ORIGINAL_ARTICLE
Upper and lower bounds for numerical radii of block shifts
For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine when either of them becomes an equality.
http://bims.iranjournals.ir/article_719_7b89cecc3c9d266bd2d8a5e08a8dc1cb.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
15
27
Numerical radius
block shift
minimum modulus
P. Y.
Wu
pywu@math.nctu.edu.tw
true
1
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan.
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan.
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan.
LEAD_AUTHOR
H.-L.
Gau
hlgau@math.ncu.edu.tw
true
2
Department of Mathematics, National Central University, Chun-gli 32001, Taiwan.
Department of Mathematics, National Central University, Chun-gli 32001, Taiwan.
Department of Mathematics, National Central University, Chun-gli 32001, Taiwan.
AUTHOR
ORIGINAL_ARTICLE
Higher numerical ranges of matrix polynomials
Let $P(\lambda)$ be an $n$-square complex matrix polynomial, and $1 \leq k \leq n$ be a positive integer. In this paper, some algebraic and geometrical properties of the $k$-numerical range of $P(\lambda)$ are investigated. In particular, the relationship between the $k$-numerical range of $P(\lambda)$ and the $k$-numerical range of its companion linearization is stated. Moreover, the $k$-numerical range of the basic $A$-factor block circulant matrix, which is the block companion matrix of the matrix polynomial $P(\lambda) = \lambda ^m I_n - A$, is studied.
http://bims.iranjournals.ir/article_720_27058d5330da2190ea9a4d45104b7f64.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
29
45
$k$-Numerical range
matrix polynomial
companion linearization
basic $A$-factor block
circulant matrix
Gh.
Aghamollaei
aghamollaei@uk.ac.ir
true
1
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
LEAD_AUTHOR
M. A.
Nourollahi
mnourollahi@bam.ac.ir
true
2
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
AUTHOR
ORIGINAL_ARTICLE
On nest modules of matrices over division rings
Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We then characterize submodules of nest modules of matrices over $D$ in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from $D$. We use this result to characterize principal submodules of nest modules. We also describe subbimodules of nest modules of matrices. As a consequence, we characterize (one-sided) ideals of nest algebras of matrices over division rings.
http://bims.iranjournals.ir/article_721_5f9a18056601fd0fd44a34d75a22addd.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
47
63
Bimodule of rectangular matrices over a division ring
(left/right) submodule
subbimodule
(one-sided) ideal
nest modules
B. R.
Yahaghi
bamdad5@hotmail.com
true
1
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan 19395-5746, Iran.
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan 19395-5746, Iran.
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan 19395-5746, Iran.
LEAD_AUTHOR
M.
Rahimi-Alangi
mrahimi40@yahoo.com
true
2
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.
AUTHOR
ORIGINAL_ARTICLE
Self-commutators of composition operators with monomial symbols on the Bergman space
Let $\varphi(z)=z^m, z \in \mathbb{U}$, for some positive integer $m$, and $C_\varphi$ be the composition operator on the Bergman space $\mathcal{A}^2$ induced by $\varphi$. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $C^*_\varphi C_\varphi, C_\varphi C^*_\varphi$ as well as self-commutator and anti-self-commutators of $C_\varphi$. We also find the eigenfunctions of these operators.
http://bims.iranjournals.ir/article_722_d5ce5eefb15ab5a75efe1e6a099e23e5.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
65
76
Bergman space
composition operator
essential spectrum
essential norm
self-commutator
A.
Abdollahi
abdollahi@shirazu.ac.ir
true
1
Department of Mathematics, Shiraz University, Shiraz, Iran.
Department of Mathematics, Shiraz University, Shiraz, Iran.
Department of Mathematics, Shiraz University, Shiraz, Iran.
LEAD_AUTHOR
S.
Mehrangiz
math.samira@yahoo.com
true
2
Department of Engineering, Khonj Branch, Islamic Azad
University, Khonj, Iran.
Department of Engineering, Khonj Branch, Islamic Azad
University, Khonj, Iran.
Department of Engineering, Khonj Branch, Islamic Azad
University, Khonj, Iran.
AUTHOR
T.
Roientan
tahereh136355@yahoo.com
true
3
Department of Mathematics, Shiraz University, Shiraz, Iran.
Department of Mathematics, Shiraz University, Shiraz, Iran.
Department of Mathematics, Shiraz University, Shiraz, Iran.
AUTHOR
ORIGINAL_ARTICLE
Linear maps preserving or strongly preserving majorization on matrices
For $A,B\in M_{nm},$ we say that $A$ is left matrix majorized (resp. left matrix submajorized) by $B$ and write $A\prec_{\ell}B$ (resp. $A\prec_{\ell s}B$), if $A=RB$ for some $n\times n$ row stochastic (resp. row substochastic) matrix $R.$ Moreover, we define the relation $\sim_{\ell s} $ on $M_{nm}$ as follows: $A\sim_{\ell s} B$ if $A\prec_{\ell s} B\prec_{\ell s} A.$ This paper characterizes all linear preservers and all linear strong preservers of $\prec_{\ell s}$ and $\sim_{\ell s}$ from $M_{nm}$ to $M_{nm}$.
http://bims.iranjournals.ir/article_723_2527aef09e5df50b63467d24125b54c8.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
77
83
Linear preserver
row substochastic matrix
matrix majorization
F.
Khalooei
f_khalooei@uk.ac.ir
true
1
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Additive maps on C$^*$-algebras commuting with $|.|^k$ on normal elements
Let $\mathcal {A} $ and $\mathcal {B} $ be C$^*$-algebras. Assume that $\mathcal {A}$ is of real rank zero and unital with unit $I$ and $k>0$ is a real number. It is shown that if $\Phi:\mathcal{A} \to\mathcal{B}$ is an additive map preserving $|\cdot|^k$ for all normal elements; that is, $\Phi(|A|^k)=|\Phi(A)|^k $ for all normal elements $A\in\mathcal A$, $\Phi(I)$ is a projection, and there exists a positive number $c$ such that $\Phi(iI)\Phi(iI)^{*}\leq
c\Phi(I)\Phi(I)^{*}$, then $\Phi$ is the sum of a linear Jordan *-homomorphism and a conjugate-linear Jordan *-homomorphism. If, moreover, the map $\Phi$ commutes with $|.|^k$ on $\mathcal{A}$, then $\Phi$ is the sum of a linear *-homomorphism and a conjugate-linear *-homomorphism. In the case when $k \not=1$, the assumption $\Phi(I)$ being a projection can be deleted.
http://bims.iranjournals.ir/article_724_15cc6b48cc93f3f328eb35b3ec30359a.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
85
98
C$^*$-algebras
additive maps
Jordan homomorphism
*-homomorphism
Y.
Guan
yarongguan721@163.com
true
1
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
AUTHOR
C.
Wang
wangcailian1224@163.com
true
2
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
Department of Mathematics, Taiyuan University of Technology, Taiyuan
030024, P.R. China.
AUTHOR
J.
Hou
jinchuanhou@aliyun.com
true
3
Department of Mathematics, Taiyuan
University of Technology, Taiyuan 030024, P.R.
China.
Department of Mathematics, Taiyuan
University of Technology, Taiyuan 030024, P.R.
China.
Department of Mathematics, Taiyuan
University of Technology, Taiyuan 030024, P.R.
China.
LEAD_AUTHOR
ORIGINAL_ARTICLE
A Haar wavelets approach to Stirling's formula
This paper presents a proof of Stirling's formula using Haar wavelets and some properties of Hilbert space, such as Parseval's identity. The present paper shows a connection between Haar wavelets and certain sequences.
http://bims.iranjournals.ir/article_725_4334fc7b24c523ffe166068cee677ed5.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
99
106
Haar wavelets
Parseval's identity
Stirling's formula
M.
Ahmadinia
mahdiahmadinia72@gmail.com
true
1
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
LEAD_AUTHOR
H.
Naderi Yeganeh
hamid@hamidnaderiyeganeh.id.ir
true
2
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
Department of Mathematics, University of Qom, P.O. Box 37185-3766, Qom, Iran.
AUTHOR
ORIGINAL_ARTICLE
Additivity of maps preserving Jordan $\eta_{\ast}$-products on $C^{*}$-algebras
Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of maps $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective, unital and satisfy $\Phi(AP+\eta PA^{*})=\Phi(A)\Phi(P)+\eta \Phi(P)\Phi(A)^{*},$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$. If $\eta$ is a non-zero complex number such that $|\eta|\neq1$, then $\Phi$ is additive. Moreover, if $\eta$ is rational<,> then $\Phi$ is $\ast$-additive.
http://bims.iranjournals.ir/article_726_99688174344f34c677f1fe618917af2c.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
107
116
Maps preserving Jordan $eta*$-product
Additive
Prime C*-algebras
A.
Taghavi
taghavi@umz.ac.ir
true
1
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
LEAD_AUTHOR
H.
Rohi
h.rohi@stu.umz.ac.ir
true
2
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
AUTHOR
V.
Darvish
v.darvish@stu.umz.ac.ir
true
3
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, P.O. Box 47416-1468,
Babolsar, Iran.
AUTHOR
ORIGINAL_ARTICLE
A note on lifting projections
Suppose $\pi:\mathcal{A}\rightarrow \mathcal{B}$ is a surjective unital $\ast$-homomorphism between C*-algebras $\mathcal{A}$ and $\mathcal{B}$, and $0\leq a\leq1$ with $a\in \mathcal{A}$. We give a sufficient condition that ensures there is a proection $p\in \mathcal{A}$ such that $\pi \left( p\right) =\pi \left( a\right) $. An easy consequence is a result of [L. G. Brown and G. k. Pedersen, C*-algebras of real rank zero, \textit{J. Funct. Anal.} {99} (1991) 131--149] that such a $p$ exists when $\mathcal{A}$ has real rank zero.
http://bims.iranjournals.ir/article_727_d877b46ffb88804bc8a9a508cad613ff.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
117
122
C*-algebra
projection
D.
Hadwin
don@unh.edu
true
1
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Toeplitz transforms of Fibonacci sequences
We introduce a matricial Toeplitz transform and prove that the Toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. We investigate the injectivity of this transform and show how this distinguishes the Fibonacci sequence among other recurrence sequences. We then obtain new Fibonacci identities as an application of our transform.
http://bims.iranjournals.ir/article_728_fac81767727c4a836e2fc2d91f8e8fed.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
123
132
Hankel transform
Fibonacci numbers
Fibonacci identities
L.
Connell
lconnell@colby.edu
true
1
111 W. Westminster, Lake Forest, IL 60045.
111 W. Westminster, Lake Forest, IL 60045.
111 W. Westminster, Lake Forest, IL 60045.
AUTHOR
M.
Levine
mjlevine@colby.edu
true
2
The Catalyst Lofts, 141 41st Street, Pittsburgh, PA 15201.
The Catalyst Lofts, 141 41st Street, Pittsburgh, PA 15201.
The Catalyst Lofts, 141 41st Street, Pittsburgh, PA 15201.
AUTHOR
B.
Mathes
dbmathes@colby.edu
true
3
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
LEAD_AUTHOR
J.
Sukiennik
justin.sukiennik@colby.edu
true
4
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
5839 Mayflower Hill, Colby College, Waterville, ME 04901.
AUTHOR
ORIGINAL_ARTICLE
A note on approximation conditions, standard triangularizability and a power set topology
The main result of this article is that for collections of entry-wise non-negative matrices the property of possessing a standard triangularization is stable under approximation. The methodology introduced to prove this result allows us to offer quick proofs of the corresponding results of [B. R. Yahaghi, Near triangularizability implies triangularizability, Canad. Math. Bull. 47, (2004), no. 2, 298--313], and [A. A. Jafarian, H. Radjavi, P. Rosenthal and A. R. Sourour, Simultaneous, triangularizability, near commutativity and Rota's theorem, Trans. Amer. Math. Soc. 347, (1995), no. 6, 2191--2199] on the approximations and triangularizability of collections of operators and matrices. In conclusion we introduce and explore a related topology on the power sets of metric spaces.
http://bims.iranjournals.ir/article_729_43d144e4ed126793c4056a889f73dcfb.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
133
153
Simultaneous triangularizability
positive matrices
standard invariant subspaces
semigroups of operators
L.
Livshits
llivshi@colby.edu
true
1
Department of Mathematics and Statistics, Colby College, Waterville, ME 04901, USA.
Department of Mathematics and Statistics, Colby College, Waterville, ME 04901, USA.
Department of Mathematics and Statistics, Colby College, Waterville, ME 04901, USA.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Addendum to: "Infinite-dimensional versions of the primary, cyclic and Jordan decompositions", by M. Radjabalipour
In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminology of linear transformations. We add an additional translation of a ring-theoretic result to give a characterization of algebraically hyporeflexive transformations and the strict closure of the set of polynomials in a transformation $T$.
http://bims.iranjournals.ir/article_730_312c207f682f4959e07b53c8cfb5db04.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
155
173
Abelian group
PID
Module
cyclic
torsion
locally algebraic
hyporeflexive
scalar-reflexive ring
strict topology
H.
Fan
hun4@unh.edu
true
1
University of New Hampshire
University of New Hampshire
University of New Hampshire
AUTHOR
D.
Hadwin
don@unh.edu
true
2
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
College of Engineering and Physical Sciences, University of New Hampshire, Durham, USA.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Infinite-dimensional versions of the primary, cyclic and Jordan decompositions
The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.
http://bims.iranjournals.ir/article_731_8937987b904454c3bdd4f748e6f5d775.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
175
183
Jordan canonical form
rational canonical form
splitting field
M.
Radjabalipour
radjabalipour@ias.ac.ir
true
1
Erfan Institute of Higher Education, Kerman, Iran.
Erfan Institute of Higher Education, Kerman, Iran.
Erfan Institute of Higher Education, Kerman, Iran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Submajorization inequalities associated with $\tau$-measurable operators
The aim of this note is to study the submajorization inequalities for $\tau$-measurable operators in a semi-finite von Neumann algebra on a Hilbert space with a normal faithful semi-finite trace $\tau$. The submajorization inequalities generalize some results due to Zhang, Furuichi and Lin, etc..
http://bims.iranjournals.ir/article_732_bff972f6998e185187ad5bd0a9ee72b7.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
185
194
Submajorization
von Neumann algebra
$tau$-measurable operators
J.
Zhao
jgzhao_dj@163.com
true
1
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China, and
College of Science, Shihezi University, Shihezi, Xinjiang, 832003, P. R. China.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China, and
College of Science, Shihezi University, Shihezi, Xinjiang, 832003, P. R. China.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China, and
College of Science, Shihezi University, Shihezi, Xinjiang, 832003, P. R. China.
LEAD_AUTHOR
J.
Wu
jlwu678@163.com
true
2
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China.
AUTHOR
ORIGINAL_ARTICLE
The witness set of coexistence of quantum effects and its preservers
One of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. In this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. We also devote to characterizing bijective morphisms on quantum effects leaving the witness set invariant. Furthermore, applying linear maps preserving commutativity, which are characterized by Choi, Jafarian and Radjavi [Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227--241.], we classify multiplicative general morphisms leaving the witness set invariant on finite dimensional Hilbert space effect algebras.
http://bims.iranjournals.ir/article_733_11e420ffa346edde192a2c50f80bc9b4.pdf
2015-12-01T11:23:20
2018-01-20T11:23:20
195
204
Positive operator matrices
Coexistence
Hilbert space effect algebras
Isomorphisms
K.
He
hk19830310@163.com
true
1
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
LEAD_AUTHOR
F. G.
Sun
472974952@qq.com
true
2
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
AUTHOR
J.
Hou
65787687@qq.com
true
3
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
AUTHOR
Q.
Yuan
875675778@qq.com
true
4
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
College of Mathematics, Institute of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi,
030024, P.R. China.
AUTHOR