Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$, respectively. For any real $\alpha \in [0,1]$, Nikiforov [12] defined the matrix $A_{\alpha}(G)$ as \[ A_{\alpha}(G) = \alpha D(G) + (1-\alpha) A(G). \] An $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that $a \leq d_{H}(v) \leq b$ for any $v \in V(G)$, where $a$ and $b$ are positive integers. In this paper, we give an upper bound of $A_{\alpha}$-spectral radius of graphs with unique perfect matching, and then present $A_{\alpha}$-spectral conditions for the existence of an $[a,b]$-factor in a graph. Our results extend the result of Fan et al. in [4] for the unique perfect matching and $[a,b]$-factor of graphs, and that of Zhao et al. in [16] for a $[1,b]$-odd factor of graphs.

Let $F(G)$, $F_{t}(G)$, $\beta(G)$, and $\beta'(G)$ be the zero forcing number, the total forcing number, the vertex covering number and the edge covering number of a graph $G$, respectively. In this paper, we first completely characterize all trees $T$ with $F(T) = (\Delta-2) \beta(T) + 1$, solving a problem proposed by Brimkov et al. in 2023. Next, we study the relationship between the zero (or total) forcing number of a tree and its edge covering number, and show that $F(T) \leq \beta'(T)-1$ and $F_{t}(T) \leq \beta'(T)$ for any tree $T$ of order $n \geq 3$. Moreover, we also characterize all trees $T$ with $F(T) = \beta'(T)-1$ and $F(T) = \beta'(T)-2$, respectively.

A multinorm one torus associated to a commutative étale algebra $L$ over a global field $k$ is of Kummer type if each factor of $L$ is a cyclic Kummer extension. In this paper we compute the Tate–Shafarevich group of such tori based on general formulas of Lee [4]. Our aim is to illustrate various invariants in Lee's formulas by this class of tori, especially from the computational aspect. We also implement an effective algorithm using SageMath which computes the Tate–Shafarevich groups when each factor of $L$ is contained in a fixed concrete bicyclic extension of a cyclotomic field $k$.

This paper studies solutions and relevant measures of a class of Sylvester-like linear matrix equations commonly encountered in control theory. Firstly, inequalities related to the singular values of solutions of a class of Sylvester-like linear matrix equations are obtained. These results improve upon existing relevant studies. Next, starting from the definition of singular values for any matrix, a lower bound for the product of solutions and their complex conjugate transpose matrices is directly obtained. Additionally, when a Hermite matrix is a solution to the matrix equation, a convergent matrix series is obtained, as the positive definite solution under certain conditions. Finally, we design two algorithms for solving the class of matrix equations, where each recursive iteration results in obtaining the upper and lower solution bounds. Numerical experiments demonstrate that our results outperform some existing studies.

In this paper, we deal with two Trudinger–Moser inequalities involving various $L^{p}$-norms on a smooth bounded domain of $\mathbb{R}^{n}$, $n \geq 3$. For any $p \gt 1$, we set \[ \lambda_{p}(\Omega) = \inf_{u \in H_{0}^{1,n}(\Omega), \; u \not\equiv 0} \frac{\|\nabla u\|_{n}^{n}}{\|u\|_{p}^{n}} \] as an eigenvalue related to the $n$-Laplacian. Based on the method of blow-up analysis, if $p_{j} \gt 1$ for all $1 \leq j \leq l$, and satisfies \[ \max_{1 \leq j \leq l} \frac{\alpha_{j}}{\lambda_{p_{j}}(\Omega)} \lt 1, \quad \sum_{j=1}^{l} \frac{\alpha_{j}}{\lambda_{p_{j}}(\Omega)} \lt 1, \] then we prove that \[ \sup_{u \in H_{0}^{1,n}(\Omega), \; \|\nabla u\|_{n} \leq 1} \int_{\Omega} e^{\alpha_{n} |u|^{\frac{n}{n-1}} \big( 1 + \sum_{j=1}^{l} \alpha_{j} \|u\|_{p_{j}}^{n} \big)^{\frac{1}{n-1}}} \, dx \] is attained, where $\alpha_{n} = n \omega_{n-1}^{1/(n-1)}$, $\omega_{n-1}$ is the surface area of the unit ball in $\mathbb{R}^{n}$. Under the same assumptions as above, we conclude that \[ \sup_{u \in H_{0}^{1,n}(\Omega), \; \|\nabla u\|_{n}^{n} - \sum_{j=1}^{l} \alpha_{j} \|u\|_{p_{j}}^{n} \leq 1} \int_{\Omega} e^{\alpha_{n} |u|^{\frac{n}{n-1}}} \, dx \] is attained.

We are concerned with a class of $(p,q)$-Laplace type biharmonic Kirchhoff equations \[ \begin{cases} M\left( \int_{\Omega} \mathcal{A}(|\Delta u|^{p}) \, dx \right) \Delta(a(|\Delta u|^{p}) |\Delta u|^{p-2} \Delta u) = \lambda f(u) + |u|^{q_{2}^{*}-2} u &\textrm{in $\Omega$}, \\ u = \Delta u = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded open set in $\mathbb{R}^{N}$ with smooth boundary, $\lambda$ is a positive real parameter, $2 \leq p \lt q \lt q_{2}^{*}$, $q_{2}^{*} = \frac{Nq}{N-2q}$ is the critical exponent, $N \gt 2q$ and $\mathcal{A}(t) = \int_{0}^{t} a(s) \, ds$ for $t \in \mathbb{R}^{+}$. Here, $M \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a Kirchhoff function, $a \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a continuous function satisfying some properties and $f \colon \mathbb{R} \to \mathbb{R}$ is a function which can have an uncountable set of discontinuity points. In this article, we study the existence of a positive weak solution for the problem above involving critical growth and a discontinuous nonlinearity via mountain pass theorem.

This paper deals with the asymptotic behavior for a porous medium equation with weighted inner source terms and a nonlinear nonlocal boundary condition. We find the influences of spacetime-varying weight functions and competitive relationship between the multiple nonlinearities on whether determining blow-up of solutions or not, and establish the blow-up rate estimate under some appropriate conditions. Especially, our results include the situation of slow, linear and fast diffusion.

We study nonnegative holomorphic sectional curvature on a compact almost Hermitian manifold. In the positive case, we show some geometric conditions for negative Kodaira dimension. In the zero case, we give some conditions of the Chern–Yamabe problem for zero Chern scalar curvature.

The purpose of this article is to investigate the boundary behaviour of the squeezing function of a general ellipsoid.