#### Table of Content

16 June 2008
Volume 43 Issue 6
Articles
The ∑Γfuzzy truth degree of formula relative to the finite theory in propositional logic system Ln
WU Hong-bo,QIAO Xi-min,
J4. 2008, 43(6):  1-8 .  doi:
Abstract ( 1218 )   PDF (263KB) ( 743 )
Related Articles | Metrics
Abstract: The theory of ∑-α-tautologies of fuzzy propositional logic was combined with the theory of truth degree in metrology of logic introduced by professor G.J.Wang, and the theory of ∑Γfuzzy truth degrees of formula relative to the finite theory in propositional logic system n was introduced. By employing the theory of ∑Γfuzzy truth degree, the concepts of Γ-pseudometric on F(S) was proposed in the propositional logic system Ln. The results obtained can complement and enhance the original theory of metrology of logic, and can give a new frame for fuzzy reasoning study.
Fractional chromatic number of one class of distance graphs
LIU Xi-kui,ZHAO Bing-qing,WANG Cai-hong,XUE Sheng-wei
J4. 2008, 43(6):  9-11 .  doi:
Abstract ( 1083 )   PDF (175KB) ( 748 )
Related Articles | Metrics
The fractional chromatic number of the distance graph G(Z,Dm,k,k+1,k+2,k+3) (where Dm,k,k+1,k+2,k+3=｛1,2,…,m｝-｛k,k+1,k+2,k+3｝) and its chromatic number for some pairs of integers, when 2k≤m≤2k+5 were discussed.
The contractible edges of the longest cycle in some 5-connected graphs
YANG Zhao-xia
J4. 2008, 43(6):  12-14 .  doi:
Abstract ( 803 )   PDF (171KB) ( 759 )
Related Articles | Metrics
Abstract: The distribution of the contractible edges on some longest cycles of some 5-connected graphs was given. The results show that some of the longest cycles of some 5-connected graphs have at least two contractible edges.
A partition of bipartite graphs with 4-cycle and 8-cycle
LI Feng,GENG Jian-yan,LI Shuo,LIANG Feng
J4. 2008, 43(6):  15-18 .  doi:
Abstract ( 1058 )   PDF (213KB) ( 861 )
Related Articles | Metrics
It was proved that if G is a balanced bipartite graph of order 4k satisfying d(x)+d(y)≥2k+1 for any two nonadjacent vertices x∈V1, y∈V2, then G contains k-2 4-cycles and one 8-cycle such that all the k-1 cycles are vertex-disjoint.
The heterochromatic girth in edge-colored graphs
LI Shuo,LI Feng,LIANG Feng
J4. 2008, 43(6):  19-20 .  doi:
Abstract ( 992 )   PDF (172KB) ( 577 )
Related Articles | Metrics
Let G be an edgecolored graph with order n, n≥3. It is proved that if for each vof G, dc(v)≥n/(2-α), where α=α=3/(s-3)ln(2+7/3), and s is an integer greater than 3, then gH(G)≤s.
Equitable coloring of planar graphs without 4,5,6-cycles
XIAO Yu-liang,MA Shuai,WU Jian-liang
J4. 2008, 43(6):  21-24 .  doi:
Abstract ( 1074 )   PDF (197KB) ( 820 )
Related Articles | Metrics
A proper vertex-coloring Φ of a graph G is called an equitable coloring of G if the numbers of vertices in any two-color class differ with each other at most one. Then any planar graph G without 4,5,6-cycles and with Δ≥9 is equitable Δ-colorable.
Fuzzy-quotient Lie super-algebras
Chen Wen-juan
J4. 2008, 43(6):  25-27 .  doi:
Abstract ( 999 )   PDF (160KB) ( 612 )
Related Articles | Metrics
The fuzzy-quotient Lie superalgebra was defined by using a fuzzy ideal. The result shows the fuzzy-quotient Lie superalgebra is Lie super-algebra. Also, the fuzzy subset of fuzzy-quotient Lie super-algebra was discussed.
Convergence theorems for generalized g-expectations
LIN Qian,SHI Yu-feng
J4. 2008, 43(6):  28-30 .  doi:
Abstract ( 1447 )   PDF (196KB) ( 820 )
Related Articles | Metrics
Some properties of the generalized g-expectations defined in l1(Ω, FT, P) were studied. Also, monotonic convergenle theorem, Fatou Lemma, and dominated convergence theorem for the generalized g-expectations were obtained.
A Turan theorem relating to the spectral radius of a graph
FENG Li-hua,YU Gui-hai
J4. 2008, 43(6):  31-33 .  doi:
Abstract ( 1321 )   PDF (152KB) ( 660 )
Related Articles | Metrics
Suppose G is a graph on n vertices. If　ρ(G)≤ρ(Tn,t) 　for some t, then e(G)≤e(Tn,t) where Tn,t is the Turan graph with k parts, which partially supports a problem raised by Nikiforov.
Mean value estimate for the greatest divisors of N which is prime to a certain integer(Ⅱ)
GUO Ru-ting
J4. 2008, 43(6):  34-36 .  doi:
Abstract ( 836 )   PDF (167KB) ( 711 )
Related Articles | Metrics
For a fixed natural squarefree number k, the mean value of the greatest divisor of N which is prime to k was studied.
Ordering of the largest eigenvalues of trees
ZHANG Sheng-li,PAN Zheng-hua,FENG Shan-zhuang
J4. 2008, 43(6):  37-43 .  doi:
Abstract ( 1351 )   PDF (263KB) ( 674 )
Related Articles | Metrics
Let Tn be a set of trees with n vertices. In this ordering, the largest eigenvalues of trees in Tnand their ordering were further investigated, and the fifteenth to thirty-third values and the corresponding trees were obtained.
Computational method for the generalized KdV equation
ZUO Jin-ming
J4. 2008, 43(6):  44-48 .  doi:
Abstract ( 1038 )   PDF (187KB) ( 629 )
Related Articles | Metrics
omputational method based on a linearized implicit scheme was proposed for the solution of the generalized Kortewegde Vries (KdV) equation.An important advantage to be gained from the linearized implicit method is unconditional stable.Numerical results portraying a single linesoliton solution and the interaction of two linesolitions were reported for the generalized KdV equation. The results show that this method has good stability and accuracy.
The solutions of boundary value problems for second order integro-differential equations in Banach spaces
DONG Li-li,LIU Yan-sheng
J4. 2008, 43(6):  49-52 .  doi:
Abstract ( 1158 )   PDF (196KB) ( 916 )
Related Articles | Metrics
By using a new comparison result and only a lower or an upper solution, the maximal or the minimal solutions of the boundary value problems for second order integro-differential equations in Banach spaces were obtained.
Existence of three solutions for some second-order four-point boundary value problems
HUANG Yu-mei,GAO De-zhi,QIN Wei,DONG Xin
J4. 2008, 43(6):  53-56 .  doi:
Abstract ( 1040 )   PDF (178KB) ( 718 )
Related Articles | Metrics
he second-order four-point boundary value problem -x″(t)=f(t,x(t),x′(t)), t∈I=［0,1］;x(0)=ax(ξ), x(1)=bx(η)　was studied, where　0<ξ<η<1,0≤a,b≤1, and f: ［0,1］×［0,∞］→［0,∞］　are non-negative continuous functions. Some degree theory arguments were used to get the multiplicity result.
The fixed point of decreasing operators in Banach space
ZHENG Yan,PANG Xin-qin
J4. 2008, 43(6):  57-60 .  doi:
Abstract ( 1018 )   PDF (187KB) ( 971 )
Related Articles | Metrics
In four different ways, in partial ordering Banach space, the existence and uniqueness of solutions for decreasing operators were discussed, and the convergent iterative sequence was also given. Some known results were improved.
Wreath product structure of the left C-wrpp semi-groups
ZHANG Xiao-min
J4. 2008, 43(6):  61-63 .  doi:
Abstract ( 1226 )   PDF (184KB) ( 577 )
Related Articles | Metrics
The curler structure of left C-wrpp semi-groups as a generalization of C-wrpp semi-groups has already been established. Another structure of left C-wrpp semi-groups was investigated by means of the wreath product introduced by Nuemann, and the wreath product structure of left C-wrpp semi-groups was obtained. This result further enriches the theory on left C-wrpp semi-groups.
(Strongly)Relative semi-compactness in L-topological spaces
HAN Hong-xia
J4. 2008, 43(6):  64-67 .  doi:
Abstract ( 984 )   PDF (164KB) ( 486 )
Related Articles | Metrics
The notions of relative semi-compactness and strongly relative semi-compactness were introduced in L-topological spaces, and some of their properties were studied.
A modified super-memory gradient method with angle property
SUN Min
J4. 2008, 43(6):  68-70 .  doi:
Abstract ( 875 )   PDF (199KB) ( 726 )
Related Articles | Metrics
A modified super-memory gradient method was proposed. The method has the following two attractive properties, one is that the iterative direction is always a sufficiently descent direction without utilizing the line search, the other is that the iterative direction always satisfies the angle property. Under mild conditions, the global convergence of the method was proved. The preliminary numerical results show the efficiency of this method.
The reverse order law for (T,S,2)-inverse of a matrix semi-tensor product
SONG Cai-qin,ZHAO Jian-li,LI Dong-fang
J4. 2008, 43(6):  71-76 .  doi:
Abstract ( 910 )   PDF (204KB) ( 753 )
Related Articles | Metrics
The necessary and sufficient condition for the validity of the reverse order law for (T, S, 2)-inverse of a matrix semitensor product was given. In addition, the equality (A⊙B)+MP=(A+MN(A⊙B))+NP((A⊙B)(B+NPIp))+MN was proved.
The Gnedenko system attended by a repairman with simple vacations
CAO Jing,YUE De-quan
J4. 2008, 43(6):  77-82 .  doi:
Abstract ( 1219 )   PDF (241KB) ( 980 )
Related Articles | Metrics
The Gnedenko system attended by a repairman with simple vacations was studied. It was assumed that the life of the operating unit and the standby, and the vacation time of the repairman were all exponential distributions, while the repair time of the unit had a general continuous distribution. By using the supplementary variables approach and generalized Markov process method, the Laplace transform of the reliability and the mean time to the first failure were attained. Meanwhile, the availability and the failure frequency of the system were obtained.
Free product of left quasi-normal bands
BI Xiao-dong
J4. 2008, 43(6):  83-86 .  doi:
Abstract ( 1020 )   PDF (180KB) ( 662 )
Related Articles | Metrics
It is proved that the maximal left quasi-normal band homomorphic image of the free product of right quasi-normal bands in the semi-group categrory is isomorphic to their free product in the left quasi-normal band category. Then, the existence of the free product in left quasinormal band was proved. Finally, the concept of commutative free product was given, and the relationship between semi-lattice free product and commutative free product was examined.
Complete convergence of weighted sums for arrays of row-wise ρ--mixing random variables
TAN Cheng-liang,WU Qun-ying,TIAN Guo-hua
J4. 2008, 43(6):  87-91 .  doi:
Abstract ( 1018 )   PDF (206KB) ( 587 )
Related Articles | Metrics
Using a Rosenthaltype inequality of partial sums for ρ-mixing random variables, the complete convergence of weighted sums for arrays of rowwise ρ--mixing random variables was studied. The results obtained can extend the corresponding results of arrays of row-wise independent random variables, and some complete convergence corollaries for weighted sums for arrays of row-wise NA and ρ*-mixing random variables.
An economic growth model of optimization and allocation with endogenous utility index of lands
JI Rong-fang,GUO Zhong-jin
J4. 2008, 43(6):  92-96 .  doi:
Abstract ( 977 )   PDF (228KB) ( 757 )
Related Articles | Metrics

By integrating both population growths with environmental factor, utility index of land resources, and production amounts of society, and introducing a utility function with both consumption and utility index of land resources was introduced, and the optimization problem with infinite level was also presented. A three-dimension dynamic system was obtained by solving the optimization problem. It is proved that this system has unique non-zero positive equilibrium, which is a saddle.