To ?nd a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To ?nd cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic pro?les. In this paper, we will ?rst illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier- Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in ?uid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.

In this paper, we study the singularities of the mean curvature ?ow from a symplectic surface or from a Lagrangian surface in a K?hler-Einstein surface. We prove that the blow-up ?ow ∑s∞ at a singular point(X₀, T₀) of a symplectic mean curvature ?ow Σ_t or of a Lagrangian mean curvature ?ow Σ_t is a nontrivial minimal surface in ?4, if ∑-∞∞ is connected.

In this paper, we mainly study the generalized Heisenberg-Virasoro algebra. Some structural properties of the Lie algebra are obtained.

Let G be a digraph with vertex set V (G) and arc set E(G) and let g = (g^-, g⁺) and f = (f^-, f⁺) be pairs of positive integer-valued functions de?ned on V (G) such that g^-(x)≤f^-(x) and g⁺(x)≤f⁺(x) for each x ∈ V (G). A (g, f)-factor of G is a spanning subdigraph H of G such that g^-(x)≤id_H(x)≤f^-(x) and g⁺(x)≤od_H(x)≤f⁺(x) for each x ∈ V (H); a (g, f)-factorization of G is a partition of E(G) into arc-disjoint (g, f)-factors. Let ?={F1,F2,?,Fm} and H be a factorization and a subdigraph of G, respectively. ? is called k-orthogonal to H if each F_i, 1≤i≤m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m-1,mf-m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k≤min{g^-(x), g⁺(x)} for any x ∈ V (G) and that every (mg,mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0≤g(x)≤f(x) for any x ∈ V (G). The results in this paper are in some sense best possible.

A k-edge-weightingw of a graph G is an assignment of an integer weight, w(e) ∈ {1, …, k}, to each edge e. An edge-weighting naturally induces a vertex coloring c by de?ning for every u ∈ V (G). A k-edge-weighting of a graph G is vertex-coloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). When k ≡ 2 (mod 4) and k≥6, we prove that if G is k-colorable and 2-connected, δ(G)≥k - 1, then G admits a vertex-coloring k-edge-weighting. We also obtain several suffcient conditions for graphs to be vertex-coloring k-edge-weighting.

By using a decomposition method, we give a criterion for the spectral gap of the reversible general jump process. This criterion enables us to obtain the lower bound for the spectral gap via Lyapunov drift condition. Some examples are presented to illustrate the results.

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we de?ne M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive de?nite. The elasticity tensor is rank-one positive de?nite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive de?nite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for ?nding M-eigenvalues are presented.

In this paper we obtain that every super-Virasoro algebra admits only triangular coboundary Lie super-bialgebra structures and this is proved mainly based on the computation of derivations from the super- Virasoro algebra to the tensor product of its adjoint module.

The method of Padématrix iteration is commonly used for computing matrix sign function and invariant subspaces of a real or complex matrix. In this paper, a detailed rounding error analysis is given for two classical schemes of the Padé matrix iteration, using basic matrix ?oating point arithmetics. Error estimations of computing invariant subspaces by the Padé sign iteration are also provided. Numerical experiments are given to show the numerical behaviors of the Pad′e iterations and the corresponding subspace computation.