Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

K(a)hler Finsler Metrics Are Actually Strongly K(a)hler

In this paper,the K(a)hler conditions of the Chern-Finsler connection in complex Finsler geometry are studied,and it is proved that K(a)hler Finsler metrics are actually strongly K(a)hler.     

Automorphisms of Ehresmann connections

We prove that a diffeomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesies, considered as parametrized curves, to geodesies) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040.     

Nickel-catalyzed multi-component connection reaction of isoprene, aldimines (lactamines), and diphenylzinc

Ni(acac)₂ catalyzes the four-component connection reaction of diphenylzinc, isoprene, aromatic aldehydes, and aromatic amines in this order and provides stereochemically homogeneous (E)-1-arylamino-1-aryl-3-methyl-5-phenyl-3-pentenes (1) in excellent yields. Aliphatic aldehydes react similarly and give (E)-1-arylamino-1-alkyl-3-methyl-5-phenyl-3-pentenes (1) in slightly reduced yields. When the alkyl groups are bulky, in addition to 1 are formed (E)-1-arylamino-1-alkyl-4-methyl-5-phenyl-3-pentenes (1′) as the minor products. Lactamines prepared in situ from five- and six-membered lactols and aromatic amines are more reactive than alkyl aldehyde aldimines and furnish (E)-4-arylamino-6-methyl-8-phenyl-6-octen-1-ols (4) and (E)-5-arylamino-7-methyl-9- phenyl-7-nonen-1-ols (5), respectively, in good yields with excellent E-stereoselectivity.     

The rainbow connection of a graph is (at most) reciprocal to its minimum degree

An edge‐colored graph Gis rainbow edge‐connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make Grainbow edge‐connected. We prove that if Ghas nvertices and minimum degree δ then rc(G)<20n/δ. This solves open problems from Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243–254). A vertex‐colored graph Gis rainbow vertex‐connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex‐connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make Grainbow vertex‐connected. One cannot upper‐bound one of these parameters in terms of the other. Nevertheless, we prove that if Ghas nvertices and minimum degree δ then rvc(G)<11n/δ. We note that the proof in this case is different from the proof for the edge‐colored case, and we cannot deduce one from the other. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 185–191, 2010     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

基于联系函数的系统综合评价模型及其应用

如何合理确定综合评价函数是系统分析评价中的关键问题.目前,所采用的方法是单纯从"数学变换"的角度建立综合评价函数,没有考虑研究对象的实际含义.因此,探讨不同的综合评价函数方法具有重要意义.根据综合评价对象的实际含义,定义了联系函数的概念,在此基础上,通过对研究对象进行两次集对同异反分析,建立基于联系函数的系统综合评价模型.实例分析表明:基于联系函数的系统综合评价模型评价结果准确、合理、可信,具有推广应用价值.     

Several formulas for special values of the Bell polynomials of the second kind and applications

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.     

Abelian-Higgs phase of SU(2) QCD and glueball energy

It is shown that SU(2) QCD admits an dual Abelian-Higgs phase, with a Higgs vacuum of a type-Ⅱ superconductor. This is done by using a connection decomposition for the gluon field and the random-direction approximation. Using a bag picture with soft wall, we presented a calculational procedure for the glueball energy based on the recent proof for wall-vortices [Nucl. Phys. B 741(2006)1].