Decomposition, or splitting, finite difference methods have been playing an important role in the numerical solution of nonsingular differential equation problems due to their remarkable efficiency, simplicity, and flexibility in computations as compared with their peers. Although the numerical strategy is still in its infancy for solving singular differential equation problems arising from many applications, explorations of the next generation decomposition schemes associated with various kinds of adaptations can be found in many recent publications. The novel approaches have been proven to be highly effective and reliable in operations. In this article, we will focus on some of the latest developments in the area. Key comments and discussion will be devoted to two particularly interesting issues in the research, that is, direct solutions of degenerate singular reactiondiffusion equations and nonlinear sine-Gordon wave equations. Numerical experiments with simulated demonstrations will be given.

In this paper, we obtain all the Leibniz 2-cocycles of the twisted N = 2 superconformal algebra ℒ, which determine its second Leibniz cohomology group.

In this paper, we give the definition of the generalized Ramond N = 2 superconformal algebras and discuss the derivation algebra and the automorphism group.

In this paper, we study the stable one-dimensional quasi-periodic C^∞ cocycles on U(N). We prove that any such cocycle on a generic irrational rotation is a limit point of reducible cocycles. The proof is based on Krikorian’s renormalization scheme and a local result of him.

Let F ∈ C([0,∞)) be a positive increasing function such that Φ(s) := sF(s) is a Young function. In general, the F-Sobolev inequality and the Φ-Orlicz-Sobolev inequality are not equivalent. In this paper, a growth condition on F is presented for these two inequalities to be equivalent. The main result generalizes the corresponding known one for F(s) = log^δ(1+s) (δ>0). As an application, some criteria are presented for the F-Sobolev inequality to hold.

Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p₁, p₂, …, p_k}, where p₁<p₂<…<p_k, then the degree pattern of G is defined by D(G) = (deg(p₁), deg(p₂), … , deg(p_k)). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions H = G and D(H) = D(G). In addition, a 1-fold Odcharacterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A₁₆ and A₂₂, i.e., the symmetric groups S₁₆ and S₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.

The domination relationship between non-negative distributions is an important question in applied probability. It has important applications in the fields of finance, insurance and risk theory. In this paper, based on class , we find the sufficient condition of dominating all light-tailed distributions and also discuss its necessity. Almost all heavy-tailed distributions often used in risk theory satisfy this condition. We also consider the domination problem between heavy-tailed distributions, and show that classes  and ℳ* have many good properties on domination problems.

The augmented cube AQ_n is a variation of the hypercube Q_n. This paper considers the panconnectivity of AQ_n (n

We give two examples to show that the strong ergodicity and the logarithmic Sobolev inequality are incomparable for ergodic birth-death processes.

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408―414].

In this paper, some sufficient and necessary conditions for the commutative product of two elements of finite groups are given. Moreover, some results are obtained by investigating the product of some elements.