Sizes of Critical Graphs with Small Maximum Degrees

In this paper, we give new lower bounds for the size of Δ-critical graphs with Δ=8,9 which improve the partial results of Luo [6] and Y. Zhao [12].     

一些特殊图类乘积的离散度

连通图的离散度是用s(G)来表示的,s(G)=max{ω(G-S)-|S|:ω(G-S)>1,SV(G)}.给出了两个完全图乘积的和一个完全图与路的乘积的离散度.还给出了两个完全图乘积的坚韧度.     

A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

A graphical approach to computing Selmer groups of congruent number curves

Let E _n:y ²=x ³−n ² x denote the family of congruent number elliptic curves. Feng and Xiong (2004) equate the nontriviality of the Selmer groups associated with E _n to the presence of certain types of partitions of graphs associated with the prime factorization of n. In this paper, we extend the ideas of Feng and Xiong in order to compute the Selmer groups of E _n. 2000 Mathematics Subject Classification Primary—11G05; Secondary—14H52, 14H25, 05C90     

Magnetic hysteresis of the magnetoresistance and the critical current density in polycrystalline YBa2Cu3O7−δ–Ag superconductors

A systematic study of the magnetic hysteresis in transport properties of polycrystalline YBa₂Cu₃O_7−δ–Ag compounds has been made based on two kinds of measurements at 77 K and under applied magnetic fields up to 30 mT: critical current density J_c(B_a) and magnetoresistance R(B_a). The R(B_a) curves show a minimum in their decreasing branch occurring at B=B_min which was found to be both the excitation current I_ex and the maximum applied magnetic field B_am dependent. In addition, for a certain value of B_am>5 mT, we have observed that B_min increases with increasing I_ex and reaches a saturation value. The J_c(B_a) curves show a maximum in decreasing applied magnetic fields occurring at B=B_max. We have also found that B_max increases with increasing B_am and reaches a saturation value. The minimum in the R(B_a) and the maximum in J_c(B_a) curves were found to be related to the trapped flux within the grains. All the experimental results are discussed within the context of the flux dynamics and transport mechanisms in these high-T_c materials.     

Optimally Stable Multivariate Bases

We show that the tensor product B-spline basis and the triangular Bernstein basis are in some sense best conditioned among all nonnegative bases for the spaces of tensor product splines and multivariate polynomials, respectively. We also introduce some new condition numbers which are analogs of component-wise condition numbers for linear systems introduced by Skeel.     

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

未确知数学方法处理物理实验数据

介绍未确知数学方法，并用未确知数学方法分析处理物理实验数据，与传统的数据处理方法比较，未确知数学方法不必对数据作任何假设，因此更合理，也更符合实际。     

Strengthened semidefinite programming bounds for codes

We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte bound for A(n,d) is the first bound in the hierarchy, and the new bound of Schrijver (IEEE Trans. Inform. Theory 51:2859–2866, 2005) is located between the first and second bounds in the hierarchy. While computing the second bound involves a semidefinite program with O(n ⁷) variables and thus seems out of reach for interesting values of n, Schrijver’s bound can be computed via a semidefinite program of size O(n ³), a result which uses the explicit block-diagonalization of the Terwilliger algebra. We propose two strengthenings of Schrijver’s bound with the same computational complexity. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

六角系统的边面点面全色数

本文得到了六角系统的边面和点面金色数．