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Frontiers of Mathematics in China

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, Volume 13 Issue 6 Previous Issue   
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RESEARCH ARTICLE
Hermitizable, isospectral complex matrices or differential operators
Mu-Fa CHEN
Front. Math. China. 2018, 13 (6): 1267-1311.  https://doi.org/10.1007/s11464-018-0716-x
Abstract   PDF (514KB)

The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth–death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.

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Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions
Xingfan CHEN, Fei GUO, Peng LIU
Front. Math. China. 2018, 13 (6): 1313-1323.  https://doi.org/10.1007/s11464-018-0736-6
Abstract   PDF (269KB)

We consider a class of asymptotically linear nonautonomous secondorder Hamiltonian systems. Using the Saddle Point Theorem, we obtain the existence result, which extends some previously known results.

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Prediction-correction method with BB step sizes
Xiaomei DONG, Xingju CAI, Deren HAN
Front. Math. China. 2018, 13 (6): 1325-1340.  https://doi.org/10.1007/s11464-018-0739-3
Abstract   PDF (530KB)

In the prediction-correction method for variational inequality (VI) problems, the step size selection plays an important role for its performance. In this paper, we employ the Barzilai-Borwein (BB) strategy in the prediction step, which is effcient for many optimization problems from a computational point of view. To guarantee the convergence, we adopt the line search technique, and relax the conditions to accept the BB step sizes as large as possible. In the correction step, we utilize a longer step length to calculate the next iteration point. Finally, we present some preliminary numerical results to show the effciency of the algorithms.

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Sharp bounds for Hardy type operators on higher-dimensional product spaces
Qianjun HE, Xiang LI, Dunyan YAN
Front. Math. China. 2018, 13 (6): 1341-1353.  https://doi.org/10.1007/s11464-018-0740-x
Abstract   PDF (282KB)

We investigate a class of fractional Hardy type operators Hβ1,β2,,βm defined on higher-dimensional product spaces n1×n2××nm and use novel methods to obtain their sharp bounds. In particular, we optimize the result due to S. M. Wang, S. Z. Lu, and D. Y. Yan [Sci. China Math., 2012, 55(12): 2469–2480].

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Exponential sums involving automorphic forms for GL(3) over arithmetic progressions
Xiaoguang HE
Front. Math. China. 2018, 13 (6): 1355-1368.  https://doi.org/10.1007/s11464-018-0732-x
Abstract   PDF (322KB)

Let f be a Hecke-Maass cusp form for SL(3; ) with Fourier coefficients Af(m; n); and let ϕ (x) be a C -function supported on [1; 2] with derivatives bounded by ϕ (j)(x)j 1. We prove an asymptotic formula for the nonlinear exponential sum Σnlmod q Af(m,n )φ(n/X)e(3 (kn))1/3/q, where e(z)=e2πiz and k +.

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Signal recovery under mutual incoherence property and oracle inequalities
Peng LI, Wengu CHEN
Front. Math. China. 2018, 13 (6): 1369-1396.  https://doi.org/10.1007/s11464-018-0733-9
Abstract   PDF (343KB)

We consider the signal recovery through an unconstrained minimiza-tion in the framework of mutual incoherence property. A sufficient condition is provided to guarantee the stable recovery in the noisy case. Furthermore, oracle inequalities of both sparse signals and non-sparse signals are derived under the mutual incoherence condition in the case of Gaussian noises. Finally, we investigate the relationship between mutual incoherence property and robust null space property and find that robust null space property can be deduced from the mutual incoherence property.

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Structured backward error for palindromic polynomial eigenvalue problems, II: Approximate eigentriplets
Changli LIU, Ren-Cang LI
Front. Math. China. 2018, 13 (6): 1397-1426.  https://doi.org/10.1007/s11464-018-0738-4
Abstract   PDF (620KB)

A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEPs) P(λ) (l=0d Al λl)x=0, Adl=ε Al,L=0,1,,[ d2],

for an approximate eigentriplet is performed, where ★ is one of the two actions: transpose and conjugate transpose, and ε{±1}. The analysis is concerned with estimating the smallest perturbation to P( λ); while preserving the respective palindromic structure, such that the given approximate eigentriplet is an exact eigentriplet of the perturbed PPEP. Previously, R. Li, W. Lin, and C. Wang [Numer. Math., 2010, 116(1): 95–122] had only considered the case of an approximate eigenpair for PPEP but commented that attempt for an approximate eigentriplet was unsuccessful. Indeed, the latter case is much more complicated. We provide computable upper bounds for the structured backward errors. Our main results in this paper are several informative and very sharp upper bounds that are capable of revealing distinctive features of PPEP from general polynomial eigenvalue problems (PEPs). In particular, they reveal the critical cases in which there is no structured backward perturbation such that the given approximate eigentriplet becomes an exact one of any perturbed PPEP, unless further additional conditions are imposed. These critical cases turn out to the same as those from the earlier studies on an approximate eigenpair.

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Acute perturbation of Drazin inverse and oblique projectors
Sanzheng QIAO, Yimin WEI
Front. Math. China. 2018, 13 (6): 1427-1445.  https://doi.org/10.1007/s11464-018-0731-y
Abstract   PDF (274KB)

For an n×n complex matrix A with ind(A) = r; let AD and Aπ = I-AAD be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) =s; it is said to be a stable perturbation of A; if I(BπAπ)2 is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs) R(Bs)N(Ar)={0} and N(Bs)R(Ar)={0}, introduced by Castro-Gonz

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Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching
Xiaobin SUN, Yingchao XIE
Front. Math. China. 2018, 13 (6): 1447-1467.  https://doi.org/10.1007/s11464-018-0735-7
Abstract   PDF (369KB)

We consider a class of stochastic differential equations driven by subordinated Brownian motion with Markovian switching. We use Malliavin calculus to study the smoothness of the density for the solution under uniform Hörmander type condition.

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Global attractiveness and exponential decay of neutral stochastic functional differential equations driven by fBm with Hurst parameter less than 1/2
Liping XU, Jiaowan LUO
Front. Math. China. 2018, 13 (6): 1469-1487.  https://doi.org/10.1007/s11464-018-0728-6
Abstract   PDF (330KB)

We are concerned with a class of neutral stochastic functional differential equations driven by fractional Brownian motion (fBm) in the Hilbert space. We obtain the global attracting sets of this kind of equations driven by fBm with Hurst parameter (0, 1/2): Especially, some suffcient conditions which ensure the exponential decay in the p-th moment of the mild solution of the considered equations are obtained. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained.

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Spectral radius of r-uniform supertrees with perfect matchings
Lei ZHANG, An CHANG
Front. Math. China. 2018, 13 (6): 1489-1499.  https://doi.org/10.1007/s11464-018-0737-5
Abstract   PDF (322KB)

A supertree is a connected and acyclic hypergraph. The set of r-uniform supertrees with n vertices and the set of r-uniform supertrees with perfect matchings on rk vertices are denoted by Tn and Tr,k, respectively. H. Li, J. Shao, and L. Qi [J. Comb. Optim., 2016, 32(3): 741–764] proved that the hyperstar Sn,r attains uniquely the maximum spectral radius in Tn. Focusing on the spectral radius in Tr,k, this paper will give the maximum value in Tr,k and their corresponding supertree.

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Positive solutions of p-th Yamabe type equations on graphs
Xiaoxiao ZHANG, Aijin LIN
Front. Math. China. 2018, 13 (6): 1501-1514.  https://doi.org/10.1007/s11464-018-0734-8
Abstract   PDF (285KB)

Let G = (V,E) be a nite connected weighted graph, and assume 1αpq. In this paper, we consider the p-th Yamabe type equation Δpu+huq1=λfuα1 on G, where Δp is the p-th discrete graph Laplacian, h<0 and f>0 are real functions dened on all vertices of G: Instead of H. Ge's approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219–2224], we adopt a new approach, and prove that the above equation always has a positive solution u>0 for some constant λ. In particular, when q = p; our result generalizes Ge's main theorem from the case of αp1 to the case of 1αp. It is interesting that our new approach can also work in the case of αp1.

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