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We apply two families of novel fractional $\theta$-methods, the FBT-$\theta$ and FBN-$\theta$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $\theta$-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $O(\tau^2+h^{r+1})$ for smooth solutions, where $\tau$ is the time mesh size and $h$ is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time., Comment: 24 pages, 9 figures

Here we investigate the two-parameter high-frequency localization for the eigenfunctions of a Schr\"{o}dinger operator with a singular inverse square potential in high-dimensional balls and spherical shells as the azimuthal quantum number $l$ and the principal quantum number $k$ tend to infinity simultaneously, while keeping their ratio as a constant, generalizing the classical one-parameter localization for Laplacian eigenfunctions [SIAM J. Appl. Math. 73:780-803, 2013]. We prove that the eigenfunctions in balls are localized around an intermediate sphere whose radius is increasing with respect to the $l$-$k$ ratio. The eigenfunctions decay exponentially inside the localized sphere and decay polynomially outside. Furthermore, we discover a novel second-order phase transition for the eigenfunctions in spherical shells as the $l$-$k$ ratio crosses a critical value. In the supercritical case, the eigenfunctions are localized around a sphere between the inner and outer boundaries of the spherical shell. In the critical case, the eigenfunctions are localized around the inner boundary. In the subcritical case, no localization could be observed, giving rise to localization breaking., Comment: 24 pages, 5 figures

The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator $I^{\alpha}$ at node $x_{n}$. In this paper, we develop the shifted convolution quadrature ($SCQ$) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter $\theta$ to cover as many numerical schemes that approximate the operator $I^{\alpha}$ with an integer convergence rate as possible. The constraint on the parameter $\theta$ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For some technique purposes when analysing the stability or convergence estimates of a method applied to PDEs, we design some novel formulas with desired properties under the framework of the $SCQ$. Finally, we conduct some numerical tests with nonsmooth solutions to further confirm our theory., Comment: 21 pages, 15 figures

In this article, we introduce two families of novel fractional $\theta$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^{\alpha}$ with a second order convergence rate. A new fractional BT-$\theta$ method connects the fractional BDF2 (when $\theta=0$) with fractional trapezoidal rule (when $\theta=1/2$), and another novel fractional BN-$\theta$ method joins the fractional BDF2 (when $\theta=0$) with the second order fractional Newton-Gregory formula (when $\theta=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $\theta$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $\theta$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis., Comment: 16 pages, 10 figures

We propose a novel stimulated emission depletion (STED) microscopy based on array detection and photon reassignment. By replacing the single-point detector in traditional STED with a detector array and utilizing the photon reassignment method to recombine the images acquired by each detector, the final photon reassignment STED (prSTED) image could be obtained. We analyze the principle and imaging characteristics of prSTED, and the results indicate that, compared with traditional STED, prSTED can improve the signal-to-noise ratio (SNR) of the image by increasing the obtained photon flux while maintaining the original spatial resolution of STED. In addition, the SNR and resolution of prSTED are strongly correlated with the intensity of depletion beam. Corresponding theoretical and experimental analysis about this feature are also conducted. In general, considering the enhanced signal strength, imaging speed and compatibility with some other imaging techniques, we believe prSTED would be a helpful promotion in biomedical imaging., Comment: 5 pages, 4 figures

Motivated by Kyprianou and Zhou (2009), Wang and Hu (2012), Avram et al. (2017), Li et al. (2017) and Wang and Zhou (2018), we consider in this paper the problem of maximizing the expected accumulated discounted tax payments of an insurance company, whose reserve process (before taxes are deducted) evolves as a spectrally negative L\'{e}vy process with the usual exclusion of negative subordinator or deterministic drift. Tax payments are collected according to the very general loss-carry-forward tax system introduced in Kyprianou and Zhou (2009). To achieve a balance between taxation optimization and solvency, we consider an interesting modified objective function by considering the expected accumulated discounted tax payments of the company until the general draw-down time, instead of until the classical ruin time. The optimal tax return function together with the optimal tax strategy is derived, and some numerical examples are also provided.

At present, deep learning has been applied more and more in monocular image depth estimation and has shown promising results. The current more ideal method for monocular depth estimation is the supervised learning based on ground truth depth, but this method requires an abundance of expensive ground truth depth as the supervised labels. Therefore, researchers began to work on unsupervised depth estimation methods. Although the accuracy of unsupervised depth estimation method is still lower than that of supervised method, it is a promising research direction. In this paper, Based on the experimental results that the stereo matching models outperforms monocular depth estimation models under the same unsupervised depth estimation model, we proposed an unsupervised monocular vision stereo matching method. In order to achieve the monocular stereo matching, we constructed two unsupervised deep convolution network models, one was to reconstruct the right view from the left view, and the other was to estimate the depth map using the reconstructed right view and the original left view. The two network models are piped together during the test phase. The output results of this method outperforms the current mainstream unsupervised depth estimation method in the challenging KITTI dataset.

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Gr\"{o}nwall inequality and apply it to the well-known L1 formula and a fractional Crank-Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction-subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis., Comment: 22 pages, 8 tables

In this paper, we first construct the $H^2$(curl)-conforming finite elements both on a rectangle and a triangle. They possess some fascinating properties which have been proven by a rigorous theoretical analysis. Then we apply the elements to construct a finite element space for discretizing quad-curl problems. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in the $H^2$(curl) norm are established. Numerical experiments are provided to confirm our theoretical results.

In this paper, we develop a straightforward $C^0$ linear finite element method for sixth-order elliptic equations. The basic idea is to use gradient recovery techniques to generate higher-order numerical derivatives from a $C^0$ linear finite element function. Both theoretical analysis and numerical experiments show that the proposed method has the optimal convergence rate under the energy norm. The method avoids complicated construction of conforming $C^2$ finite element basis or nonconforming penalty terms and has a low computational cost.

In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, better than the global rate for the one-point and two-point interpolations. Here $N$ represents the degree of interpolation polynomial. It is proved that the $\alpha$-th fractional derivative of $(u-u_N)$ with $k<\alpha

In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalisation of orthogonal {\em ball polynomials} in primitive variables with a tuning parameter $c>0$, through a "perturbation" of the Sturm-Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions., Comment: 25 pages