This paper is a continuation of our previous paper [Front. Math. China, 2017, 12(5): 1023{1043] where global algorithms for computing the maximal eigenpair were introduced in a rather general setup. The efficiency of the global algorithms is improved in this paper in terms of a good use of power iteration and two quasi-symmetric techniques. Finally, the new algorithms are applied to Hua's economic optimization model.

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube {{\displaystyle Q}}_n, and obtain the following results. Let n\ge \mathrm{3},\hspace{0.17em}M\subset E({{\displaystyle Q}}_n), and F\subset E({{\displaystyle Q}}_n)\backslash M with \mathrm{1}\le \left|F\right|\le \mathrm{2}n-\mathrm{4}-\left|M\right|. If M is a matching and every vertex is incident with at least two edges in the graph {{\displaystyle Q}}_n-F, then all edges of M lie on a Hamiltonian cycle in {{\displaystyle Q}}_n-F. Moreover, if \left|M\right|=\mathrm{1} or \left|M\right|=\mathrm{2}, then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for \left|M\right|=\mathrm{1}.

Considering the Navier-Stokes-Landau-Lifshitz-Maxwell equations, in dimensions two and three, we use Galerkin method to prove the existence of weak solution. Then combine the a priori estimates and induction technique, we obtain the existence of smooth solution.

We consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra g; which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold T or a bounded domain in {ℝ}^n into a unit sphere {{\displaystyle S}}_g(\mathrm{1}) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of {ℝ}^{\mathrm{3}} and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

We show that there is no localization for the 4-order Schrödinger operator {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f and Beam operator {{\displaystyle B}}_{t}f, more precisely, on the one hand, we show that the 4-order Schrödinger operator {{\displaystyle S}}_{t,\mathrm{4}}f does not converge pointwise to zero as t\to \mathrm{0} provided f\in H^s(ℝ) with compact support and 0<s<1/4 by constructing a counterexample in ℝ. On the other hand, we show that the Beam operator Btf also has the same property with the 4-order Schrödinger operator {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f. Hence, we find that the Hausdorff dimension of the divergence set for {{\displaystyle \mathcal{S}}}_{t,\mathrm{4}}f and {{\displaystyle B}}_{t}f is {{\displaystyle \alpha }}_{\mathrm{1},{{\displaystyle S}}_{\mathrm{4}}}(s)={{\displaystyle \alpha }}_{\mathrm{1},B}(s)=\mathrm{1} as 0<s<1/4.

The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix. We characterize the connected graphs of order n and size n + k (5≤k≤8 and n>k + 5) with the minimal least eigenvalue.

We completely classify the normalized integral table algebra (A, B) generated by a faithful real element of degree 2 and having four linear elements.

We consider a branching random walk with an absorbing barrier, where the associated one-dimensional random walk is in the domain of attraction of an α-stable law. We shall prove that there is a barrier and a critical value such that the process dies under the critical barrier, and survives above it. This generalizes previous result in the case that the associated random walk has finite variance.

We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process.

Let ω(n) is the number of distinct prime factors of the natural number n,we consider two cases where \ell is even and odd natural numbers, and then we prove a more general form of the classical Erdős-Kac theorem.

For any N\ge \mathrm{2} and \alpha ={(\alpha }_1,\mathrm{...},{\alpha }_{N+1})\in {(0,\infty )}^{N+1}, let {\mu }_{\alpha }^{(N)} be the Dirichlet distribution with parameter α on the set {\mathrm{\Delta }}^{(N)}:={\{x\in [0,1]}^N:{\displaystyle {\sum }_{\mathrm{1}\le i\le N}}{x}_i\le \mathrm{1}\}. The multivariate Dirichlet diffusion is associated with the Dirichlet form

{\mathcal{E}}_{\alpha }^{(N)}(f,f):={\displaystyle \sum _{n=\mathrm{1}}^N}{\displaystyle {\int }_{{\mathrm{\Delta }}^{(N)}}}(\mathrm{1}-{\displaystyle \sum _{\mathrm{1}\le i\le N}}{x}_i)x_n{({\partial }_{n}f)}^{\mathrm{2}}(x){\mu }_{\alpha }^{(N)}(\mathrm{d}x)

with Domain \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}) being the closure of {C^1(\mathrm{\Delta }}^{(N)}). We prove the Nash inequality

{\mu }_{\alpha }^{(N)}(f^{\mathrm{2}})\le C{\mathcal{E}}_{\alpha }^{(N)}{(f,f)}^{p/(p+\mathrm{1})}{\mu }_{\alpha }^{(N)}{(|f|)}^{\mathrm{2}/(p+\mathrm{1})},\mathrm{\hspace{0.17em}}f\in \mathcal{D}({\mathcal{E}}_{\alpha }^{(N)}),{\mu }_{\alpha }^{(N)}(f)=0,

for some constant C>0 and p={({\alpha }_{N+\mathrm{1}}-\mathrm{1})}^{+}+{\displaystyle {\sum }_{i=\mathrm{1}}^N\mathrm{1}}\vee (\mathrm{2}{\alpha }_i), where the constant p is sharp when {\max }_{\mathrm{1}\le i\le N}\mathrm{\hspace{0.17em}}{\alpha }_i\le 1/2 and {\alpha }_{N+\mathrm{1}}\ge \mathrm{1}. This Nash inequality also holds for the corresponding Fleming-Viot process.

For a coupled system of wave equations with Dirichlet boundary controls, this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.

Let G be a group, and let α be a regular automorphism of order p² of G, where p is a prime. If G is polycyclic-by-finite and the map ϕ : G →G defined by g^ϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then C_G(α^p) and G/[G,α^p] are both nilpotent-by-finite.