Jan 2019, Volume 13 Issue 6

    

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• RESEARCH ARTICLE

  Hermitizable, isospectral complex matrices or differential operators

  Mu-Fa CHEN

  2018, 13(6): 1267-1311. https://doi.org/10.1007/s11464-018-0716-x

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  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.

• RESEARCH ARTICLE

  Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions

  Xingfan CHEN, Fei GUO, Peng LIU

  2018, 13(6): 1313-1323. https://doi.org/10.1007/s11464-018-0736-6

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  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.

• RESEARCH ARTICLE

  Prediction-correction method with BB step sizes

  Xiaomei DONG, Xingju CAI, Deren HAN

  2018, 13(6): 1325-1340. https://doi.org/10.1007/s11464-018-0739-3

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  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.

• RESEARCH ARTICLE

  Sharp bounds for Hardy type operators on higher-dimensional product spaces

  Qianjun HE, Xiang LI, Dunyan YAN

  2018, 13(6): 1341-1353. https://doi.org/10.1007/s11464-018-0740-x

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  We investigate a class of fractional Hardy type operators {\mathcal{H}}_{{\beta }_1,{\beta }_2,\dots ,{\beta }_m} defined on higher-dimensional product spaces {ℝ}^{n_1}\times {ℝ}^{n_2}\times \cdots \times {ℝ}^{n_m} 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].

• RESEARCH ARTICLE

  Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

  Xiaoguang HE

  2018, 13(6): 1355-1368. https://doi.org/10.1007/s11464-018-0732-x

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  Let f be a Hecke-Maass cusp form for SL(3;\mathrm{\mathbb{Z}}) with Fourier coefficients A_f(m; n); and let \varphi (x) be a C^{\infty } -function supported on [1; 2] with derivatives bounded by {\varphi }^{(j)}(x)\ll j 1. We prove an asymptotic formula for the nonlinear exponential sum {\mathrm{\Sigma }}_{n\equiv l\mathrm{mod}q}{A}_f(m,n)\phi (n/X)e{\left(3\left(kn\right)\right)}^{1/3}/q, where e(z)={\mathrm{e}}^{\mathrm{2}\pi iz} and k\in {\mathrm{\mathbb{Z}}}^{+}.

• RESEARCH ARTICLE

  Signal recovery under mutual incoherence property and oracle inequalities

  Peng LI, Wengu CHEN

  2018, 13(6): 1369-1396. https://doi.org/10.1007/s11464-018-0733-9

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  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.

• RESEARCH ARTICLE

  Structured backward error for palindromic polynomial eigenvalue problems, II: Approximate eigentriplets

  Changli LIU, Ren-Cang LI

  2018, 13(6): 1397-1426. https://doi.org/10.1007/s11464-018-0738-4

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  A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEPs)P(\lambda )\equiv \left({\displaystyle \sum _{l=\mathrm{0}}^d{{\displaystyle A}}_l{\lambda }^l}\right)x=\mathrm{0},{{\displaystyle A}}_{d-l}=\epsilon {{\displaystyle A}}_l^{★},L=\mathrm{0},\mathrm{1},\dots ,\left[{\displaystyle \frac{d}{\mathrm{2}}}\right],

  for an approximate eigentriplet is performed, where ★ is one of the two actions: transpose and conjugate transpose, and \epsilon \in \{\pm \mathrm{1}\}. The analysis is concerned with estimating the smallest perturbation to P(\lambda); 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.

• RESEARCH ARTICLE

  Acute perturbation of Drazin inverse and oblique projectors

  Sanzheng QIAO, Yimin WEI

  2018, 13(6): 1427-1445. https://doi.org/10.1007/s11464-018-0731-y

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  For an n×n complex matrix A with ind(A) = r; let A^D and A^{\pi } = I-AA^D 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-{(B^{\pi }-A^{\pi })}^{\mathrm{2}} is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs) R(B^s)\cap N(A^r)=\left\{0\right\} and N(B^s)\cap R(A^r)=\left\{0\right\}, introduced by Castro-Gonz

• RESEARCH ARTICLE

  Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching

  Xiaobin SUN, Yingchao XIE

  2018, 13(6): 1447-1467. https://doi.org/10.1007/s11464-018-0735-7

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  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.

• RESEARCH ARTICLE

  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

  2018, 13(6): 1469-1487. https://doi.org/10.1007/s11464-018-0728-6

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  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 \hslash \in (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.

• RESEARCH ARTICLE

  Spectral radius of r-uniform supertrees with perfect matchings

  Lei ZHANG, An CHANG

  2018, 13(6): 1489-1499. https://doi.org/10.1007/s11464-018-0737-5

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  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 T_n and T_r,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 T_n. Focusing on the spectral radius in T_r,k, this paper will give the maximum value in T_r,k and their corresponding supertree.

• RESEARCH ARTICLE

  Positive solutions of p-th Yamabe type equations on graphs

  Xiaoxiao ZHANG, Aijin LIN

  2018, 13(6): 1501-1514. https://doi.org/10.1007/s11464-018-0734-8

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  Let G = (V,E) be a nite connected weighted graph, and assume \mathrm{1}\le \alpha \le p\le q. In this paper, we consider the p-th Yamabe type equation -{\Delta p}_{}u+{huq-\mathrm{1}}^{}={\lambda fu\alpha -\mathrm{1}}^{} on G, where {\Delta p}_{} 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 \lambda \in ℝ. In particular, when q = p; our result generalizes Ge's main theorem from the case of \alpha \ge p\rangle \mathrm{1} to the case of \mathrm{1}\le \alpha \le p. It is interesting that our new approach can also work in the case of \alpha \ge p\rangle \mathrm{1}.