In this paper, we study the finite element method for a nonsmooth elliptic equation. Error analysis is presented, including a priori and a posteriori error estimates as well as superconvergence analysis. We also propose two algorithms for solving the underlying equation. Numerical experiments are employed to confirm our error estimations and the efficiency of our algorithms.

Let ℱ be a saturated formation containing the class of supersolvable groups. In this paper, we use c-normal subgroups and ss-quasinormal subgroups to give criteria for a group to belong to ℱ.

In this paper, we study the second-order differentiability of solutions with respect to parameters in a class of delay differential equations, where the evolution of the delay is governed explicitly by a differential equation involving the state variable and the parameters. We introduce the notion of locally complete triple-normed linear space and obtain an extension of the well-known uniform contraction principle in such spaces. We then apply this extended principle and obtain the second-order differentiability of solutions with respect to parameters in the W¹,p-norm (1≤p<∞).

A subgroup H of a group G is said to be g-s-supplemented in G if there exists a subgroup K of G such that HK ⊴ G and H ∩ K≤H_sG, where H_sG is the largest s-permutable subgroup of G contained in H. By using this new concept, we establish some new criteria for a group G to be soluble.

In this paper, we mainly study the geometry of conformal minimal immersions of two-spheres in a complex Grassmann manifold G(2,4). At first, we give a precise description of any non-±holomorphic harmonic 2-sphere in G(2,4) with the linearly full holomorphic maps (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ: S² → G(2, 4) with constant curvature has constant Kähler angle. Furthermore, ϕ is either , which is totally geodesic, with constant Gauss curvature 2/5 and constant Kähler angle given by t = 3/2 or , which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V0, V1, V2, is the Veronese sequence.

In this paper, we study the evolution of a noncompact hypersurface moving by mean curvature minus an external force field. We prove that the flow has a long-time smooth solution for a kind of special external force fields if the initial hypersurface is a Lipschitz entire graph with linear growth.

Let 1 be the set of fundamental cycles of breadth-first-search trees in a graph G, and let 2 be the set of the sums of two cycles in 1. Then we show the following: (1) contains a shortest Π-twosided cycle in a Π-embedded graph G. This implies the existence of a polynomially bounded algorithm to find a shortest Π-twosided cycle in an embedded graph and thus solves an open problem of Mohar and Thomassen [Graphs on Surfaces, 2001, p. 112]. (2)  contains all the possible shortest even cycles in a graph G. Therefore, there are at most polynomially many shortest even cycles in any graph. (3) Let 0 be the set of all the shortest cycles of a graph G. Then is a subset of  . Furthermore, many types of shortest cycles are contained in . Infinitely many examples show that there are exponentially many shortest odd cycles, shortest Π-onesided cycles and shortest Π-twosided cycles in some (embedded) graphs.

Let d be the smallest generator number of a finite p-group P, and let be a set of maximal subgroups of P such that . In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in , for each prime p, and Sylow p-subgroups P of G is studied.

This paper considers a heat system with localized sources and local couplings subject to null Dirichlet boundary conditions, for which both total and single point blow-up are possible. The aim of the paper is to identify the total and single point blow-up via a complete classification for all the nonlinear parameters in the model. As preliminaries of the paper, simultaneous versus non-simultaneous blow-up of solutions is involved, too. The results are then compared with those for another kind of heat system coupled via localized sources in a previous paper of the authors.

This paper is devoted to a study on the linear stability of some basic normal forms of symplectic matrices. Some necessary and sufficient conditions are given.