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 1. A Sufficient Condition for the Genus of an Annulus Sum of Two 3-manifolds to Be Non-degenerate Li Feng-ling    Lei Feng-chun《东北数学》,2010年第26卷第1期 Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2. 2. Heegaard splittings 的曲面连通和 张明星《数学研究及应用》,2009年第29卷第3期 Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures. 3. 平环和的Heegaard亏格的一个下界 李风玲  雷逢春《数学研究及应用》,2011年第31卷第4期 Let Mi, i = 1,2, be a compact orientable 3-manifold, and Ai an incompressible annulus on a component Fi of OMi. Suppose A1 is separating on F1 and A2 is non-separating on F2. Let M be the annulus sum of M1 and M2 along A1 and A2. In the present paper, we give a lower bound for the genus of the annulus sum M in the condition of the Heegaard distances of the submanifolds M1 and M2 4. 沿着不连通曲面的Heegaard分解的不可稳定化的融合积 高绪涛  郭启龙《数学研究及应用》,2013年第33卷第4期 Let M be a 3-manifold, F= {F1 , F2 , . . . , Fn } be a collection of essential closed surfaces in M (for any i, j ∈ {1, ..., n}, ifi≠j, Fi is not parallel to Fj and Fi ∩Fj = φ) and0 M be a collection of components of M. Suppose M-UFi ∈FFi×(-1, 1) contains k components M1 , M2 , . . . , Mk . If each M i has a Heegaard splitting ViUSiWi with d(Si) > 4(g(M1 ) + ··· + g(Mk )), then any minimal Heegaard splitting of M relative to 0M is obtained by doing amalgamations and self-amalgamations from minimal Heegaard splittings or -stabilization of minimal Heegaard splittings of M1 , M2 , . . . , Mk . 5. Topologically minimal surfaces versus self-amalgamated Heegaard surfaces Qiang E  FengChun Lei《中国科学 数学(英文版)》,2014年第57卷第11期 Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface. 6. A Note on Heegaard Genus of Self-amalgamated3-Manifold Qilong GUO  Ruifeng QIU  Yanqing ZOU《数学年刊B辑(英文版)》,2015年第36卷第1期 Let M be a connected orientable compact irreducible 3-manifold. Suppose that ?M consists of two homeomorphic surfaces F1 and F2, and both F1 and F2 are compressible in M. Suppose furthermore that g(M,F1) = g(M) + g(F1), where g(M,F1) is the Heegaard genus of M relative to F1. Let Mf be the closed orientable 3-manifold obtained by identifying F1 and F2 using a homeomorphism f : F1 → F2. The authors show that if f is sufficiently complicated, then g(Mf) = g(M,?M) + 1. 7. Geometry of rectangular block triangular matrices Li Ping Huang  Su Wen Zou《数学学报(英文版)》,2009年第25卷第12期 Let D be any division ring, and let T（mi,ni,k） be the set of k × k （k ≥ 2） rectangular block triangular matrices over D. For A, B ∈ T（mi,ni,k）, if rank（A - B） = 1, then A and B are said to be adjacent and denoted by A -B. A map T ： T（mi,ni,k） -〉 T（mi,ni,k） is said to be an adjacency preserving map in both directions if A - B if and only if φ（A） φ（B）. Let G be the transformation group of all adjacency preserving bijections in both directions on T（mi,ni,k）. When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D. 8. A SOLUTION TO ONE OF KANO'S CONJECTURES CONCERNING k-th MAXIMAL SPANNING TREES 刘振宏《应用数学学报(英文版)》,1988年第3期 Let (G) be the collection of all spanning trees of a connected and weighted graph G,and F_1, F_2,…,F_m the partition of (G) such that F_i is the set of i-th maximal spanning trees of G.Kano conjectured that for any A∈F_1 and every integer k,1≤k≤m,there exists T∈F_k such that|T/A| k—l.This paper gives the conjecture a very simple proof,and related results. 9. 大距离Heegaard分解的三穿孔球面和 王树新  倪楠《数学研究及应用》,2014年第34卷第2期 设\$M_i~(i=1,2)\$是一个紧致可定向的三维流形, \$F_i\$是\$M_i\$边界上的一个不可压缩曲面, \$M=M_{1}\cup_{f}M_{2}\$, 其中\$f\$是\$F_1\$到\$F_2\$一个同胚,对于具有特定条件的相粘曲面\$F_i\$, 如果\$M_i\$具有一个Heegaard距离至少是\$2(g(M_1)+g(M_2))+1\$的Heegaard分解,则\$g(M)=g(M_1)+g(M_2)\$. 10. Powers of general digital sums Feng Juan Chen《数学学报(英文版)》,2010年第26卷第6期 Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi（0）=0.write S（n）=i=0∑tfi（ai）.In this paper, we give the asymptotic formula for x^-1∑n≤xS（n）^k,where k is a positive integer. 11. TWO-STAGE MULTISPLITTING OF SYMMETRIC POSITIVE SEMIDEFINITE MATRICES 《高等学校计算数学学报(英文版)》,2000年第Z1期 Main resultsTheorem 1 Let A be symmetric positive semidefinite.Let (?) be a diagonally compen-sated reduced matrix of A and Let (?)=σI+(?)(σ>0) be a modiffication(Stieltjes) matrixof (?).Let the splitting (?)=M-(?) be regular and M=F-G be weak regular,where M andF are symmetric positive definite matrices.Then the resulting two-stage method corre-sponding to the diagonally compensated reduced splitting A=M-N and inner splitting M=F-G is convergent for any number μ≥1 of inner iterations.Furthermore,the iteration 12. 与微分多项式分担值相关的正规定则(英文) 邓炳茂  雷春林  杨德贵《数学季刊》,2013年第1期 Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D. 13. CODIMENSION 1 IMMERSIONS OF THE CIRCLE AND SURFACES 李邦河《数学物理学报(B辑英文版)》,1988年第4期 Let f: S~1→M~2 or M~2→N~3 be a map, we classify the immersions homotopic to f, where M~2 and N~3 are arbitrary surface and 3-manifolds respectively. 14. A Lower Bound of the Genus of a Self-amalgamated 3-manifolds LI Xu    Lei Feng-chun《东北数学》,2011年第27卷第1期 Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h ： Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M＇ has a Heegaard splitting with sufficiently high distance. 15. ON MAXIMAL MATCHINGS OF CONNECTED GRAPHS 许宝刚《数学物理学报(B辑英文版)》,2004年第24卷第4期 Let I with |I| = k be a matching of a graph G (briefly, I is called a k-matching). If I is not a proper subset of any other matching of G, then I is a maximal k-matching and m(gk, G) is used to denote the number of maximal k-matchings of G. Let gk be a k-matching of G, if there exists a subset {e1, e2,…, ei} of E(G) \ gk, i (?)1, such that (1) for any j ∈ {1, 2,…,i}, gk + {ej} is a (k + l)-matching of G; (2) for any f ∈ E(G) \ (gk ∪ {e1,e2,…,ei}), gk + {f} is not a matching of G; then gk, is called an i wings k-matching of G and mi(gk,G) is used to denote the number of i wings k-matchings of G. In this paper, it is proved that both mi(gk,G) and m(gk,G) are edge reconstructible for every connected graph G, and as a corollary, it is shown that the matching polynomial is edge reconstructible. 16. Diameter preserving surjection on alternate matrices Li Ping Huang《数学学报(英文版)》,2009年第25卷第9期 Let F be a field with ｜F｜ ≥ 3, Km be the set of all m × m （m ≥ 4） alternate matrices over F. The arithmetic distance of A, B ∈ Km is d（A, B） ：= rank（A - B）. If d（A, B） = 2, then A and B are said to be adjacent. The diameter of Km is max{d（A, B） ： A, B ∈ km}. Assume that φ ： Km→Km is a map. We prove the following are equivalent： （a） φ is a diameter preserving surjection in both directions, （b） φ is both an adjacency preserving surjection and a diameter preserving map, （c） φ is a bijective map which preserves the arithmetic distance. 17. 关于不变集的一个注记 瞿成勤  苏维宜  等《东北数学》,2000年第16卷第4期 § 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＲｎｂｅｎｄｉｍｅｎｓｉｏｎａｌＥｕｃｌｉｄｅａｎｓｐａｃｅ,ａｎｄ (ｆ1,… ,ｆｍ,Ｒｎ)ａｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔｉｏｎｓｙｓｔｅｍ .Ｉｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｔｈｅｒｅｅｘｉｓｔｓａｕｎｉｑｕｅｎｏｎ ｅｍｐｔｙｃｏｍｐａｃｔｓｅｔＥｓｕｃｈｔｈａｔＥ =∪ｍｉ=1 ｆｉ(Ｅ) .ＷｅｃａｌｌｔｈｅｓｅｔＥｉｎｖａｒｉｎｔｓｅｔｆｏｒ (ｆ1,… ,ｆｍ,Ｒｎ) .　Ｌｅｔ (ｆ1,… ,ｆｍ,Ｒｎ)ｂｅａｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔ… 18. Diffeomorphisms with C 1-stably average shadowing Manseob Lee  Xiao Wen《数学学报(英文版)》,2013年第29卷第1期 Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting. 19. （g, f）-Factorizations Randomly Orthogonal to a Subgraph in Graphs HaoZHAO GuiZhenLIU XiaoXiaYAN《数学学报(英文版)》,2005年第21卷第2期 Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integervalued functions defined on V(G) such that 2k - 2 ≤g(x)≤f(x) for all x∈V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg m-1,mf-m 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions. 20. 循环群\$Z_{2}\$的模向量不变式 南基洙  赵辉芳《数学研究及应用》,2011年第31卷第6期 Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.