Consider nonautonomous evolution equation $\dot{u} = A(t) u(t) + Bg(u)(t)$ in which the family of operators $(A(t))_{t \geq 0}$ generates the evolution family $(\mathcal{U}(t,s))_{t \geq s \geq 0}$ of $(X,Y,\varphi)$-type, i.e., $\|\mathcal{U}(t,0)x\|_{Y} \leq \varphi(t) \|x\|_{X}$, $t > 0$, for certain couple of Banach spaces $(X,Y)$ and real-valued, positive function $\varphi$ satisfying $\lim_{t \to \infty} \varphi(t) = 0$. Inspired by Serrin's technique, we develop a unified approach toward the problems on the existence of periodic solutions to above equation. As illustrations of our abstract results, we give applications to the existence and uniqueness of periodic solutions to Oseen–Navier–Stokes and damped wave equations, as well as the existence of local stable manifolds nearby the periodic solution to the damped wave equations.

The aim of this paper is to give existence and uniqueness results for solutions to the Cauchy problem of semilinear heat equations on stratified Lie groups $\mathbb{G}$ with the small initial data belonging to the critical Sobolev space. We consider a power type nonlinearity that behaves like $|u|^{\alpha}$ or $|u|^{\alpha-1}u$ ($\alpha > 1$).

We provide a certain direct-sum decomposition of reflexive modules over (one-dimensional) Arf local rings. We also see the equivalence of three notions, say, integrally closed ideals, trace ideals, and reflexive modules of rank one (i.e., divisorial ideals) up to isomorphisms in Arf rings. As an application, we obtain the finiteness of indecomposable reflexive modules, up to isomorphism, for analytically irreducible Arf local rings.

Let $\varphi \colon \mathbb{R}^{n} \times [0,\infty) \to [0,\infty)$ be an anisotropic growth function and $A$ a general expansive matrix on $\mathbb{R}^{n}$. Let $H_{A}^{\varphi}(\mathbb{R}^{n})$ be the anisotropic Musielak–Orlicz Hardy space associated with $A$. In this paper, a general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H_{A}^{\varphi}(\mathbb{R}^{n})$. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $\theta$-means, from $H_{A}^{\varphi}(\mathbb{R}^{n})$ to the Musielak–Orlicz space $L^{\varphi}(\mathbb{R}^{n})$. As applications, some norm and almost everywhere convergence results of the $\theta$-means, which generalize the well-known Lebesgue's theorem, are presented. Finally, the corresponding conclusions of two well-known specific summability methods, that is, Bochner–Riesz and Weierstrass means, are also obtained.

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

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.

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.

The existence and asymptotic behavior of solutions a fourth-order partial differential equation with a $p$-Laplacian diffusion and a nonlinear source are studied by using potential well theory. When the initial functionals satisfy $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \gt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \geq 0$, the existence and exponential decay result of weak solutions are given. For $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \lt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \lt 0$, we obtain the blow-up behavior at a finite time for weak solutions. For $\mathcal{F}(w_{0}) \gt d$, we show the global existence for small initial datum and blow-up for big initial datum. Moreover, the uniqueness holds for bounded solutions. In addition, we show that the $p$-Laplacian term has an essential effect to the source function so that we add some growth conditions to $g(w)$.

In 1991, Chen proposed a conjecture which is the relationship between biharmonic submanifolds and harmonic submanifolds in Euclidean space and quite a few related studies have supported it. Around the same time, it was proved that Chen's conjecture does not extend to submanifolds in Minkowski space. In this paper, as part of these researches, we investigate biharmonic marginally trapped ruled surfaces in Minkowski $m$-space and then construct some examples about them in which Chen's conjecture does not hold.

This paper is devoted to studying a class of fractional $(p,q)$-Laplacian problems with subcritical and critical Hardy potentials: \[ \begin{cases} (-\Delta)^{s_{1}}_{p} u + \nu (-\Delta)^{s_{2}}_{q} u = \lambda \frac{|u|^{r-2} u}{|x|^{a}} + \frac{|u|^{p^{*}_{s_{1}}(b)-2} u}{|x|^{b}} &\textrm{in $\Omega$}, \\ u = 0 &\textrm{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, and $p_{s_{1}}^{*}(b) = \frac{(N-b)p}{N-ps_{1}}$ denotes the fractional critical Hardy–Sobolev exponent. More precisely, when $\nu = 1$ and $\nu \gt 0$ is sufficiently small, using some asymptotic estimates and the Mountain Pass Theorem, we establish the existence results for the above fractional elliptic equation under some suitable hypotheses, respectively, which are gained over a wider range of parameters.

The techniques of high dimensionality reduction are important tools in machine learning and data science. The method of singular value decomposition (SVD) is a popular method in dimensionality reduction and image compression. However, it suffers from heavily computational overhead in practice, especially for images with high-resolution. In order to achieve the efficiency and the accuracy, we propose a refinement of approximate invariant subspaces of matrices (REIS) algorithm based on SVD. The theoretical contribution of our paper is threefold. Firstly, we describe the properties of the SVD of the matrices and discuss how to apply SVD to do image compression. Secondly, we introduce the method of REIS based on SVD for image compression in the high-resolution images. The core of REIS is adapted to large and real matrices in $\mathbb{R}^{n \times n}$, through some nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton's method. Thirdly, some measurement tools are provided such as compression ratio, mean square error, peak signal to noise ratio and structural similarity index to compare the performance of the compression factors and the quality of the compressed images. Numerical examples for testing some real world image sets are presented to illustrate the feasibility of our proposed algorithm.

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.

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.

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.

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.

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.