A graph is subeulerian if it is a spanning subgraph of an eulerian graph. All subeulerian graphs were characterized by Boesch, Suffel, Tindell in 1977. Later, a simple proof of their theorem was given by Jaeger. A digraph $D$ is eulerian if and only if $D$ is connected and $d^{+}(x) = d^{-}(x)$ for every vertex $x \in V(D)$. An orientation $\overrightarrow{G}$ of a graph $G$ is a digraph obtained from $G$ by replacing each edge $xy$ of $G$ with an arc $xy$ or $yx$. An oriented graph is an orientation of a simple graph. An oriented graph is said to be subeulerian if it is a spanning subdigraph of an eulerian oriented graph.

In this paper, we initiated the study of subeulerian oriented graphs. We give a necessary and sufficient condition for an orientated digraph to be subeulerian. We refine this condition in order to give necessary and sufficient condition for an orientation of a forest to be subeulerian. Furthermore, we prove that if $G$ is a graph of order $n$ with $n \geq \max \{ 4\Delta(G)-1, 3 \}$, then every orientation of $G$ is subeulerian. In particular, we show that if $G$ is a graph of odd order $n$ with $\Delta(G) \leq n/4$, then every orientation of $G$ is a spanning subdigraph of a regular tournament.

Let $R$ be a prime ring of characteristic different from $2$ and $3$, $Q_{r}$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $L$ is a non-central Lie ideal of $R$. If there exists an integer $k \geq 1$ such that $[F(x)x-xG(x),x]_{k} = 0$ for any $x \in L$, then

1. either there exist $a \in Q_{r}$ and $\lambda \in C$ such that $F(x) = xa$ and $G(x) = (a+\lambda)x$ for all $x \in R$;

2. or there exists a field $K$ such that $R \subseteq M_{2}(K)$, the ring of $2 \times 2$ matrices over $K$, and one of the following conclusions occurs: (a) $K$ is a finite field; (b) there exist $a \in Q_{r}$ and $\lambda \in C$ such that $F(x) + G(x) = ax + xa + \lambda x$ for all $x \in R$.

Ideal submanifolds have been studied from various aspects since Chen invented $\delta$-invariants in early 1990s (see [12] for a survey). In centroaffine differential geometry, Chen's invariants denoted by $\delta^{\sharp}$ are used to determine an optimal bound for the squared norm of the Tchebychev vector field of a hypersurface. We point out that a hypersurface attaining this bound is said to be an ideal centroaffine hypersurface. In this paper, we deal with $\delta^{\sharp}(2,2)$-ideal centroaffine hypersurfaces in $\mathbb{R}^{5}$ and in particularly, we focus on $4$-dimensional $\delta^{\sharp}(2,2)$-ideal centroaffine hypersurfaces of type $1$.

In this paper, a class of viscoelastic wave equations with exponential source and the strong damping $-\Delta_{p} u_{t}$ is concerned. The local existence of the weak solution is established by combining the theory of monotone operators, and nonlinear semigroups with Banach's fixed point theorem. The global existence of the weak solution is also proved by using the potential well theory. In addition, for $p \equiv 2$, the blow-up in finite time is discussed under low and arbitrarily high initial energy based on Levine's concavity method, respectively. Meanwhile, the lifespan of the weak solution is also derived.

Resonant regime in two-phase flow models causes tough problems for constructing computational Riemann solvers. The nonlinear algebraic equations that determine the states could be in implicit form. In this work, we develop computational algorithms for the states in Riemann solvers, which include a method of pattern search of the intersection points of composite wave curves associated with different characteristic fields. Numerical tests show that numerical schemes employing these algorithms provides us with a good accuracy. The schemes are also shown to be well-balanced in subsonic regions.

In this article, we investigate the maximal bilinear Riesz means $S_{\ast}^{\alpha}$ associated to the sublaplacian on the Heisenberg group. We prove that the operator $S_{\ast}^{\alpha}$ is bounded from $L^{p_{1}} \times L^{p_{2}}$ into $L^{p}$ for $2 \leq p_{1}, p_{2} \leq \infty$ and $1/p = 1/p_{1} + 1/p_{2}$ when $\alpha$ is large than a suitable smoothness index $\alpha(p_{1},p_{2})$. For obtaining a lower index $\alpha(p_{1},p_{2})$, we define two important auxiliary operators and investigate their $L^{p}$ estimates, which play a key role in our proof.

In the present paper, we devote our efforts to a backward problem for an anomalous diffusion model with multi-term time fractional derivatives. Such a problem is ill-posed. For this purpose, we introduce a fractional-order quasi-reversibility regularization method that is a new perturbation related to the time fractional derivative into the original equation. Based on some properties of the multinomial Mittag-Leffler function as well as the Fourier method, we theoretically give some regularity results of the regularized solution, and prove the corresponding convergence rate under the a-priori regularization parameter choice rule in the general dimensional case. Finally, several numerical examples are given to demonstrate the effectiveness of the proposed method. The numerical results are well in line with our expectations.

Erdős–Rényi graph is a random graph in which the probability of a connection between two nodes follows a Bernoulli distribution independently. The stochastic block models (SBM) are an extension of the Erdős–Rényi graph by dividing nodes into $K$ subsets, known as blocks or communities. Let $\widetilde{A}_{N} = (\widetilde{A}_{ij}^{(N)})$ be an $N \times N$ normalized adjacency matrix of the SBM with $K$ blocks of any sizes, and let $\mu_{\widetilde{A}_{N}}$ be the empirical spectral density of $\widetilde{A}_{N}$.

In this paper, we first showed that if the connecting probabilities between nodes of different blocks are zero, then $\lim_{N \to \infty} \mu_{\widetilde{A}_{N}} = \mu$ exists almost surely, and we gave the explicit formulas for $\mu$ and its Stieltjes transform, respectively. Second, we showed under a suitable condition on the maximum of connecting probability between nodes in different blocks, say by $\zeta_{0}$, $\mu_{\widetilde{A}_{N}}$ converges both in probability and expectation as first $N \to \infty$ and then $\zeta_{0} \to 0$.

In this paper, a new concept of the subdifferential is defined for nondifferentiable (not necessarily) locally Lipschitz functions. Namely, the concept of $E$-subdifferential and the notion of $E$-subconvexity are introduced for $E$-convex functions. Thus, the notion of an $E$-subdifferentiable $E$-convex function is introduced and some properties of this class of nondifferentiable nonconvex functions are studied. The necessary optimality conditions in $E$-subdifferentials terms of the involved functions are established for a new class of nondifferentiable optimization problems. The introduced concept of $E$-subconvexity is used to prove the sufficiency of the aforesaid necessary optimality conditions for nondifferentiable optimization problems in which the involved functions are $E$-subdifferentiable $E$-convex.