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16 February 2016
Volume 51 Issue 2
α
-skew Armendariz rings relative to a monoid
GENG Dao-hong, WANG Yao, REN Yan-li
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 1-5. doi:
10.6040/j.issn.1671-9352.0.2015.213
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The concept of
α
-skew
M
-Armendariz rings is introduced. The relationship between
α
-skew
M
-Armendariz rings and related rings is investigated, and it is proved that the skew monoid ring extension of a Baer ring(right p.p.-ring, right zip ring)is a Baer ring(right p.p.-ring, right zip ring)in case the ring is an
α
-skew
M
-Armendariz.
Power series weak McCoy rings
LI Min, WANG Yao, REN Yan-li
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 6-11. doi:
10.6040/j.issn.1671-9352.0.2015.150
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The concept of a power series weak McCoy ring is introduced. It is shown that(1)if every
R
i
(i∈I)
is a power series weak McCoy ring, then ∏
i∈I
R
i
is a power series weak McCoy ring;(2)If
R
is a nil-semicommutative ring, then
R［x］
is a power series weak McCoy ring if and only if
R
is a power series weak McCoy ring;(3)If
R
is a
α
-compatible nil-semicommutative ring, then
R［x;α］
is a power series weak McCoy ring.
Non-solvabla elements of semigroup P
Γ
(Λ×Λ)of binary relations determined by the semilattice
Γ
on the set
Λ
LIN Ping-feng
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 12-15. doi:
10.6040/j.issn.1671-9352.0.2015.060
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Let
Λ
be an arbitrary nonempty set, and
Γ
be a semilattice on the set
Λ
. Let P
Γ
(Λ×Λ)is a semigroup of binary relations determined by the semilattice
Γ
on the set
Λ
. In the semigroup P
Γ
(Λ×Λ), a necessary and sufficient condition of non-solvable elements is obtained,and a class of non-solvable elements is found under certain conditions.
On algebras with all
τ
-rigid modules projective
XIE Zong-zhen, ZHANG Xiao-jin
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 16-20. doi:
10.6040/j.issn.1671-9352.0.2015.185
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For a special class of algebras, a method on constructing indecomposable
τ
-rigid modules from simple modules is given. As a result, it is proved that a basic and connected algebra
A
with radical square zero is local if all
τ
-rigid A-modules are projective.
Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations
LIU Wen-yue, SUN Tong-jun
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 21-28. doi:
10.6040/j.issn.1671-9352.0.2015.190
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A numerical method for solving optimal boundary control problems governed by elliptic equations is considered. In order to avoid large amounts of calculation produced by traditional numerical methods. An iterative non-overlapping domain decomposition method is established. The whole domain is divided into many non-overlapping subdomains, and the optimal boundary control problem is decomposed into local problems in these subdomains. Robin conditions are used to communicate the local problems on the interfaces between subdomains. The iterative scheme for solving these local problems is studied, and prove the convergence of the scheme is proved. Finally, a numerical example to prove the validity of the scheme is presented.
Stochastic suppression on explosive solutions of a class of nonlinear impulsive differential systems by noise
ZHANG Chun-yan, HAO Sheng-nan, FENG Li-chao
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 29-36. doi:
10.6040/j.issn.1671-9352.0.2015.271
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We investigate the problem of stochastic suppression on explosive solutions of nonlinear impulsive differential systems by noise. For a given nonlinear impulsive differential system satisfying one-sided polynomial growth condition which may explode at a finite time, we introduce a polynomial stochastic noise
σ|x(t)|
β
x(t)
d
B(t)(B(t)
is a Brownian motion)such that there exists a unique global solution for the corresponding stochastically perturbed impulsive differential system. The global solution is bounded in the sense of the moment and the trajectory with large probability and the global solution grow at most polynomial.
Uniqueness of stationary solutions to a simplified energy-transport model for semiconductors
DONG Jian-wei, LOU Guang-pu, WANG Yan-ping
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 37-41. doi:
10.6040/j.issn.1671-9352.0.2015.146
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A stationary simplified energy-transport model for semiconductors is studied in a one-dimensional bounded domain. The uniqueness of solutions to the model is proved under some conditions by using some inequality techniques.
On complete moment convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables
ZHANG Li-jun, GUO Ming-le
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 42-49. doi:
10.6040/j.issn.1671-9352.0.2015.151
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1257
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The complete moment convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables was investigated. By using Rosenthal type inequality and truncation method, the sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables were obtained. By using the sufficient conditions, we extend the corresponding results from the complete convergence to the complete moment convergence. Therefore, we complement the property of asymptotically almost negatively associated random variables.
Almost sure central limit theorem for self-normalized products of some partial sums of NA random variables
XU Feng, WU Qun-ying
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 50-57. doi:
10.6040/j.issn.1671-9352.0.2015.106
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Let {
X,X
n
}
n∈N
be a stationary sequence of NA positive random variables. We proved an almost sure central limit theorem for the self-normalized products of some partial sums(∏
k
i=1
(S
k,i
/((k-1
)μ)))
μ/(βV
k
)
,
where
β
>0 was a constant,
E(X)=μ, S
k,i
=∑
k
j=
1
X
j
-X
i
,
1
≤i≤k, V
2
k
=∑
k
i=
1
(X
i
-μ)
2
.
The results generalize not only on the weigh of the almost sure central limit theorem but also in the range of random variables.
Type of dirichlet series on
p-
precise order in right half plane
WU Shi-gan
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 58-63. doi:
10.6040/j.issn.1671-9352.0.2015.138
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Discusses the relationship between the type(lower type)of holomorphic function represented by Dirichlet series and random Dirichlet series on
p
-precise order in right half plane with coefficient and index.
The operator theory on complete partially ordered objects in a topos
LU Tao, WANG Xi-juan, HE Wei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 64-71. doi:
10.6040/j.issn.1671-9352.0.2015.481
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The concept of the projective operators between complete partially ordered objects in a topos is investigated. The characterizations of the closure operator and the kernel operator in a topos are given, which are common generalizations of the classical lattice theory.
MacWilliams identity and self-dual codes of linear codes over Z
2
a
+uZ
2
a
SONG Xian-mei, XIONG Lei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 72-78. doi:
10.6040/j.issn.1671-9352.1.2015.005
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The linear codes over
R=Z
2
a
+uZ
2
a
with
u
2
=u
are discussed. MacWilliams identity for the complete Gray weight enumerator is investigated firstly. Then, the generator matrices of self-dual codes over
R
and the number of distinct self-dual codes of even length n over Z
2
3
+uZ
2
3
are given. The torsion codes over R are discussed and the generator matrices of torsion codes and the relationship between the torsion codes and the residue codes are also obtained.
Gorenstein injective objects in Abelian categories
CHENG Hai-xia, YIN Xiao-bin
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 79-84. doi:
10.6040/j.issn.1671-9352.0.2015.076
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Let A be an abelian category with enough injective objects and X a full subcategory of A. The definitions of X -Gorenstein injective object and Y
X
-Gorenstein injective object are given, where Y
X
={Y∈Ch(A)|Y is acyclic and Kerd
n
Y
∈X. Under certain conditions, these two Gorenstein injective objects are related in a nice way. In particular, if I(A)⊆X, X∈Ch(A)is Y
X
-Gorenstein injective if and only if X
i
is X -Gorenstein injective for each
i
, when X is a self-orthogonal class. Subsequently, the relationships between X -Gorenstein injective dimension and Y
X
-Gorenstein injective dimension are considered.
Radial distribution of Julia sets of derivatives of solutions to complex differential equations
ZHANG Guo-wei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 85-88. doi:
10.6040/j.issn.1671-9352.0.2015.188
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1573
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The radial distributions of Julia sets of derivatives of entire solutions to some complex differential equations are studied. We obtain their radial distributions include intervals, of which lower bound also can be found.
Gorenstein injective modules over upper triangular matrix Artin algebras
WANG Chao
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 89-93. doi:
10.6040/j.issn.1671-9352.0.2015.235
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1515
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Let Λ=(
A M
0
B
)be an Artin algebra. The definition of the cocompatible bimodule is introduced, and the category Ginj(Λ)of the finitely generated Gorenstein injective Λ-modules under the condition that
M
is a cocompatible
(A,B)
-bimodule are described.
New upper bound on the chromatic number of the square of a planar graph
ZHU Hai-yang, GU Yu, LÜ Xin-zhong
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 94-101. doi:
10.6040/j.issn.1671-9352.0.2014.594
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Let
V(G), E(G),
Δ
(G)
and
χ(G)
denote the vertex set, the edge set, the maximum degree and the chromatic number of a graph
G
, respectively. The square
G
2
of a graph
G
is defined such that
V(G
2
)=V(G),
and
uv∈E(G
2
)
if and only if 1
≤d
G
(u,v)≤
2
.
It is proved that
χ(G
2
)≤
Δ
(G)+
8 if
G
is a planar graph with Δ
(G)
≤6 and girth
g(G)
≥5.
〈,
(-overQ)
〉-fuzzy filter and its characterization of the non-commutative residual lattices on the interval sets
QIAO Xi-min, WU Hong-bo
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 102-107. doi:
10.6040/j.issn.1671-9352.0.2015.116
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Taking the thought of interval sets as the research tool, the theories of non-commutative residual lattices are discussed on the interval sets and filters. The concepts of non-commutative residual lattices are introduced on interval sets and fuzzy filters of non-commutative residual lattices on interval sets. The algebraic structure of the 〈,
(-overQ)
〉-fuzzy filters of non-commutative residual lattices is provided on interval sets, and several equivalent characterization are received. A detailed demonstration for the necessary and sufficient conditions of the representation theorem is given.
Metric structure in modal logic system S4
GONG Jia-an, WU Hong-bo
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 108-113. doi:
10.6040/j.issn.1671-9352.0.2015.135
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The theory of(
n
)truth degrees of formulas is proposed in modal logic system S4. Some properties of(
n
)truth are investigated in modal logic system S4. The theory of(
n
)similarity degrees among modal formulas is proposed and a pseudo-metric is defined thereform on the set of all formulas. The(
n
)modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic system S4.
Characterization of
L
-fuzzifying topological spaces
YANG Xiao-fei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 114-118. doi:
10.6040/j.issn.1671-9352.0.2014.486
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1323
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The concept of fuzzifying topological filter-convergence spaces is introduced, and then it is proved that fuzzifying topological filter-convergence spaces are categorically isomorphic to
L
-fuzzifying topological spaces. The relationship between fuzzifying topological filter-convergence spaces and other limit structures is established. Finally, it is shown that fuzzifying topological filter-convergence theory is compatible with fuzzifying topological net-convergence theory.
