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16 February 2016
Volume 51 Issue 2
α
skew Armendariz rings relative to a monoid
GENG Daohong, WANG Yao, REN Yanli
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 15. doi:
10.6040/j.issn.16719352.0.2015.213
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406
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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 Yanli
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 611. doi:
10.6040/j.issn.16719352.0.2015.150
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482
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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 nilsemicommutative 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 nilsemicommutative ring, then
R［x;α］
is a power series weak McCoy ring.
Nonsolvabla elements of semigroup P
_{Γ}
(Λ×Λ)of binary relations determined by the semilattice
Γ
on the set
Λ
LIN Pingfeng
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 1215. doi:
10.6040/j.issn.16719352.0.2015.060
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463
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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 nonsolvable elements is obtained,and a class of nonsolvable elements is found under certain conditions.
On algebras with all
τ
rigid modules projective
XIE Zongzhen, ZHANG Xiaojin
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 1620. doi:
10.6040/j.issn.16719352.0.2015.185
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463
)
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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 Amodules are projective.
Iterative nonoverlapping domain decomposition method for optimal boundary control problems governed by elliptic equations
LIU Wenyue, SUN Tongjun
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 2128. doi:
10.6040/j.issn.16719352.0.2015.190
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372
)
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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 nonoverlapping domain decomposition method is established. The whole domain is divided into many nonoverlapping 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 Chunyan, HAO Shengnan, FENG Lichao
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 2936. doi:
10.6040/j.issn.16719352.0.2015.271
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601
)
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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 onesided polynomial growth condition which may explode at a finite time, we introduce a polynomial stochastic noise
σx(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 energytransport model for semiconductors
DONG Jianwei, LOU Guangpu, WANG Yanping
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 3741. doi:
10.6040/j.issn.16719352.0.2015.146
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596
)
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A stationary simplified energytransport model for semiconductors is studied in a onedimensional 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 Lijun, GUO Mingle
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 4249. doi:
10.6040/j.issn.16719352.0.2015.151
Abstract
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398
)
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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 selfnormalized products of some partial sums of NA random variables
XU Feng, WU Qunying
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 5057. doi:
10.6040/j.issn.16719352.0.2015.106
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425
)
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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 selfnormalized products of some partial sums(∏
^{k}
_{i=}
1
(S
_{k,i}
/((k
1
)μ)))
^{μ/(βVk)}
,
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 Shigan
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 5863. doi:
10.6040/j.issn.16719352.0.2015.138
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460
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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 Xijuan, HE Wei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 6471. doi:
10.6040/j.issn.16719352.0.2015.481
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607
)
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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 selfdual codes of linear codes over Z
_{2a}
+uZ
_{2a}
SONG Xianmei, XIONG Lei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 7278. doi:
10.6040/j.issn.16719352.1.2015.005
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412
)
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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 selfdual codes over
R
and the number of distinct selfdual codes of even length n over Z
_{23}
+uZ
_{23}
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 Haixia, YIN Xiaobin
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 7984. doi:
10.6040/j.issn.16719352.0.2015.076
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480
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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 selforthogonal 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 Guowei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 8588. doi:
10.6040/j.issn.16719352.0.2015.188
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574
)
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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): 8993. doi:
10.6040/j.issn.16719352.0.2015.235
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(
535
)
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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 Λmodules 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 Haiyang, GU Yu, LÜ Xinzhong
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 94101. doi:
10.6040/j.issn.16719352.0.2014.594
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467
)
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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 noncommutative residual lattices on the interval sets
QIAO Ximin, WU Hongbo
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 102107. doi:
10.6040/j.issn.16719352.0.2015.116
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519
)
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Taking the thought of interval sets as the research tool, the theories of noncommutative residual lattices are discussed on the interval sets and filters. The concepts of noncommutative residual lattices are introduced on interval sets and fuzzy filters of noncommutative residual lattices on interval sets. The algebraic structure of the 〈,
(overQ)
〉fuzzy filters of noncommutative 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 Jiaan, WU Hongbo
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 108113. doi:
10.6040/j.issn.16719352.0.2015.135
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497
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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 pseudometric 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 Xiaofei
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(2): 114118. doi:
10.6040/j.issn.16719352.0.2014.486
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503
)
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The concept of fuzzifying topological filterconvergence spaces is introduced, and then it is proved that fuzzifying topological filterconvergence spaces are categorically isomorphic to
L
fuzzifying topological spaces. The relationship between fuzzifying topological filterconvergence spaces and other limit structures is established. Finally, it is shown that fuzzifying topological filterconvergence theory is compatible with fuzzifying topological netconvergence theory.