Let $R$ be a ring. The unit graph of $R$, denoted by $G(R)$, is the simple graph defined on all elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x+y$ is a unit of $R$. The diameter of a simple graph $G$, denoted by $\operatorname{diam}(G)$, is the longest distance between all pairs of vertices of the graph $G$. In the present paper, we prove that for each integer $n \geq 1$, there exists a ring $R$ such that $n \leq \operatorname{diam}(G(R)) \leq 2n$. We also show that $\operatorname{diam}(G(R)) \in \{ 1,2,3,\infty \}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with $\operatorname{diam}(G(R)) = 1,2,3$ and $\infty$, respectively. This extends [12, Theorem 2 and Corollary 1].

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1} = n_1$, and non-negative integers $j$ and $r$ with $j \leq m$, the following two expressions \begin{gather*} \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} q^{j(k^2+k) - (2r+1)k} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k}, \\ \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} (-1)^k q^{\binom{k}{2} + j(k^2+k) - 2rk} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k} \end{gather*} are Laurent polynomials in $q$ with integer coefficients, where $[n] = 1+q+\cdots+q^{n-1}$ and ${n \brack k} = \prod_{i=1}^k (1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by Guo and Zeng. We also confirm some related conjectures of Guo and Zeng by establishing their $q$-analogues. Several conjectural congruences for sums involving products of $q$-ballot numbers $\left( {2n \brack n-k} - {2n \brack n-k-1} \right)$ are proposed in the last section of this paper.

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of $\mathscr{C}$; (2) special preenveloping ideals of $\mathscr{C}$; (3) additive subfunctors of $\mathbb{E}$ having enough special injective morphisms; and (4) additive subfunctors of $\mathbb{E}$ having enough special projective morphisms. Moreover, we show that if $(\mathscr{C},\mathbb{E}, \mathfrak{s})$ is an extriangulated category with enough injective objects and projective morphisms, then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of $\mathscr{C}$; (2) all additive subfunctors of $\mathbb{E}$ having enough special injective objects.

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.

In this paper, we study the parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ and the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function. We proved that if the kernel $\Omega$ associated to parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ is homogeneous of degree zero and satisfies the Lipschitz continuous condition, or the kernel $K$ associated to the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function satisfies the Hörmander condition, then they are bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$ with $mn/(mn+\gamma) \lt p_1, \ldots, p_m \leq 1$ and $1/p = 1/p_1 + \cdots + 1/p_m$.

Using Frazier and Jawerth's $\varphi$-transform, we characterize weighted generalized Carleson measure spaces $\dot{C}MO^{\alpha,q}_{p,w}$ for a weight $w$ and show that the definition of this space is well-defined by a Plancherel-Pôlya inequality. Note that $\dot{C}MO^{0,2}_{1,w}$ is the weighted $BMO$ space.

Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Hölder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.

In this paper we study several dynamical properties of a Riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed $3$-manifold $M$. Carrière [6] classified such pairs $(M,\mathcal{L})$. Using the classification we describe in detail recurrence points, $\omega$-limit sets and attractors. Finally, using the fact that the Poincaré map on a transversal surface for a Riemannian $1$-dimensional foliation is an isometry, we show the nonhyperbolicity of $(M,\mathcal{L})$.

Two classes of plate equations with past history and strong time-dependent delay in the internal feedback are considered. Our results contain the global well-posedness and exponential stability of the two systems. We prove the global well-posedness of a system with rotational inertia without any restrictions on $\mu_1$, $\mu_2$, and the system without rotational inertia under the assumption $|\mu_2| \leq \mu_1$. For the system with rotational inertia, we establish exponential stability to the plate equation with the memory term only to control the delay term if the amplitude of the time delay term is small, and the stability result also holds for the plate equation with strong anti-damping. For the system without rotational inertia, we obtain the exponential stability under the assumption $|\mu_2| \lt \sqrt{1-d} \mu_1$.

In this paper, we investigate abstract critical point theorems for continuously Gâteaux differentiable functionals satisfying the Cerami condition via the generalized Ekeland variational principle developed by C.-K. Zhong. As applications of our results, under certain assumptions, we show the existence of at least one or two weak solutions for nonlinear elliptic equations with variable exponents \[ -\operatorname{div} (\varphi(x, \nabla u)) + V(x) |u|^{p(x)-2} u = \lambda f(x,u) \quad \textrm{in } \mathbb{R}^{N}, \] where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with a continuous function $p \colon \mathbb{R}^{N} \to (1,\infty)$, $V \colon \mathbb{R}^{N} \to (0,\infty)$ is a continuous potential function, $\lambda$ is a real parameter, and $f \colon \mathbb{R}^{N} \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function. Especially, we localize precisely the intervals of $\lambda$ for which the above equation admits at least one or two nontrivial weak solutions by applying our critical points results.

We consider the following singular Kirchhoff type equation with Hardy-Sobolev critical exponent \[ \begin{cases} \displaystyle -\left( a + b \int_{\Omega} |\nabla u|^2 \, dx \right) \Delta u = \frac{u^{3}}{|x|} + \frac{\lambda}{|x|^{\beta} u^{\gamma}}, & x \in \Omega, \\ u > 0, & x \in \Omega, \\ u = 0, & x \in \partial \Omega, \end{cases} \] where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with smooth boundary $\partial \Omega$, $0 \in \Omega$, $a,b,\lambda \gt 0$, $0 \lt \gamma \lt 1$, and $0 \leq \beta \lt (5+\gamma)/2$. Combining with the variational method and perturbation method, two positive solutions of the equation are obtained.

In this paper, we focus on certain functions as scalarization for six types of set relations and discuss calculation algorithms for them between polyhedral sets, while those between polytopes have been already investigated. A major difference between polyhedral sets and polytopes is in boundedness. Polyhedral sets are no longer necessarily bounded. Methods for calculating types (1), (2), (4), (6) are easily available by a similar way to existing ideas. However, those for types (3) and (5), which are actually the most famous and long-standing types, require some technical ways approaching to the value of them by using the fact that finitely generatedness and polyhedrality coincide and can be algorithmically switched in finite-dimensional spaces. As a result, we show all types are reduced to a finite number of linear programming problems. Also, we demonstrate our methods through an example and give detailed calculation process.