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20 June 2022
Volume 57 Issue 6
Crossed representation categories of multiplier Hopf
T
-coalgebras
LIU Hui-li, YANG Tao
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 1-7. doi:
10.6040/j.issn.1671-9352.0.2021.520
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Let A=⊕
p∈G
A
p
be a multiplier Hopf
T
-coalgebra over a group of
G
. Considering its crossed left
A-G
-modules, and show the crossed left
A-G
-module category is a monoidal category, and that a family of multipliers
R={R
p,q
∈M(A
p
⊗A
q
)}
p,q∈G
is a quasitriangular structure on
A
if and only if the crossed left
A-G
-module category over
A
is a braided monoidal category with the braiding
c
defined by
R
.
The
(m,n
)-cotorsion dimensions of modules and rings
WANG Ya-li, ZHAO Ren-yu
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 8-14. doi:
10.6040/j.issn.1671-9352.0.2021.413
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437
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Let
m, n
be two given positive integers. It is shown that over slightly(
m,n
)-coherent rings, the(
m,n
)-cotorsion dimensions of modules and rings share many nice properties as the classical homological dimensions. Some equivalent characterizations that slightly(
m,n
)-coherent rings are von-Neumann regular rings are given.
Greens relations on a class of semiring which multiplicative reduct is an idempotent semigroup
WANG Jun-ling, SHAO Yong
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 15-22. doi:
10.6040/j.issn.1671-9352.0.2021.135
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607
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The multiplicatively idempotent semirings satisfying the identities
x+x+x=x,
2
x+
2
y=
2
(x+y)
are studied. The characterizations of the binary relations(·overL)∧(+overD),(·overL)∧(+overL),(·overL)∧(+overR),(+overL)∧(·overD)related to the Greens relation of the multiplicative semigroups(additive semigroups)of the semirings are given, and the sufficient and necessary conditions which make these binary relations be congruences are obtained. Moreover, the classes of semirings which are determined by these congruences are proved to be semiring varieties.
Completions of
S
-semigroups
LIU Min, LI Yu-lin
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 23-30. doi:
10.6040/j.issn.1671-9352.0.2021.566
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667
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The concept of
S
-semigroup quantale completions for
S
-semigroups is introduced. It is proved that all
S
-semigroup quantale completions of an
S
-semigroup
SA can be fully characterized as compatible quotients of the power-set
S
-semigroup quantale corresponding to
S
A
. Three kinds of classical completion methods are given. Furthermore,
S
-algebra completions of double residuated
S
-semigroups are considered. The quotient with respect to the largest compatible nucleus on the down-set
S
-semigroup quantale is proved to be an
S
-algebra completion for an arbitrary double residuated
S
-semigroup.
Characterizations of monoids by
K
-separation property
WANG Ya-ting, QIAO Hu-sheng
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 31-35. doi:
10.6040/j.issn.1671-9352.0.2021.523
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551
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Let
S
be a monoid,
K
be an arbitrary non-empty subset of
S
. This paper studies the
K
-separation properties of Rees factors and amalgamated coproducts, gives the characterizations of
K
-separation property by pullback diagrams, and discusses the homological classification problem of
K
-separation property on free(projective, principally weakly flat)Rees factor
S
-acts.
Weighted neighbor toughness of graphs
WENG Ting-ting, WEI Zong-tian
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 36-43. doi:
10.6040/j.issn.1671-9352.0.2021.330
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554
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Introduced neighbor toughness into the weighted graph, and the concept of weighted neighbor toughness of the graph is proposed. On the basis of giving some basic graphs weighted neighbor toughness, the extreme value problem of weighted neighbor toughness of several types of graphs was focused. The results show that the parameter values are related to the structure of the graph, the size of the weights, and the way of weighting, so the invulnerability of the network can be described more accurately.
Degree-based graph entropy on graph operations
WU Chuan-shu, ZHAO Hai-xing, DENG Bo
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 44-53. doi:
10.6040/j.issn.1671-9352.0.2021.718
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740
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Graph invariants are widely used to construct entropy-based metrics to describe the structures of complex networks. In particular, graph entropy based on vertex degrees is often used to measure graph structure information after graph operations.The degree-based graph entropy calculation on some graph operations containing the symmetric difference, Cartesian product, tensor product, Corona product of graphs are presented. These results are applied to calculate the degree-based graph entropy of molecular graphs such as nano-structure and hypercubes.
Vertex-distinguishing Ⅰ-total colorings of
mC
14
ZHAO Ya-di, CHEN Xiang-en
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 54-60. doi:
10.6040/j.issn.1671-9352.0.2020.592
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645
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The problems of the optimal vertex-distinguishing Ⅰ-total colorings(VDITC)and the optimal vertex-distinguishing Ⅵ-total colorings(VDVITC)of graph
mC
14
are proved by constructing a matrix which is composed of color sets and empty set as the elements and using the methods of distributing color sets in advance and describing coloring explicitly. Thus vertex-distinguishing Ⅰ-total chromatic numbers and the vertex-distinguishing Ⅵ-total chromatic numbers of graph
mC
14
are determined. The results show that the VDITC conjecture and VDVITC conjecture are valid for graph
mC
14
.
Total colorings of 3-degenerate graphs
YANG Teng-fei, XU Chang-qing
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 61-63. doi:
10.6040/j.issn.1671-9352.0.2020.556
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678
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A
k
-total coloring of a graph
G
is a coloring of the vertices and edges with
k
colors, such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph
G
is the smallest integer
k
such that
G
has a
k
-total coloring, denoted by
χ″(G)
. Behzad and Vizing independently proposed the Total Coloring Conjecture that
χ″(G)≤Δ(G)+
2 for any graph
G
. In this paper, it is proved that the 3-degenerate graph with
Δ(G)
≥5 satisfies the total coloring Conjecture.
Simple 3-designs of PSL(2,2
n)with block size 9
WEI Meng-meng, LI Wei-xia, XING Jian-min
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 64-73. doi:
10.6040/j.issn.1671-9352.0.2021.565
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The number of the orbits of 9-subsets is determined by the action of PSL(2,2
n)on 9-subsets of the projective line. Consequently, all simple 3-designs with block size 9 are obtained, under the action of PSL(2,2
n).
Optimal scale reduction based on graph theory in consistent multi-scale decision tables
JIN Ming, CHEN Jin-kun
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 74-83. doi:
10.6040/j.issn.1671-9352.0.2021.691
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Multi-scale decision table is a model based on real-world data with multi-scale background. It is a difficult problem that how to carry out optimal scale reduction of multi-scale decision tables. By constructing the identification matrix on multi-scale, the properties of the identification matrix are put forward, and the relationship between the matrix and the optimal scale reduction is given. A fast algorithm for optimal scale reduction is presented by combining the identification matrix with graph theory. Finally, the effectiveness of the proposed algorithm is verified via numerical experiments.
Asymptotic behavior of a stochastic SIQS model with Markov switching
WANG Yan-mei, LIU Gui-rong
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 84-93. doi:
10.6040/j.issn.1671-9352.0.2021.793
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A stochastic SIQS epidemic model with Markovian switching and saturated incidence is investigated. First, the existence and uniqueness of the global positive solution of the model are proved by constructing the suitable Lyapunov functions. Then by using the ergodic property of Markov chains, the sufficient conditions of extinction and persistence in the mean of the disease are obtained. At last, the theoretical results are verified by numerical simulations. The results show that if one of the subsystems is stochastically persistent, and another is stochastically extinct, then the hybrid system may be either stochastically extinct or persistent, and the result depends on the probability that the Markov chain remains in each status. Telegraph noise has a major impact on disease transmission. It can be seen that isolation has an inhibitory effect on disease transmission, so the isolation of infected individuals is more helpful to control the spread of the disease.
Generalization of Gronwalls inequality and applications
WANG Xiao-huan, LÜ Guang-ying, DAI Li-jie
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 94-101. doi:
10.6040/j.issn.1671-9352.0.2021.451
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The classical Gronwalls inequality is generalized and a Gronwalls inequality with multiple singularity points is obtained. Then the existence and uniqueness of solutions to stochastic fractional order differential equations is obtained by using the obtained inequality.
Reconstruction of rational polynomial Coons surface patches throuth Bézier triangular geodesic
WANG Shu-juan, YANG Huo-gen, CHAI Ying
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2022, 57(6): 102-110. doi:
10.6040/j.issn.1671-9352.0.2021.613
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474
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For the quintic Bézier triangular curve that satisfies certain constraints, four rational polynomial Coons surfaces which interpolate the triangular curve as the geodesics is proposed. Firstly, the corner compatibility constraints of the cross-boundary tangent vector of the interpolated surface along the boundary geodesics are analyzed. Secondly, based on the rational Hermite polynomial basis represented by the barycentric coordinate, the interpolation operator for interpolating two adjacent boundary geodesics is designed. Finally, the construction scheme of rational polynomial Coons surface for interpolating triangular geodesics is presented. The construction algorithm of Coons surface through Bézier triangular geodesics proposed in this paper is simple and easy to implement, and the calculation results show the feasibility of the algorithm.
