Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

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, Volume 14 Issue 6

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Improved global algorithms for maximal eigenpair
Mu-Fa CHEN, Yue-Shuang LI
Front. Math. China. 2019, 14 (6): 1077-1116.

Abstract   PDF (1061KB)

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.

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Matchings extend to Hamiltonian cycles in hypercubes with faulty edges
Xie-Bin CHEN
Front. Math. China. 2019, 14 (6): 1117-1132.

Abstract   PDF (311KB)

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube Qn, and obtain the following results. Let n3,ME(Qn), and FE(Qn)\M with 1|F|2n4|M|. If M is a matching and every vertex is incident with at least two edges in the graph QnF, then all edges of M lie on a Hamiltonian cycle in QnF. Moreover, if |M|=1 or |M|=2, then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for |M|=1.

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Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations
Boling GUO, Fengxia LIU
Front. Math. China. 2019, 14 (6): 1133-1161.

Abstract   PDF (320KB)

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.

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Global weak solutions to Landau-Lifshitz equations into compact Lie algebras
Zonglin JIA, Youde WANG
Front. Math. China. 2019, 14 (6): 1163-1196.

Abstract   PDF (368KB)

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 n into a unit sphere Sg(1) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of 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.

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On 4-order Schrödinger operator and Beam operator
Dan LI, Junfeng LI
Front. Math. China. 2019, 14 (6): 1197-1211.

Abstract   PDF (279KB)

We show that there is no localization for the 4-order Schrödinger operator St,4f and Beam operator Btf, more precisely, on the one hand, we show that the 4-order Schrödinger operator St,4f does not converge pointwise to zero as t0 provided fHs() with compact support and 0<s<1/4 by constructing a counterexample in . On the other hand, we show that the Beam operator Btf also has the same property with the 4-order Schrödinger operator St,4f. Hence, we find that the Hausdorff dimension of the divergence set for St,4f and Btf is α1,S4(s)=α1,B(s)=1 as 0<s<1/4.

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Minimal least eigenvalue of connected graphs of order n and size m = n + k (5≤k≤8)
Xin LI, Jiming GUO, Zhiwen WANG
Front. Math. China. 2019, 14 (6): 1213-1230.

Abstract   PDF (617KB)

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.

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Normalized integral table algebras generated by a faithful real element of degree 2 and having 4 linear elements
Yu LI, Guiyun CHEN
Front. Math. China. 2019, 14 (6): 1231-1258.

Abstract   PDF (359KB)

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

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Critical survival barrier for branching random walk
Jingning LIU, Mei ZHANG
Front. Math. China. 2019, 14 (6): 1259-1280.

Abstract   PDF (339KB)

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.

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Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Lévy processes with periodic mean
Guangjun SHEN, Qian YU, Yunmeng LI
Front. Math. China. 2019, 14 (6): 1281-1302.

Abstract   PDF (313KB)

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.

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Generalization of Erdős-Kac theorem
Yalin SUN, Lizhen WU
Front. Math. China. 2019, 14 (6): 1303-1316.

Abstract   PDF (283KB)

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

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Nash inequality for diffusion processes associated with Dirichlet distributions
Feng-Yu WANG, Weiwei ZHANG
Front. Math. China. 2019, 14 (6): 1317-1338.

Abstract   PDF (355KB)

For any N 2 and α=(α1,...,αN+1)(0,)N+1, let μα(N) be the Dirichlet distribution with parameter α on the set Δ(N):={x[0,1]N:1iNxi1}. The multivariate Dirichlet diffusion is associated with the Dirichlet form


with Domain D(Eα(N)) being the closure of C1(Δ(N)). We prove the Nash inequality


for some constant C>0 and p=(αN+11)++i=1N1(2αi), where the constant p is sharp when max1iNαi1/2 and αN+11. This Nash inequality also holds for the corresponding Fleming-Viot process.

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Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations
Yanyan WANG
Front. Math. China. 2019, 14 (6): 1339-1352.

Abstract   PDF (255KB)

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.

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Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources
Hanyu WEI, Tiecheng XIA
Front. Math. China. 2019, 14 (6): 1353-1366.

Abstract   PDF (211KB)

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.

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Regular automorphisms of order p2
Tao XU, Heguo LIU
Front. Math. China. 2019, 14 (6): 1367-1373.

Abstract   PDF (240KB)

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 gϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then CG(αp) and G/[G,αp] are both nilpotent-by-finite.

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14 articles