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.

Kohnen proved a non-vanishing result for $L$-functions associated to Hecke eigenforms of integral weights on the full group. In this paper, we show a non-vanishing result for the averages of $L$-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms of weight $k \in \frac{1}{2} \mathbb{Z}$ on the full group. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize the result of Kohnen to $\Gamma_{0}(N)$ and prove the analogous result for Jacobi forms.

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.

In this paper, we explore the multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearities \[ \begin{cases} -\Delta u + V(\epsilon x)u = \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = c, \end{cases} \] where $\mu \gt 0$, $c \gt 0$, $2 \lt q \lt 2+4/N \lt p \lt 2N/(N-2)$, $N \geq 3$, $\epsilon \gt 0$ is a parameter and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. By employing the minimization techniques and the truncated argument, we obtain that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\epsilon$ is sufficiently small.

We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $\lambda_{1}$ and $\lambda_{2}$ on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-$(p,q)$ conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions which is weaker than the well-known Ambrosetti–Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters $\lambda_{1}$ and $\lambda_{2}$.

The aim of this paper is to study the stability of perturbed set optimization problems via general ordering sets. Firstly, for a set optimization problem (SOP) via general ordering sets, four kinds of concepts about the minimal solutions of (SOP) are given. Then, some properties of the four kinds of solution sets and the level set of objective mappings are investigated. Finally, by employing the recession cone technique, sufficient conditions of upper Painlevé–Kuratowski convergence of minimal approximate solution sets, Painlevé–Kuratowski convergence of weak minimal approximate solution sets of (SOP) are obtained, where the feasible set is perturbed. Some examples are given to illustrate the mainly results in the paper.

The Lusztig correspondence is a bijective mapping between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^{*})^{0}$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^{*}$. In this article we discuss the unicity and ambiguity of such a bijective correspondence. In particular, we show that the Lusztig correspondence for a classical group can be made to be unique if we require it to be compatible with the parabolic induction and the finite theta correspondence.

This paper is to derive the precise asymptotic spreading behavior for an epidemic model with nonlocal dispersal. The proof is based on a Liouville type theorem on the positive bounded entire solutions. This Liouville theorem holds for a general class of reaction-diffusion systems with nonlocal dispersal which can be useful for reaction-diffusion systems arising in ecology and epidemiology.

This paper deals with a pseudo-parabolic equation involving variable exponents under homogeneous Dirichlet boundary value condition. The authors first develop the potential well method to prove a threshold result on the existence or nonexistence of global solutions to the equation when the initial energy is less than the mountain pass level $d$. In the case of high energy initial data, a new characterization for the nonexistence of global solution is also given. These results extend and improve some recent results obtained by Di et al. [9].

Let $(A,\mathbb{N}^{2},\alpha)$ be a dynamical system consisting of a $C^{*}$-algebra $A$ and an action $\alpha$ of $\mathbb{N}^{2}$ on $A$ by automorphisms. Let $A \times_{\alpha}^{\mathrm{piso}} \mathbb{N}^{2}$ be the partial-isometric crossed product of the system. We apply the fact that it is a full corner of a crossed product by the group $\mathbb{Z}^{2}$ in order to give a complete description of its primitive ideal space.