Hasse's norm theorem forK2

In this paper, we prove Hasse's principle onK ₂ for a certain two-dimensional semi-global field. And, as a bi-product, we have the isomorphism theorem of the class field theory of such a field.     

关于m-HPK(n1,n2,n3,n4)-残差图

本文研究了3-维超平面完备残差图以及m 重3-维超平面完备残差图. 利用容斥原理以及集合的运算性质等方法, 分别获得了3-维超平面完备残差图和m 重3-维超平面完备残差图的最下阶以及二者的唯一极图, 将文献[1] 中定义的残差图从平面推广到超平面上.     

Universal embeddings for theD4(2) hexagon andJ2near-octagon

The universal embeddings overF₂ of the generalized hexagon for³D₄(2)and the near-octagon forJ₂are determined to be the 28-dimensional adjoint modules: for typeD₄in the first case, and in the second from the 14-dimensional adjointF₄-module forG₂(4)containingJ₂.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

On the P4-components of graphs

Two edges are called P₄-adjacent if they belong to the same P₄ (chordless path on four vertices). P₄-components, in our terminology, are the equivalence classes of the transitive closure of the P₄-adjacency relation. In this paper, new results on the structure of P₄-components are obtained. On the one hand, these results allow us to improve the complexity of orienting P₄-comparability graphs and of recognizing P₄-indifference graphs from O(n⁵) and O(n⁶) to O(m²). On the other hand, by combining the modular decomposition with the substitution of P₄-components, a new unique tree representation for arbitrary graphs is derived which generalizes the homogeneous decomposition introduced by Jamison and Olariu (SIAM J. Discrete Math. 8 (1995) 448–463).     

On B6- and B7-groups

A finite group G is said to be a Bn-group if any n-element subset A = {a₁, a₂,..., a_n} of G satisfies \left|A^2\right|=\left|\left\{a_i{a}_j|1\le i,j\le n\right\}\right|\le n(n+1)/2. In this paper, the characterizations of the B₆- and B₇-groups are given.     

F_2+vF_2上的循环码

本文研究了环F_2+vF_2上的循环码.利用标准形生成元集刻画了环F_2+vF_2上的循环码的代数结构,证明了环F_2+vF_2上的每一个非零的循环码均有唯一的标准形生成元集,进而得到了每一个循环码均是由一个多项式生成的.     

Frustrated quantum two-dimensional J1-J2-J3 antiferromagnet in a spherically symmetric self-consistent approach

In the framework of a spherically symmetric self-consistent approach to two-time retarded spin-spin Green’s functions, we develop the theory of a two-dimensional frustrated J ₁ -J ₂ -J ₃ quantum S=1/2 antiferromagnet. We show that taking the damping of spin fluctuations into account is decisive in forming both the spin-liquid state and the state with long-range order. In particular, the existence of damping allows explaining the scaling behavior of the susceptibility χ(q, ω) of the CuO ₂ cuprate plane, the behavior of the spin spectrum in the two-plane case, and the occurrence of an incommensurable χ(q, ω) peak. In the case of the complete J₁-J ₂ -J ₃ model, in a single analytic approach, we find continuous transitions between three phases with long-range order (“checkerboard,” stripe, and helical (q, q) phases) through the spin-liquid state. We obtain good agreement with cluster computations for the J ₁ -J ₂ -J ₃ model and agreement with the neutron scattering data for the J ₁ -J ₂ model of cuprates.     

The G2 sphere of a 4-manifold

We bring to light a G ₂ structure existing on the unit sphere tangent bundle S _M of any given orientable Riemannian 4-manifold M. The associated 3-form φ is co-calibrated if, and only if, M is an Einstein manifold—a result which leads to new examples of co-calibrated G ₂ spaces. We hope to be contributing both to the knowledge of special geometries and to the study of 4-manifolds.     

On K1 and K2 of Algebraic Surfaces

In this paper we study higher Chow groups of smooth, projective surfaces over a field k of characteristic zero, using some new Hodge theoretic methods which we develop for this purpose. In particular we investigate the subgroup of CH ^r+1 (X,r) with r = 1,2 consisting of cycles that are supported over a normal crossing divisor Z on X. In this case, the Hodge theory of the complement forms an interesting variation of mixed Hodge structures in any geometric deformation of the situation. Our main result is a structure theorem in the case where X is a very general hypersurface of degree d in projective 3-space for d sufficiently large and Z is a union of very general hypersurface sections of X. In this case we show that the subgroup of CH ^r+1 (X,r) we consider is generated by obvious cycles only arising from rational functions on X with poles along Z. This can be seen as a generalization of the Noether–Lefschetz theorem for r = 0. In the case r = 1 there is a similar generalization by Müller-Stach, but our result is more precise than it, since it is geometric and not only cohomological. The case r = 2 is entirely new and original in this paper. For small d, we construct some explicit examples for r = 1 and 2 where the corresponding higher Chow groups are indecomposable, i.e. not the image of certain products of lower order groups. In an appendix Alberto Collino constructs even more indecomposable examples in CH ³ (X,2) which move in a one-dimensional family on the surface X.Contribution to appendix.     

Zeros of abelian integrals for the reversible codimension four quadratic centersQ3r ∩Q4

We study the number of zeros of Abelian integrals for the reversible codimension four quadratic centersQ ₃ ^R ∩Q ₄, when we perturb such systems inside the class of all polynomial systems of degreen. Supported by grants from NSF of China (No. 10101031), Guangdong Natural Science Foundation (No. 001289), and NSF of Sun Yat-sen University for younger teachers. The author wishes to thank Prof. R. Conti and Prof. G. Villari for their discussions, and Dipartimento di Matematica “U. Dini”, Università Degli Studi di Firenze for its support and hospitality during the period when this paper was elaborated. Y. Zhao is grateful to the referee for helpful suggestions.     

P4k?1-factorization of bipartite multigraphs

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 proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k−1-factorization of λK _m,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.     

环F2+uF2+vF2上自对偶码的两种构造方法

本文研究环R=F2+uF2+vF2上的自对偶码问题.利用Rn到F3n2的Gray映射及R上的自对偶码C的Gray像为F2上自对偶码,获得了R上任何偶长度的自对偶码存在性的结论.最后,给出了R上两种构造自对偶码的方法.     

环F3+vF3上的循环码与常循环码

本文研究了环F₃+vF₃上的循环码与常循环码.通过环F₃+vF₃与域F₃上的循环码之间关系,证明了环F₃+vF₃上循环码是由一个多项式生成的.最后,用类似的方法,得到了环F₃+vF₃上v-常循环码也是由一个多项式生成的.     

ON DEGENERATE WEAKLY （K1, K2）-QUASIREGULAR MAPPINGS

The authors first give the definition of degenerate weakly （K1,K2）-quasiregular mappings using the technique of exterior power and exterior differential forms, and then, using the method of McShane extension, a useful inequality is obtained, which can be used to derive the self-improving regularity.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

不含有5-圈和k4平面图的森林分解

本文研究了不含有5-圈和K₄的平面图的森林分解问题.利用权转移法,证明了任意不含有5-圈和K₄的平面图能分解成三个森林,且其中有一个森林的最大度不超过2,这一结果推广了文献[2,3]中的结论.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.     

Cocycle d'Euler etK2

In this paper we construct a canonical 2-cocycle on the groupP SL(2,k) with values in the Witt groupW(k) of the fieldk. This allows us to produce anatural homomorphism :H ₂(SL(2,k); Z)I ²(k), whereI ²(k) is the square of the fundamental ideal. We prove that this homomorphism is in fact a lift of Milnor's symbol.