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The proof of Ushio's conjecture concerning path factorization of complete bipartite graphs

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Science china mathematics 2006年第3期49卷 289-299页

作者： du beiliang & wang jian department of mathematics, suzhou university, suzhou 215006, china Nantong Vocational College, Nantong 226007, china department of mathematics Suzhou University Suzhou 215006 China Nantong Vocational College Nantong 226007 China

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, wang and Ushio gave a necessary and sufficient condition for existence of Pv-factorization of Km,n. When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio's conjecture is true when v = 4k - 1. In this paper we shall show that Ushio Conjecture is true when v = 4k + 1, and then Ushio's conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P4k+1-factorization of Km,n is (i) 2km ≤ (2k + 1)n,(ii) 2kn ≤ (2k + 1)m, (iii) m + n ≡ 0 (mod 4k + 1), (iv) (4k + 1)mn/[4k(m + n)] is an integer.

关键词： complete bipartite graph, factorization, Ushio Conjecture.

P_(4k-1)-factorization of bipartite multigraphs

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Science china mathematics 2006年第7期49卷 961-970页

作者： wang jian & du beiliang Nantong Vocational College, Nantong 226007, china department of mathematics, suzhou university, suzhou 215006, china 1. Vocational College Nantong 226007 China 2. department of mathematics Suzhou University Suzhou 215006 China

LetλKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pν-factorization ofλKm,n is a set of edge-disjoint Pν-factors ofλKm,n which partition the set of edges ofλKm,n. Whenνis an even number, Ushio, wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pν-factorization ofλKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true forν= 3. In this paper we will show that the conjecture is true whenν= 4k-1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization ofλKm,n is (1) (2k-1)m≤2kn, (2) (2k-1)n≤2km, (3)m + n = 0 (mod 4k-1), (4)λ(4k-1)mn/[2(2k-1)(m + n)] is an integer.

关键词： bipartite multigraph, factor, factorization.

K_(p,q)-factorization of complete bipartite graphs

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Science china mathematics 2004年第3期47卷 473-479页

作者： du beiliang wang jian department of mathematics, suzhou university, suzhou 215006, china Nantong Vocational College, Nantong 226007, china 1. department of mathematics Suzhou University 215006 Suzhou China 2. Nantong Vocational College 226007 Nantong China

Let Km,n be a completebipartite graph with two partite sets having m and n vertices,respectively. A Kp,q-factorization of Km,n is a set ofedge-disjoint Kp,q-factors of Km,n which partition theset of edges of Km,n. When p=1 and q is a prime number,wang, in his paper 'On K1,k-factorizations of a completebipartite graph' (Discrete Math, 1994, 126: 359-364),investigated the K1,q-factorization of Km,n and gave asufficient condition for such a factorization to exist. In the paper'K1,k-factorizations of complete bipartite graphs' (DiscreteMath, 2002, 259: 301-306), du and wang extended wang's resultto the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have aKp,q-factorization. As a special case, it is shown that theMartin's BAC conjecture is true when p:q=k:(k+1) for any positiveinteger k.

关键词： complete bipartite graph factorization HUBMFS 2 scheme

A RANDOM FUNCTIONAL CENTRAL LIMIT THEOREM FOR PROCESSES OF PROduCT SUMS OF LINEAR PROCESSES GENERATED BY MARTINGALE DIFFERENCES

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Chinese Annals of mathematics,Series B 2003年第4期24卷 449-456页

作者： wang YUEBAO YANG YANG ZHOU HAIYANG department of mathematics, suzhou university, suzhou 215006, jiangsu, china. department of mathematics, suzhou university, suzhou 215006, jiangsu, china. suzhou Univ Dept Math Suzhou 215006 Peoples R China

A random functional central limit theorem is obtained for processes of partial sums andproduct sums of linear processes generated by non-stationary martingale differences. It devel-ops and improves some corresponding results on processes of partial sums of linear processesgenerated by strictly stationary martingale differences, which can be found in [5].

关键词： Ilandom functional central limit theorem Processes of product sums

The spectrum of path factorization of bipartite multigraphs

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Science china mathematics 2007年第7期50卷 1045-1054页

作者： jian wang~1 Bei-liang du~(2+) 1 Nantong Vocational College,Nantong 226007,china 2 department of mathematics,suzhou university,suzhou 215006,china 1. Nantong Vocational College Nantong 226007 China 2. department of mathematics Suzhou University Suzhou 215006 China

Let λK m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P v-factorization of λK m,n is a set of edge-disjoint P v-factors of λK m,n which partition the set of edges of λK m,n . When v is an even number, Ushio, wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P v-factorization of λK m,n . When v is an odd number, we have proposed a conjecture. Very recently, we have proved that the conjecture is true when v = 4k ? 1. In this paper we shall show that the conjecture is true when v = 4k + 1, and then the conjecture is true. That is, we will prove that the necessary and sufficient conditions for the existence of a P 4k+1-factorization of λK m,n are (1) 2km ? (2k + 1)n, (2) 2kn ? (2k + 1)m, (3) m + n ≡ 0 (mod 4k + 1), (4) λ(4k + 1)mn/[4k(m + n)] is an integer.

关键词： bipartite multigraph factorization 05B30 05C70

Identifying the Active Site of Water Oxidation Catalyst Based on Defective Fe-Doped Ni Oxyhydroxide

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CCS Chemistry 2024年第2期6卷 365-376页

作者： Yunxuan Ding jian du Tao wang Licheng Sun Institute of Natural Sciences Westlake Institute for Advanced StudyHangzhou 310024 Center of Artificial Photosynthesis for Solar Fuels Department of ChemistrySchool of ScienceWestlake UniversityHangzhou 310024

Electrochemical water splitting plays a crucial role in storing sustainable energy into H2 as an ideal fuel,in which the anodic oxygen evolution reaction(OER)is the ***,Fe-doped NiOOH(Ni_(1−x)Fe_(x)OOH)is a promising OER catalyst under alkaline ***,the specific active sites responsible for their good performances remain ***,we identify a nickel-oxygen radical(Ni–O·)and an iron-oxo(Fe=O)group as two essential states for adsorbed ***,both thermochemical O–O coupling and electrochemical OER elementary steps are considered on the Ni_(1−x)Fe_(x)OOH *** mechanistic studies with a combined density functional theory simulation and experimental analysis unravel that both the Ni and Fe centers can act as the active sites for the defective Ni_(1−x)Fe_(x)OOH to catalyze OER,depending on the nature of the *** presence of Fe can benefit the stabilization of the defective surface and influence the electronic property of the nearby Ni *** work not only identifies the intrinsic active sites of Ni_(1−x)Fe_(x)OOH,but also illustrates the essential roles of defects in catalysis,which sheds light on the design of more efficient OER catalysts in the future.

关键词： defect water nucleophilic attack oxygen evolution reaction heterogeneous catalysis density functional calculations

Grammatical Complexity of One-DimensionalMaps with Mutiple Monotone Intervals

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Communications in Nonlinear Science and Numerical Simulation 1999年第3期 3-5页

作者： Yi wang(department of mathematics, suzhou university, suzhou 215006, china)

***-studiedmodelandattractedtheattentionbeforeearlyof1980’***,NlilnorandThurston[ljpioneeredthestudyofpiecewisemonotonemaps(whichcontainstheunimodalmapsasaspecialcase)anddevelopeda... 详细信息

***-studiedmodelandattractedtheattentionbeforeearlyof1980’***,NlilnorandThurston[ljpioneeredthestudyofpiecewisemonotonemaps(whichcontainstheunimodalmapsasaspecialcase)anddevelopeda...

关键词： piecewise monotonic map kneading sequence formal language grammatical complexity longest prefix-suffix

Some Results for Classical Risk Process with Stochastic Return on Investments

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Acta Mathematicae Applicatae Sinica 2002年第4期18卷 685-692页

作者： Guo-jing wang, Rong Wudepartment of mathematics, suzhou university, suzhou 215006, china department of mathematics, Nankai Univercity, Tianjin 300071, china department of mathematics Suzhou University Suzhou 215006 China department of mathematics Nankai Univercity Tianjin 300071 China

In this paper, we discuss the classical risk process with stochastic return on investment. We prove some properties of the ruin probability, the supremum distribution before ruin and the surplus distribution at the ti... 详细信息

关键词： Ruin probability supremum distribution before ruin surplus distribution at the time of ruin integro-differential equation

Synthesis and Structural Characterization of [(h^5-C_5Me_5)_2Mo_2Fe_2S_4Br_2]

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Chinese Journal of Structural Chemistry 2002年第1期21卷 34-37页

作者： YU Hong JI Shun-Jun Lang jian-Ping② (department of Chemistry and Chemical Engineering, suzhou university, suzhou 215006, china) department of Chemistry and Chemical Engineering Suzhou University Suzhou 215006

The reactions of [(h5-C5Me5)Mo(StBu)3] with equimolar FeBr3 in THF afforded a neutral tetranuclear cluster [(h5-C5Me5)2Mo2Fe2S4Br2] 1. The structure of 1 was determined by single-crystal X-ray diffraction analysis. 1 crystallizes in the monoclinic space group P21/n (No. 14) with a = 9.691(2), b = 15.204(2), c = 18.850(2) ? b = 96.44(1), V = 2759.8(7) ?, Z = 4, Mr = 862.08, Dc = 2.075 g/cm3, F(000) = 1680, m (MoKa) = 51.31 cm-1 and T = 296.2 K. With the use of 3667 observed reflections (I > 3.0s(I)), the structure was refined to R = 0.041 and Rw = 0.051. 1 is composed of a distorted Mo2Fe2S4 cubane core with two Br atoms and two h5-C5Me5 groups coordinating at the Fe and Mo sites, respectively. The lengths of MoMo and FeFe bonds are 2.826 and 2.773 ? respectively the mean length of MoFe bond is 2.751 ?

关键词： synthesis crystal structure molybdenum cluster iron cluster