartite system \C^{d+1}\otimes\C^{d+1}\otimes\C^{d+1}, while arbitrary quantum states cannot be completely hidden in the correlations between subsystems of a bipartite system [PRL 98, 080502 (2007)].Comment: This is a revision about arXiv:2004.14540. In the present version, $k$ and $j$ old Eq. (2.2) have been exchanged and the followed three equations have been correcte

Extremely large magnetoresistance in topologically trivial semimetal α\alpha-WP2_2

Extremely large magnetoresistance (XMR) was recently discovered in many non-magnetic materials, while its underlying mechanism remains poorly understood due to the complex electronic structure of these materials. Here, we report an investigation of the $\alpha$-phase WP$_2$, a topologically trivial semimetal with monoclinic crystal structure (C2/m), which contrasts to the recently discovered robust type-II Weyl semimetal phase in $\beta$-WP$_2$. We found that $\alpha$-WP$_2$ exhibits almost all the characteristics of XMR materials: the near-quadratic field dependence of MR, a field-induced up-turn in resistivity following by a plateau at low temperature, which can be understood by the compensation effect, and high mobility of carriers confirmed by our Hall effect measurements. It was also found that the normalized MRs under different magnetic fields has the same temperature dependence in $\alpha$-WP$_2$, the Kohler scaling law can describe the MR data in a wide temperature range, and there is no obvious change in the anisotropic parameter $\gamma$ value with temperature. The resistance polar diagram has a peanut shape when field is rotated in $\textit{ac}$ plane, which can be understood by the anisotropy of Fermi surface. These results indicate that both field-induced-gap and temperature-induced Lifshitz transition are not the origin of up-turn in resistivity in the $\alpha$-WP$_2$ semimetal. Our findings establish $\alpha$-WP$_2$ as a new reference material for exploring the XMR phenomena.Comment: 18 pages, 12 figure

EVD Surgical Guidance with Retro-Reflective Tool Tracking and Spatial Reconstruction using Head-Mounted Augmented Reality Device

Augmented Reality (AR) has been used to facilitate surgical guidance during External Ventricular Drain (EVD) surgery, reducing the risks of misplacement in manual operations. During this procedure, the pivotal challenge is the accurate estimation of spatial relationship between pre-operative images and actual patient anatomy in AR environment. In this research, we propose a novel framework utilizing Time of Flight (ToF) depth sensors integrated in commercially available AR Head Mounted Devices (HMD) for precise EVD surgical guidance. As previous studies have proven depth errors for ToF sensors, we first conducted a comprehensive assessment for the properties of this error on AR-HMDs. Subsequently, a depth error model and patient-specific model parameter identification method, is introduced for accurate surface information. After that, a tracking procedure combining retro-reflective markers and point clouds is proposed for accurate head tracking, where head surface is reconstructed using ToF sensor data for spatial registration, avoiding fixing tracking targets rigidly on the patient's cranium. Firstly, $7.580\pm 1.488 mm$ ToF sensor depth value error was revealed on human skin, indicating the significance of depth correction. Our results showed that the ToF sensor depth error was reduced by over $85\%$ using proposed depth correction method on head phantoms in different materials. Meanwhile, the head surface reconstructed with corrected depth data achieved sub-millimeter accuracy. Experiment on a sheep head revealed $0.79 mm$ reconstruction error. Furthermore, a user study was conducted for the performance of proposed framework in simulated EVD surgery, where 5 surgeons performed 9 k-wire injections on a head phantom with virtual guidance. Results of this study revealed $2.09 \pm 0.16 mm$ translational accuracy and $2.97\pm 0.91 ^\circ$ orientational accuracy