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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Two new eigenvalue localization sets for tensors and theirs applications

Jianxing Zhao
  • Corresponding author
  • College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
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/ Caili Sang
Published Online: 2017-10-09 | DOI: https://doi.org/10.1515/math-2017-0106

Abstract

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

Keywords: Nonnegative tensors; Tensor eigenvalue; Localization set; Positive definite; Spectral radius

MSC 2010: 15A18; 15A42; 15A69

1 Introduction

For a positive integer n, n ≥ 2, N denotes the set {1, 2, …, n}. ℂ(ℝ) denotes the set of all complex (real) numbers. We call 𝓐 = (ai1im) a complex (real) tensor of order m dimension n, denoted by 𝓐 ∈ ℂ[m,n](ℝ[m,n]), if ai1imC(R),

where ijN for j = 1, 2, ⋯, m. A tensor of order m dimension n is called the unit tensor, denoted by 𝓘, if its entries are δi1im for i1, ⋯, imN, where δi1im=1,if i1==im,0,otherwise.

𝓐 is called nonnegative if ai1im ≥ 0. 𝓐 = (ai1im) ∈ ℝ[m,n] is called symmetric [1] if ai1im=aπ(i1im),πΠm,

where Πm is the permutation group of m indices. 𝓐 = (ai1im) ∈ ℝ[m,n] is called weakly symmetric [2] if the associated homogeneous polynomial Axm=i1,,imNai1imxi1xim

satisfies ∇ 𝓐xm = m𝓐xm−1. It is shown in [2] that a symmetric tensor is necessarily weakly symmetric, but the converse is not true in general.

To an n-vector x = (x1, x2⋯, xn)T, real or complex, we define the n-vector: Axm1=i2,,imNaii2imxi2xim1in

and x[m1]=(xim1)1in.

Definition 1.1

([1, 3]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. A pair (λ, x) ∈ ℂ × (ℂn ∖ {0}) is called an eigenvalue-eigenvector (or simply eigenpair) of 𝓐 if Axm1=λx[m1].

(λ, x) is called an H-eigenpair if both of them are real.

Definition 1.2

([1, 3]). Let 𝓐 = (ai1im) ∈ ℝ[m,n]. A pair (λ, x) ∈ ℂ × (ℂn ∖ {0}) is called an E-eigenpair of 𝓐 if Axm1=λxandxTx=1.

(λ, x) is called an Z-eigenpair if both of them are real.

We define the Z-spectrum of 𝓐, denoted 𝓩(𝓐) to be the set of all Z-eigenvalues of 𝓐. Assume 𝓩(𝓐) ≠ 0, then the Z-spectral radius [2] of 𝓐, denoted ϱ(𝓐), is defined as ϱ(A):=sup{|λ|:λZ(A)}.

It is shown in [1] that a real even-order symmetric tensor 𝓐 = (ai1im) is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. However, when m and n are very large, it is not easy to compute all H-eigenvalues (Z-eigenvalues) of 𝓐. Then we can try to give a set in the complex which includes all H-eigenvalues (Z-eigenvalues) of 𝓐. If this set is in the right-half complex plane, then we can conclude that all H-eigenvalues (Z-eigenvalues) are positive, consequently, 𝓐 is positive definite; for details, see [1, 47].

There are other applications of (E-)eigenvalue inclusion sets, for example we can use them to obtain the lower and upper bounds for the H-eigenvalues (Z-spectral radius) of (nonnegative) tensors and the minimum eigenvalue of 𝓜-tensors; for details, see [819].

In 2005, Qi [1] presented the following Geršgorin-type eigenvalue localization set for real symmetric tensors, which can be easily extended to general tensors [4, 20].

Theorem 1.3

([1, Theorem 6]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then σ(A)Γ(A)=iNΓi(A),

where σ(𝓐) is the set of all eigenvalues of 𝓐 and Γi(A)={zC:|zaii|ri(A)},ri(A)=δii2im=0|aii2im|.

To get a tighter eigenvalue localization set than Γ(𝓐), Li et al. [4] proposed the following Brauer-type eigenvalue localization set for tensors.

Theorem 1.4

([4, Theorem 2.1]). Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then σ(A)K(A)=i,jN,jiKi,j(A),

where Ki,j(A)={zC:(|zaii|rij(A))|zajj||aijj|rj(A)},rij(A)=δii2im=0,δji2im=0|aii2im|=rj(A)|aijj|.

To reduce computations, Li et al. [4] gave an S-type eigenvalue localization set by breaking N into disjoint subsets S and S, where S is the complement of S in N.

Theorem 1.5

([4, Theorem 2.2]). Let 𝓐 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N. Then σ(A)KS(A)=iS,jS¯Ki,j(A)iS¯,jSKi,j(A).

In 2017, Wang et al. established the following Z-eigenvalue localization set for a real tensor 𝓐, which is completely different from eigenvalue localization sets and can be generalized to an E-eigenvalue localization set easily.

Theorem 1.6

([8, Theorem 3.1]). Let 𝓐 = (ai1im) ∈ ℝ[m,n]. Then Z(A)Γ^(A)=iNΓ^i(A),

where Γ^i(A)={zC:|z|Ri(A)},Ri(A)=i2imN|aii2im|.

The main aim of this paper is to give a new eigenvalue localization set for tensors, which is tighter than those in Theorems 1.3-1.5, and a new E-eigenvalue localization set for tensors, which is tighter than that in Theorem 1.6. As applications, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained based on the eigenvalue localization set, and a new upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained based on the E-eigenvalue localization set.

2 A new eigenvalue localization set for tensors and its applications

In this section, we propose a new eigenvalue localization set for tensors and establish the comparisons between this set with those in Theorems 1.3-1.5. As an application of this set, we give a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor.

Theorem 2.1

Let 𝓐 = (ai1im) ∈ ℂ[m,n]. Then σ(A)K(A)=iNjN,jiKi,j(A),

where Ki,j(A)={zC:(|zaii|rij(A))|zajj||aijj|rj(A)}.

Proof

Let λ be an eigenvalue of 𝓐 with corresponding eigenvector x = (x1, ⋯, xn)T ∈ ℂn ∖ {0}, i.e., Axm1=λx[m1].(1)

Let |xp| = max{|xi| : iN}. Then, |xp| > 0. From (1), we have (λapp)xpm1=δpi2im=0,δji2im=0api2jmxi2xim+apjjxjm1,jN,jp.

Taking modulus in the above equation and using the triangle inequality give |λapp||xp|m1δpi2im=0,δji2im=0|api2im||xi2||xim|+|apjj||xj|m1δpi2im=0,δji2im=0|api2im||xp|m1+|apjj||xj|m1=rpj(A)|xp|m1+|apjj||xj|m1,

equivalently, (|λapp|rpj(A))|xp|m1|apjj||xj|m1.(2)

If |xj| = 0, by |xp| > 0, we have |λapp|rpj(A)0. Then (|λapp|rpj(A))|λajj|0|apjj|rj(A),

which implies that λ ∈ 𝓚p,j(𝓐) ⊆ 𝓚(𝓐). otherwise, |xj| > 0. Similarly, from (1), we can obtain |λajj||xj|m1rj(A)|xp|m1(3)

Multiplying (2) with (3) and noting that |xp|m−1|xj|m−1 > 0, we have (|λapp|rpj(A))|λajj||apjj|rj(A),

then λ ∈ 𝓚p,j(𝓐) ⊆ 𝓚(𝓐) . From the arbitrariness of j, we have λjN,jpKp,j(A). Furthermore, λiNjN,jiKi,j(A).

Next, a comparison theorem is given for Theorems 1.3-1.5 and Theorem 2.1.

Theorem 2.2

Let 𝓐 = (ai1im) ∈ ℂ[m,n], S be a nonempty proper subset of N. Then K(A)KS(A)K(A)Γ(A).

Proof

Let S be the complement of S in N. According to Theorem 2.3 in [4], 𝓚S(𝓐) ⊆ 𝓚(𝓐) ⊆Γ(𝓐) . Hence, we only prove 𝓚(𝓐) ⊆ 𝓚S(𝓐) . Let z ∈ 𝓚(𝓐), then there exists i0N, such that z ∈ 𝓚i0,j(𝓐), ∀ jN, ji0. If i0S, then for any jS, we have zi0S,jS¯Ki0,j(A)KS(A). If i0S, then for any jS, we have zi0S¯,jSKi,j(A)KS(A). The conclusion follows. □

Remark 2.3

Theorem 2.2 shows that this set in Theorem 2.1 is tighter than those in Theorem 1.3, Theorem 1.4 and Theorem 1.5, that is, 𝓚(𝓐) can capture all eigenvalues of 𝓐 more precisely than Γ(𝓐), 𝓚(𝓐) and 𝓚S(𝓐).

As shown in [1, 47], an eigenvalue localization set can provide a checkable sufficient condition for the positive (semi-)definiteness of tensors. As an application of Theorem 2.1, we give a checkable sufficient condition for the positive (semi-)definiteness of tensors.

Theorem 2.4

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be an even-order symmetric tensor with akk > 0 for all kN. IfiN, ∃ jN, ji, (aiirij(A))ajj>|aijj|rj(A),

then 𝓐 is positive definite.

Proof

Let λ be an H-eigenvalue of 𝓐. By Theorem 2.1, we have λ ∈ 𝓚(𝓐), that is, there is i0N, for any jN, ji0, (|λai0i0|ri0j(A))|λajj||ai0jj|rj(A).

Suppose that λ ≤ 0. Then for i0N, ∃ j0, such that ai0i0 > 0, aj0j0 > 0, and (|λai0i0|ri0j0(A))|λaj0j0|(ai0i0ri0j0(A))aj0j0>|ai0j0j0|rj0(A).

This is a contradiction. Hence, λ > 0, and 𝓐 is positive definite. The conclusion follows. □

Similar to the proof of Theorem 2.4, the following sufficient condition is easily obtained.

Theorem 2.5

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be an even-order symmetric tensor with akk ≥ 0 for all kN. IfiN, ∃ jN, ji, (aiirij(A))ajj|aijj|rj(A),

then 𝓐 is positive semi-definite.

Remark 2.6

When n = 2, Theorem 2.4 is the same as Theorem 4.1 and Theorem 4.2 in [4]. When n ≥ 3, it is easy to see that the conditions of Theorem 2.4 for determining the positive definiteness of tensors are weaker than those in Theorem 4.1 and Theorem 4.2 in [4].

Next, an example is given to verify the fact in Remark 2.6.

Example 2.7

Let 𝓐 = (aijkl) ∈ ℝ[4,3] be a symmetric tensor with elements defined as follows: a1111=12.1,a2222=4.6,a3333=3.6,a1112=0.1,a1113=0.15,a1122=0.2,a1123=0.2,a1133=0,a1222=0.1,a1223=0.3,a1233=0.1,a1333=0.15,a2223=0.1,a2233=0.1,a2333=0.2.

By computations, we get that (a1111r13(A))a3333=29.7>0.5550=|a1333|r3(A);(4) (a2222r23(A))a3333=1.0800>0.7400=|a2333|r3(A);(5) (a3333r31(A))a1111=0.6050>0.6000=|a3111|r1(A);(6) (a3333r32(A))a2222=0<0.45=|a3222|r2(A).(7)

Let S = {1, 2}, S = {3}. Because (7) holds, we can not use Theorem 4.1 and Theorem 4.2 in [4] to determine the positiveness of 𝓐 under this division. But from (4)-(6) and Theorem 2.4, we can determine that 𝓐 is positive definite. In fact, all the H-eigenvalues of 𝓐 are 2.9074, 3.1633, 3.7705, 4.6282 and 12.4216. By Theorem 5 in [1], 𝓐 is positive definite.

3 A new E-eigenvalue localization set for tensors and its applications

In this section, we give an S-type E-eigenvalue localization set for tensors, and establish the comparison between this set with that in Theorem 1.6. For simplification, we first denote some notations. Given a nonempty proper subset S of N, let ΔN={(i2,i3,,im):eachijNforj=2,,m},ΔS={(i2,i3,,im):eachijSforj=2,,m},

and then ΔS¯=ΔNΔS.

This implies that for a tensor 𝓐 = (ai1im) ∈ ℝ[m,n], we have that for iS, Ri(A)=i2,,imN|aii2im|=RiΔS(A)+RiΔS¯(A),

where RiΔS(A)=(i2,,im)ΔS|aii2im|,RiΔS¯(A)=(i2,,im)ΔS¯|aii2im|.

Theorem 3.1

Let 𝓐 = (ai1im) ∈ ℝ[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then σE(A)ΩS(A)=iS,jS¯Ωi,jS(A)iS¯,jSΩi,jS¯(A),

where σE(𝓐) is the set of all E-eigenvalues of 𝓐 and Ωi,jS(A)={zC:|z|(|z|RjΔS¯(A))Ri(A)RjΔS¯(A)},Ωi,jS¯(A)={zC:|z|(|z|RjΔS¯¯(A))Ri(A)RjΔS¯(A)}.

Proof

Let λ be an E-eigenvalue of 𝓐 with corresponding eigenvector x = (x1,⋯,xn)T ∈ ℂn ∖ {0}, i.e., Axm1=λx,||x||2=1.(8)

Let |xp| = max{|xi| : iS} and |xq| = max{|xj| : jS}. Then, at least one of |xp| and |xq| is nonzero. We next distinguish two cases to prove.

Case I. Suppose that |xq||xp|, then |xq|=maxjN|xj| and 0<|xq|m1|xq|1. From (8), we have λxq=(i2im)ΔSaqi2imxj2xim+(i2im)ΔS¯aqi2imxi2xim.

Taking modulus in the above equation and using the triangle inequality give |λ||xq|m1|λ||xq|(i2im)ΔS|aqi2im||xi2||xim|+(i2im)ΔS¯|aqi2im||xi2||xim|(i2im)ΔS|aqi2im||xp|m1+(i2im)ΔS¯|aqi2im||xq|m1=RqΔS(A)|xp|m1+RqΔS¯(A)|xq|m1,

i.e., (|λ|RqΔS¯(A))|xq|m1RqΔS(A)|xp|m1.(9)

If |xp| = 0, by |xq| > 0, we have |λ|RqΔS¯(A)0. Then (|λ|RqΔS¯(A))|λ|0RqΔS(A)Rp(A),

which implies that λΩp,qS(A)ΩS(A). If |xp| > 0, from (8), we can obtain |λ||xp|m1|λ||xp|i2imN|api2im||xi2||xim|Rp(A)|xq|m1.(10)

Multiplying (9) with (10) and noting that |xp|m−1|xq|m−1 > 0, we have (|λ|RqΔS¯(A))|λ|RqΔS(A)Rp(A),

which leads to λΩp,qS(A)ΩS(A).

Case II. Suppose that |xp| ≥ |xq|, then |xp|=maxiN|xi| and 0 < |xp|m−1 ≤ |xp| ≤ 1. Similar to (9), we can obtain (|λ|RpΔs¯¯(A))|xp|m1RpΔS¯(A)|xq|m1.(11)

If |xq| = 0, by |xp| > 0, we have |λ|RpΔs¯¯(A)0. Then (|λ|RpΔS¯¯(A))|λ|0RpΔS¯(A)Rq(A),

which implies that λΩq,pS¯(A)ΩS(A). If |xq| > 0, similar to (10), we have |λ||xq|m1Rq(A)|xp|m1.(12)

Multiplying (11) with (12) and noting that |xp|m−1|xq|m−1 > 0, we have (|λ|RpΔS¯¯(A))|λ|RpΔS¯(A)Rq(A),

which leads to λΩq,pS¯(A)ΩS¯(A). The conclusion follows from Cases I and II. □

Theorem 3.2

Let 𝓐 = (ai1im) ∈ ℝ[m,n], S be a nonempty proper subset of N, S be the complement of S in N. Then ΩS(A)Γ^(A).

Proof

Let λ ∈ ΩS(𝓐) . Then λiS,jS¯Ωi,jS(A) or λiS¯,jSΩi,jS¯(A).

Without loss of generality, suppose that λiS,jS¯Ωi,jS(A) (we can prove it similarly if λiS¯,jSΩi,jS¯(A)).

Then there are iS and jS such that λΩi,jS(A), i.e., |λ|(|λ|RjΔS¯(A))Ri(A)RjΔS(A).(13)

If Rj(A)RjΔS(A)=0, then λ=0 or |λ|RjΔS¯(A)Rj(A). Hence, λRi (𝓐) ⋃ Rj(𝓐). otherwise, from (13), we have |λ|Ri(A)|λ|RjΔS¯(A)RjΔS(A)1.

Furthermore, |λ|Ri(A)1

or |λ|RjΔS¯(A)RjΔS(A)1,

which implies that λRi (𝓐)⋃ Rj(𝓐) . □

Remark 3.3

From Theorem 3.2, we known that the set ΩS(𝓐) in Theorem 3.1 localizes all E-eigenvalues of a tensor 𝓐 more precisely than the set Γ^(A) in Theorem 1.6.

Next, based on Theorem 3.1, we give an S-type upper bound for the Z-spectral radius of a weakly symmetric nonnegative tensor.

Theorem 3.4

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be a weakly symmetric nonnegative tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then ϱ(A)ΨS(A)=maxmaxiS,jS¯ΨijS(A),maxiS,jS¯ΨijS¯(A),

where ΨijS(A)=12RjΔS¯(A)+[(RjΔS¯)2+4Ri(A)RjΔS(A)]12,ΨijS¯(A)=12RjΔS¯¯(A)+[(RjΔS¯¯)2+4Ri(A)RjΔS¯(A)]12.

Proof

By Lemma 4.4 in [8], ϱ(𝓐) is the largest Z-eigenvalue of 𝓐. From Theorem 3.1, we know that ϱ(𝓐) ∈ ΩS(𝓐) . Then ϱ(A)iS,jS¯Ωi,jS(A) or ϱ(A)iS¯,jSΩi,jS¯(A).

We next distinguish two cases to prove.

Case I: If ϱ(A)iS,jS¯Ωi,jS(A), then there exists i ∈ S, jS, such that ϱ(A)(ϱ(A)RjΔS¯(A))Ri(A)RjΔS(A).

Then ϱ(A)12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12.

Furthermore, ϱ(A)maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12.

Case II: If ϱ(A)iS¯,jSΩi,jS¯(A), similar to the proof of Case I, we can obtain ϱ(A)maxiS¯,jS12RjΔS¯¯(A)+[(RjΔS¯¯(A))2+4Ri(A)RjΔS¯(A)]12.

The conclusion follows from Cases I and II. □

Theorem 3.5

Let 𝓐 = (ai1im) ∈ ℝ[m,n] be a weakly symmetric nonnegative tensor, S be a nonempty proper subset of N, S be the complement of S in N. Then ΨS(A)maxiNRi(A).(14)

Proof

Here, we only prove that when ΨS(A)=maxiS,jS¯ΨijS(A), (14) holds. Similarly, we can also prove that (14) holds if ΨS(A)=maxiS¯,jSΨijS¯(A). Next, we divide two cases to prove.

Case I: For any iS,jS, if Ri(𝓐) ≤ Rj(𝓐), then ΨS(A)=maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Rj(A)RjΔS(A)]12=maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4(RjΔS¯(A)+RjΔS(A))RjΔS(A)]12=maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A)+2RjΔS(A))2]12=maxiS,jS¯12RjΔS¯(A)+RjΔS¯(A)+2RjΔS(A)=maxjS¯Rj(A).

Case II: For any iS,jS, if Rj(𝓐) ≥ Ri(𝓐), then 0 ≤ RjΔS(A)Ri(A)RjΔS¯(A), and ΨS(A)=maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)RjΔS(A)]12maxiS,jS¯12RjΔS¯(A)+[(RjΔS¯(A))2+4Ri(A)(Ri(A)RjΔS¯(A))]12=maxiS,jS¯12RjΔS¯(A)+[(2Ri(A)RjΔS¯(A))2]12=maxiS,jS¯12RjΔS¯(A)+2Ri(A)RjΔS¯(A)=maxiSRi(A)maxiNRi(A).

The conclusion follows from Cases I and II. □

Remark 3.6

Theorem 3.5 shows that the upper bound in Theorem 3.4 is better than Corollary 4.5 of [9].

Now, we show that the upper bound in Theorem 3.4 is sharper than those in [813] in some cases by the following example.

Example 3.7

Let 𝓐 = (aijk) ∈ ℝ[3,3] with entries defined as follows: A(:,:,1)=0201.512020,A(:,:,2)=10.5200222.53,A(:,:,3)=020.5232.5121.

It is not difficult to verify that 𝓐 is a weakly symmetric nonnegative tensor By computation, we obtain (ϱ(𝓐),x) = (7.3450, (0.3908, 0.6421, 0.6596)). By Corollary 4.5 of [9] and Theorem 3.3 of [10], we both have ϱ(A)14.

By Theorem 3.5 of [11], we have ϱ(A)13.9189.

By Theorem 4.6 of [8], we have ϱ(A)13.9133.

By Theorem 4.7 of [8], we have ϱ(A)13.8167.

By Theorem 4.5 of [8] and Theorem 6 of [12], we both have ϱ(A)13.5000.

By Theorem 2.9 of [13], we have ϱ(A)12.9790.

Let S = {1},S = {2,3}. By Theorem 3.4, we obtain ϱ(A)11.5440.

which shows that the upper bound in Theorem 3.4 is sharper.

4 Conclusions

In this paper, we give a new eigenvalue localization set 𝓚(𝓐) and prove that 𝓚(𝓐) is tighter than those in [1] and [4]. Based on this set, we obtain a weaker checkable sufficient condition to determine the positive (semi-)definiteness for an even-order real symmetric tensor. Meanwhile, we present an S-type E-eigenvalue localization set ΩS(𝓐) and prove that ΩS(𝓐) is tighter than that in [8]. As an application, we obtain an S-type upper bound ΨS(𝓐) for the Z-spectral radius of weakly symmetric nonnegative tensors, and show that ΨS(𝓐) is sharper than those in [813] in some cases by a numerical example. Then an interesting problem is how to pick S to make ΨS(𝓐) as small as possible. But this is difficult when n is large. In the future, we will focus on this problem.

Acknowledgement

The authors are very indebted to the reviewers for their valuable comments and corrections, which improved the original manuscript of this paper. This work is supported by National Natural Science Foundation of China (No.11501141), Foundation of Guizhou Science and Technology Department (Grant No.[2015]2073) and Natural Science Programs of Education Department of Guizhou Province (Grant No.[2016]066).

References

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About the article

Received: 2017-01-27

Accepted: 2017-08-29

Published Online: 2017-10-09


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1267–1276, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0106.

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© 2017 Zhao and Sang. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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