In this paper, we prove that every $n$-vertex connected $K_{1,5}$-free graph $G$ with $\sigma_{4}(G) \geq n-1$ contains a spanning tree with at most $5$ leaves and branch vertices in total. Moreover, the degree sum condition “$\sigma_{4}(G) \geq n-1$” is best possible.

Let $s_{b}(n)$ denote the sum-of-digits function in base ($b \geq 2$) of the integer $n$. The aim of this work is to address a question proposed by Gelfond in 1967/1968. Specifically, we prove that sum-of-digits function of $k$-free integers of a monomial is equidistributed in arithmetic progressions.

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.

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.

Computational exact Riemann solvers are constructed for a spray model. The model is characterized by a system of balance laws with nonconservative terms in the equations for conservations of momentum and energy. The proposed scheme is combined from an upwind scheme for the equation of the volume fraction evolution and a Godunov-type scheme for the governing equations. The numerical results show that the scheme provides us with the convergence and it possesses a good accuracy, even when resonant phenomena occur. It is interesting to see that the proposed scheme is well-balanced, and preserves the positivity of density, of volume fraction, and the $C$-property.

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 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 $g(w)$.

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.

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.

The predator fertility function $\mathcal{M}(a)$ is thought to be a piecewise function related to the predator development time $\tau_{2}$, and a predator-prey model with predator-age structure and prey reaction time delay $\tau_{1}$ is investigated. By calculating the stability switching curves and using the theory of integrated semigroups and the Hopf bifurcation theory for abstract Cauchy problems with non-dense domain, we obtain that oscillations are caused by $\tau_{1}$ and $\tau_{2}$. Numerical simulations and a summary are presented at the end.