For a graph $G$, denote its domination number and independent domination number by $\gamma(G)$ and $i(G)$. For a positive integer $k$, a subset $S \subseteq V(G)$ is a $k$-independent set of $G$ if $\Delta(G[S]) \lt k$. The $k$-independence number $\alpha_{k}(G)$ is the cardinality of a maximum $k$-independent set of $G$. In this paper, we prove that for any nontrivial tree $T$, $i(T) \leq \frac{k+1}{2k} \alpha_{k}(T)$ and $\gamma(T) \leq \frac{k}{2k-1} \alpha_{k}(T)$, respectively. This extends the recent result for $k = 2$ by Dehgardi et al. Both bounds are best possible and the extremal trees are characterized.

A Pisot number is a real algebraic integer $\alpha \gt 1$ such that all of its Galois conjugates, other than $\alpha$ itself, lie inside the unit circle. An anti-Pisot number is a real algebraic integer $\alpha \gt 1$, such that exactly one Galois conjugate of $\alpha$ lies inside the unit circle, and $\alpha$ has at least one Galois conjugate, other than $\alpha$ itself, outside the unit circle. We call the minimal polynomials of these algebraic integers, respectively, Pisot and anti-Pisot polynomials. In this article, we find infinite families of Pisot (anti-Pisot) polynomials $f(x)$ such that $\{ 1, \alpha, \alpha^2, \ldots, \alpha^{n-1} \}$ is a basis for the ring of integers of $\mathbb{Q}(\alpha)$, where $\alpha$ is a Pisot (anti-Pisot) number and $\deg(f) = n$, for certain $n$. We refer to these polynomials as monogenic Pisot (anti-Pisot) polynomials.

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} u = |x|^a |u|^{p-1} u \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending the celebrated result of [14], we obtain a classification result on finite Morse index solutions to the fractional elliptic equation above with Hardy potential. In particular, a critical exponent $p$ of Joseph–Lundgren type is derived in the supercritical case studying a Liouville type result for the $s$-harmonic extension problem.

This paper deals with a chemotaxis model with nonlinear signal production in a smoothly bounded domain. When there is no logistic growth source, the solutions of the system are globally bounded. This is also true if the logistic damping effect is strong enough. We extend recent research on single-species and one stimulus obtained by Tao et al. (2019, J. Math. Anal. Appl.) to two species chemotaxis system with two chemicals by creating an extra subtle inequality. We also partially extended some other related work.

In this paper, we introduce three notions of topological entropy of a free semigroup action generated by proper maps for noncompact subsets, which extends the notions defined by Ju et al. [13] and Ma et al. [17]. By using the one-point compactification as a bridge, we study the relations of the entropies between two dynamical systems. We then introduce three skew-product transformations, and for a particular subset, the relationship between the upper capacity topological entropy of a free semigroup action generated by proper maps, and the upper capacity topological entropy of a skew-product transformation is given. As applications, we examine the multifractal spectrum of a locally compact separable metric space, and it is shown that the irregular set has full upper capacity topological entropy of a free semigroup action generated by proper maps.

In this paper we establish the Gamma-convergence of generalized gradient flows with conjugate type. It provided a criteria for obtaining the convergence of generalized gradient flows that correspond to a sort of $C^1$-order $\Gamma$-convergence of energy functionals and a kind of bounded symmetric positive definite linear operators.

In this paper, we establish two embedding inequalities for the weighted Sobolev space and the weighted homogeneous endpoint Besov space by using the weighted Hausdorff capacity. To do this, we shall determine the dual spaces of weighted Choquet and weighted homogeneous Besov spaces.

In this paper, we study a predator-prey system with general response function and nonlocal diffusion in high dimensional space and investigate the propagation properties of its solution. More precisely, we study the invasion speed of the predator into habitat of the aborigine prey and obtain the existence of the planar waves by constructing the upper and lower solutions. Finally, we present some numerical simulations to support our results.

One way of HIV infection spreading is through the cell division of infected cells by mitosis‎ ‎expressed in mathematical models as a logistic process‎. ‎Cell-to-cell transmission is another factor in the spread and speed of disease‎. ‎In this work‎, ‎we present a five-dimensional Ordinary Differential Equation model (ODE) with the logistic form for proliferation of uninfected cells‎, ‎cell-to-cell and virus-to-cell transmission rate‎, ‎two types of cellular and humoral immune responses‎, ‎the cure rate for returning infected cells to non-infectious cells‎, ‎and two treatment rates‎, ‎one for reducing infectious cells and the other for blocking free viruses‎. ‎We discuss the positivity and boundedness of solutions‎, ‎free-equilibrium points‎, ‎steady-state equilibrium points‎, ‎and stability by the Routh Hurwitz criterion‎. ‎The rate of reproduction is analyzed, ‎and the useful parameters for increasing or decreasing it are identified‎. ‎Numerical simulations are performed to investigate the dynamic behavior of model responses to treatment effects on disease.

The article, A Note on Number Knots and the Splitting of the Hilbert Class Field by Yih-Jeng Yu which appeared in Volume 24, Number 2 (April Issue 2020, pp. 283–300) of the Taiwanese Journal of Mathematics has been retracted due to numerous unfixable errors. In particular, the main result of the paper, Corollary 3.10, is unfortunately false.