In this paper, we consider the geometric mean of min matrices. We establish a closed form for the geometric mean of two positive definite min matrices. We further show that the set of min matrices is geodesically convex in the Cartan–Hadamard manifold of positive definite matrices. We also present monotonic one-parameter families of min matrices.

The single weighted differentiation-composition operator $W_{u,\varphi}^{(k)}$ on various spaces of analytic functions has been investigated for decades, i.e., $W_{u,\varphi}^{(k)}f = u \cdot f^{(k)} \circ \varphi$. However, the study of the finite sum of weighted differentiation-composition operators with different orders are far from complete, i.e., \[ T_{\mathbf{u},\varphi}^{(n)}f = u_{0} \cdot f \circ \varphi + u_{1} \cdot f' \circ \varphi + \cdots + u_{n} \cdot f^{(n)} \circ \varphi. \] In this paper, we completely solve the problem about boundedness and compactness of the operator $T_{\mathbf{u},\varphi}^{(n)}$ between two Bergman spaces over the upper half-plane. We show a rigidity property of $T_{\mathbf{u},\varphi}^{(n)}$. Specifically, the boundedness and compactness of the sum $T_{\mathbf{u},\varphi}^{(n)}$ is equivalent to those of each $W_{u_{k},\varphi}^{(k)}$, $0 \leq k \leq n$.

In this paper, we concern with the existence of solutions of the complex $m$-Hessian type equation $-\chi(u) H_{m}(u) = \mu$ in the class $\mathcal{E}_{m,\chi}(f,\Omega)$ if there exists subsolution in this class, where the given boundary value $f \in \mathcal{N}_{m}(\Omega) \cap MSH_{m}(\Omega)$.

In this paper, we consider a representation of the competing Finsler double phase equation, i.e., a representation of the competing Finsler double phase equation in the Heisenberg groups $\mathbb{H}^{n}$. We prove the existence of a generalized variational solution for the nonlinear Dirichlet problem driven by the Finsler–Laplacian competing operator in the Heisenberg group. This solution differs from the concept of the weak solution. To achieve this, we employ a Galerkin type procedure as part of our methodology.

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.

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 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 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.