In this article, we introduce a class of multilinear fractional integral operators with generalized kernels that are weaker than the Dini kernel condition. We establish the boundedness of multilinear fractional integral operators with generalized kernels on weighted Lebesgue spaces and variable exponent Lebesgue spaces, as well as the boundedness of multilinear commutators generated by multilinear fractional integral operators with generalized kernels and $\operatorname{BMO}$ functions. Even when the generalized kernels condition goes back to the Dini kernel condition, the conclusions on the commutators remain new.

In this first part we describe the group $\operatorname{Aut}_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S) = 1$), in the initial case $\chi(\mathcal{O}_{S}) = 0$.

In particular, in the case where $\operatorname{Aut}_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $\operatorname{Aut}_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2$, $\mathbb{Z}/3$, $(\mathbb{Z}/2)^{2}$. We also show with easy examples that the groups $\mathbb{Z}/2$, $\mathbb{Z}/3$ do effectively occur.

Respectively, in the case where $\operatorname{Aut}_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.

We determine the automorphism groups of the orbifold vertex operator algebras associated with the coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $3C$, $5C$, $11A$, and $23A$. These orbifold vertex operator algebras appear in a classification given by Lam and Shimakura.

The main purpose of the present paper is to provide a partial classification, performed with respect the weak-combinatorics, of free arrangements consisting of lines and one smooth conic with quasi-homogeneous ordinary singularities.

In this paper, we consider the $L^{\infty}$ bounds of the eigenfunctions for some diffusion operators. In case of classical Laplace eigenvalue problem $\Delta u = -\lambda u$ with Dirichlet boundary condition, we obtain the polynomial bound which is different from the result derived by heat kernel estimates. We then study the nonlocal dispersal eigenvalue problem $\int_{\Omega} J(x-y) u(y) \, dy - u(x) = -\lambda u(x)$ with bounded domain $\Omega$ and obtain an exponential bound by means of Fourier transform and nonlocal estimates.

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.

In this paper, a modified quasi-boundary value regularization method is proposed to solve an inverse source problem for time fractional diffusion equation. First, the regularization problem is proposed and the existence and the uniqueness of the regularized solution are proven. Then based on some source condition for the source term and the selection strategies for the regularization parameter, the a prior convergence rate and the posteriori convergence rate of regularization solution are derived by using the eigenfunction expansion method. Next, an all-in-one matrix form with block-Toeplitz structure is obtained by using the finite difference discretization for the regularized problem, and an efficient preconditioning technique is adopted to approximately solve the regularization solution. Finally, several numerical examples are presented to evaluate the effectiveness of the inversion method.

In this paper, we will study the eigenvalue ratios $\frac{\lambda_{n}}{\lambda_{m}}$ and the eigenvalue gaps $\lambda_{n} - \lambda_{m}$ of the Sturm–Liouville problem \[ -(p(x) y')' + q(x)y = \lambda w(x)y. \] We prove that if $q \leq 0$ and $\theta'' \geq 0$, then $\frac{\lambda_{n}}{\lambda_{m}} \leq \big( \frac{n}{m} \big)^{2}$ and if $q \geq 0$ and $\theta'' \leq 0$, then $\frac{\lambda_{n}}{\lambda_{m}} \geq \big( \big\lfloor \frac{n}{m} \big\rfloor \big)^{2}$. In the case where $q \equiv 0$ and $p = 1$, we prove that if $4ww'' \geq 5(w')^{2}$ then $\lambda_{n} - \lambda_{m} \leq \big[ \big( \frac{n}{m} \big)^{2} - 1 \big] \frac{(m\pi)^{2}}{w_{m}}$ and if $4ww'' \leq 5(w')^{2}$, then $\lambda_{n} - \lambda_{m} \geq \big[ \big( \big\lfloor \frac{n}{m} \big\rfloor \big)^{2} - 1 \big] \frac{(m\pi)^{2}}{w_{M}}$, where $\theta(x) = (p(x) w(x))^{1/4}$, $w_{m} = \inf_{x \in [0,1]} w(x)$ and $w_{M} = \max_{w \in [0,1]} w(x)$.

We study the existence and nonexistence of normalized solutions for the following fractional Choquard equations with Hardy–Littlewood–Sobolev lower critical exponent and nonlocal perturbation: \[ \begin{cases} (-\Delta)^{s} u + \lambda u = \gamma (I_{\alpha} \ast |u|^{\frac{\alpha}{N}+1}) |u|^{\frac{\alpha}{N}-1} u + \mu (I_{\alpha} \ast |u|^{q}) |u|^{q-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, \mathrm{d}x = c^{2}, \end{cases} \] where $N \geq 3$, $s \in (0,1)$, $\alpha \in (0,N)$, $\gamma,\mu,c \gt 0$ and $2_{\alpha} := \frac{N+\alpha}{N} \lt q \leq 2_{\alpha,s}^{\ast} := \frac{N+\alpha}{N-2s}$. $I_{\alpha}$ is the Riesz potential and $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier. By precisely restricting parameters $\gamma$, $\mu$ and $c$, using constrained variational method and introducing new relevant arguments, we establish several existence and nonexistence results. In particular, we consider the case $q = 2_{\alpha,s}^{\ast}$ which corresponds to equations involving double critical exponents, and the Hardy–Littlewood–Sobolev subcritical approximation method is used to solve the case.

The alternating direction implicit (ADI) scheme is used to numerically solve the 2D subdiffusion equation with initial singularity. The time derivative is defined by the commonly used Caputo fractional derivative, and discretised by the L1 scheme on nonuniform mesh. The finite difference method (FDM) is applied to spatial discretization. The local error analyses of fully discrete scheme under the $L^{2}$-norm and $H^{1}$-norm are strictly established. By selecting the milder grading parameter $r \gt 2-\alpha$, the time convergence rate can reach $O(M^{-\min \{ 2-\alpha, 2\alpha \}})$ in positive time. In order to verify the correctness of the theoretical analysis, some numerical results are presented.

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.

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.

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.

In the mathematical literature, there are various approaches to the concept of chaos, also in the local aspects. Points of chaos (connected with homoclinic points of Poincare), points of distributional chaos and points focusing entropy are known. It can be shown in [14], by using topological tools that these concepts are significantly different. In this paper we join them, considering new kind of points (strongly focusing chaos) combinig them with problem of disruptions of semigroups of functions. The paper ends with open problems.

We survey on algebraically elliptic varieties in the sense of Gromov.

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

In the present paper, we study a singular $p(x)$-Kirchhoff equation with combined effects of variable singular, $\beta(x)$, and sublinear, $q(x)$, nonlinearities. Using the Ekeland's variational principle and a constrained minimization, we show the existence of a positive solution for the case variable singularity $\beta(x)$ assumes its values in the interval $(1,\infty)$. We provide an example to illustrate our results.

We discuss many interesting properties of the initial-boundary value problem for the heat equation $u_{t}(x,t) = u_{xx}(x,t)$ on $(x,t) \in (0,\infty) \times (0,\infty)$. In particular, we can prescribe the space, time, and space-time oscillation limits (limsup and liminf) of $u(x,t)$ by choosing suitable initial data $h(x)$ and boundary dada $g(t)$. We also consider singular initial-boundary value problem and oblique initial-boundary value problem for the heat equation.

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.

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.

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.

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 $48$.