In the present paper, we study the geometry of plane quartics with large automorphism groups. We show results devoted to smooth plane quartics that are invariant under the action of the elementary Abelian group of type $[2,2,2]$. Then we study geometric properties of the smooth plane quartic having automorphism group of order $48$.

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. In this paper we characterize modules whose first nonzero Fitting ideal has a maximal radical or is a prime ideal. Moreover, a number of examples concerning this characterization are given.

Color image deblurring problem, one of the primary goals in color image processing, can be mathematically represented by quaternion linear systems. In this paper, we propose a new quaternion generalized conjugate residual method for solving the general quaternion linear systems. The main advantages are that the proposed method saves three-quarters of the theoretical costs compared to the traditional GCR iterations for the real representation of quaternion linear systems, and also converges faster and smoother than the quaternion biconjugate gradient (QBiCG) method. Finally, we provide several numerical examples to illustrate the effectiveness of the proposed method compared with some existing methods.

In this paper, a compact finite difference scheme is constructed and studied for the fourth-order subdiffusion equation with the Riemann–Liouville fractional integral. The Caputo time-fractional derivative term and the Riemann–Liouville fractional integral term are discretized by L1-2 discrete formula and second order convolution quadrature rule, respectively. By using the discrete energy method, the Cholesky decomposition method and the reduced-order method, the stability and convergence are attained. And the convergence orders are reached second-order in time and fourth-order in space. Numerical examples verify the theoretical analysis.

This paper concerns the following planar Schrödinger–Poisson system \[ \begin{cases} -\Delta u + V(x) u + \phi u = |u|^{p-2} u &\textrm{in $\mathbb{R}^{2}$}, \\ \Delta \phi = u^{2}, \end{cases} \] where $p \geq 3$. By developing some new analytic techniques and variational methods, we establish a local compactness splitting lemma, and prove that this system possesses ground state solutions. We extend the case where $V(x)$ is a constant coefficient to the case where $V(x)$ is a variable coefficient. Some related results are improved.

Motivated by the recent work of Abraham and Nickl on the statistical Calderón problem [2], we revisit the increasing stability phenomenon in the inverse boundary value problem for the stationary wave equation with a potential using the Bayesian approach. In this paper, rather than the Dirichlet-to-Neumann map, we consider another type of boundary measurements called the impedance-to-Neumann map. Its graph forms a subset of Cauchy data. We show the consistency of the posterior mean with a contraction rate demonstrating the phenomenon of increasing stability.

This work is concerned with the three-dimensional generalized incompressible MHD equations with nonlinear damping terms (polynomial damping and exponential damping). We first establish the existence and uniqueness, and then we study the asymptotic behavior of weak solutions for these systems via attractors. The novelty is that the strength of nonlinearities and the degree of dissipations can work together to yield the global existence and uniqueness of the weak solutions of these systems. Since our systems might be not uniqueness, we could not use directly the classical scheme of the dynamical system to find attractors. Therefore, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and its properties.

In this paper, we give an approximation characterization, embedding properties and the duality of matrix weighted modulation spaces.

The spectral radius of nonnegative tensors and the minimum $H$-eigenvalues of strong $\mathcal{M}$-tensors are two types of tensor eigenvalues with important research significance, which promotes the tensor eigenvalue inequality to become an important component in tensor analysis. In this paper, based on Brualdi-type and Brauer-type eigenvalue inclusion sets of tensors, some Brualdi-type inequalities on the spectral radius for Hadamard product of two nonnegative tensors and some Brauer-type inequalities on the minimum $H$-eigenvalue for the Fan product of two strong $\mathcal{M}$-tensors are provided, respectively. In addition, the theoretical comparisons between the newly obtained inequalities and some previous ones are investigated. Finally, some numerical examples are reported to show the feasibility and effectiveness of our theoretical results.