Weak Graph Map Homotopy and Its Applications∗

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order MP-homotopy group of a pointed simple graph. Moreover, they show that the 1-order MP-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.     

磁致伸缩材料弱磁场响应特性的实验研究

针对磁致伸缩材料在弱磁场传感器领域的应用需要,采用迈克耳逊干涉原理实验测量了零应力条件下Tb-Dy-Fe材料和Fe-Ga合金的磁场响应灵敏度,以及不同应力下Fe-Ga合金的磁场响应特性和温度响应特性.实验结果表明：在零应力,外加磁场16 mT条件下,Fe-Ga合金的磁场响应灵敏度远高于Tb-Dy-Fe材料,更合适作为弱磁场传感器敏感材料;同时,在1.2 MPa预应力和26 mT偏置磁场下,Fe-Ga合金材料具有较好的磁场响应灵敏度和较大的饱和磁致伸缩系数,因而处在最佳工作状态.所得到的材料的磁场和温度响应曲线可作为弱磁场传感器参量设计的参考依据.     

PbII 4,4,4-trifluoro-1-naphthyl-1,3-butanedione complexes of 1,10-phenanthroline and 2,2′-bipyridine ligands

Lead(II) 4,4,4-trifluoro-1-naphthyl-1,3-butanedione (Htfnb) complexes of 1,10-phenanthroline (phen) and 2,2′-bipyridine (bpy), [Pb₂(bpy)₂(tfnb)₂] (1) and [Pb₂(phen)₂(tfnb)₂] (2), have been synthesized and characterized by elemental analysis, IR, spectroscopy and X-ray crystallography. The self-assembly of 1 and 2 is likely to be caused by C–H ··· F–C, C–H ··· O and π–π stacking interactions.     

Use of the Bell–Evans–Polanyi Principle to predict regioselectivity of nucleophilic aromatic photosubstitution reactions

Regioselectivity of nucleophilic aromatic photosubstitution has been shown experimentally to depend upon activation energies of the competing transition states. Computational means of determining relative activation energies were sought, therefore, in order to predict regioselectivity. Optimization of the three triplet transition states of 2-chloro-4-nitroanisole with hydroxide ion gave energies of insufficient accuracy to predict regioselectivity. Computed enthalpy changes from the first triplet transition state to the triplet σ-complexes correlated precisely with the experimental activation energies. This exemplifies the Bell–Evans–Polanyi Principle, and it provides an accurate means of assessing regioselectivity.     

Global Stability Results for the Weak Vector Variational Inequality

In this paper, we consider the global stability of solutions of a Weak Vector Variational Inequality in a finite-dimensional Euclidean space. Upper semi-continuity of the solution set mapping is established. And by a scalarization method, we derive a sufficient condition that guarantees the lower semi-continuity of the solution set mapping for the Weak Vector Variational Inequality     

Weak Galerkin finite element methods combined with Crank-Nicolson scheme for parabolic interface problems

This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K),\;{\cal P}_{k-1}(\partial K),\;\big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results.     

弱Lp空间的Marcienkiewicz型内插定理及其应用

在弱$L_p$空间之间建立一类Marcienkiewicz型内插定理并且给出它在现代调和分析理论中的某些应用.     

On the asymptotic number of inequivalent binary self-dual codes

Let Ψn be the number of inequivalent self-dual codes in . We prove that , where . Let Δn be the number of inequivalent doubly even self-dual codes in . We also prove that .     

Weak solutions of the forward problem in EEG for different conductivity values

The potential distribution on the scalp produced by current sources in the brain can be measured by an EEG recorder. The relationship between these sources and the scalp potential distribution may be described by a well-known mathematical model where some simplifications are usually introduced. The head is modeled as a multicompartment nested set and the conductivity of the different tissues is approximated by a positive piecewise constant function. This simplified model is used to solve the forward problem (FP), i.e., to calculate the scalp potential for a current source configuration. In this work, we prove that the weak solutions of the FP are continuous with respect to the conductivity values, that is, the difference between the scalp potentials is small if the conductivity values are closed enough. We present numerical examples that illustrates this property.