An $H$-packing $\mathcal{F}$ of a graph $G$ is a set of edge-disjoint subgraphs of $G$ in which each subgraph is isomorphic to $H$. The leave $L$ or the remainder graph $L$ of a packing $\mathcal{F}$ is the subgraph induced by the set of edges of $G$ that does not occur in any subgraph of the packing $\mathcal{F}$. If a leave $L$ contains no edges, or simply $L = \phi$, then $G$ is said to be $H$-decomposable, denoted by $H \mid G$. In this paper, we prove a conjecture made by Chartrand, Saba and Mynhardt [13]: If $G$ is a graph of size $q(G) \equiv 0 \pmod{3}$ and $\delta(G) \geq 2$, then $G$ is $H$-decomposable for some graph $H$ of size $3$.

This paper gives a short derivation for a result by Faudree and Schelp that the Turán number $\operatorname{ex}(n;P_{k+1})$ of a path of $k+1$ vertices is equal to $q \binom{k}{2} + \binom{r}{2}$, where $n = qk+r$ and $0 \leq r \lt k$, with the set $\operatorname{EX}(n;P_{k+1})$ of extremal graphs determined.

In 1999, C. J. Smyth proved that, for all $d \geq 4$, there are Salem numbers of degree $2d$ and trace $-1$, and that the number of them is greater than $0.1387d/(\log \log d)^2$. He gave also all Salem numbers of trace $-1$ and degree $2d = 8,10,12,14$. In this paper, we complete the table of the minimal polynomials of all Salem numbers of trace $-1$ and degree $2d = 16, 18, 20$, and we conjecture a new lower bound of the numbers of such Salem numbers.

We study the finiteness of associated primes and support of the generalized local cohomology modules $H^i_{I,J}(M,N)$ concerning Grothendieck's conjecture and Huneke's question. The paper also discusses the relationship between the vanishing and the finiteness of the generalized local cohomology modules.

We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper we consider the following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least $N-2$ integers in $\alpha_1, \ldots, \alpha_N$ then there exists one non-CSC HCMU metric on $S^2$ satisfying the condition stated above no matter where the given points are.

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.

We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

In this paper, we study the dynamics of a stochastic fractional reaction-diffusion equation with multiplicative noise in three spatial dimension. By proving the well-posedness and conducting a priori estimates for the solutions of the considered equation we obtain simultaneously the existence and the regularity of random attractors of the random dynamical systems for the equation. The main approach here is to establish an abstract result on existence of bi-spatial random attractors for random dynamical systems. Moreover, we also give estimate of the Hausdorff dimension in $L^{2}$ space for the obtained random attractor.

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, \\ u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrödinger wave function. Namely, we are still looking for solutions which concentrate at the points $x^j \in \Omega$, $j = 1,\ldots,m$ as the parameter $\rho$ tends to $0$. We find sufficient conditions under which, as $\rho$ tend to $0$, there exists an explicit class of solutions which admit a concentration behavior with a prescribed bubble profile around some given $m$-points in $\Omega$, for any given integer $m$. These are the so-called singular limits. The candidate $m$-points of concentration must be nondegenerate (in essential way) critical points of a suitable finite dimensional functional explicitly and the higher order Green's function with respect to the imposed boundary conditions.

In this paper, we deal with the asymmetric Orlicz zonotopes by using the method of shadow system. We establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.

Let $\mu$ be a non-negative Radon measure satisfying the polynomial growth condition. In this paper, the authors characterize the set of pointwise multipliers on a BMO type space $\operatorname{RBMO}(\mu)$ introduced by Tolsa.

A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and $\dim_{H} R(f)$ is the solution of the Bowen equation of topological pressure. These results can be applied to a large class of symbolic systems, including $\beta$-shifts, $S$-gap shifts, and their factors.

We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$ random chosen individuals. In this paper, we study the distributions of $D_k(t)$ and $X_k(t)$ and their limit distributions as $t \to \infty$.