Logarithmic Frobenius structures and Coxeter discriminants

We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.     

Case studies on the application of high-resolution electron channelling contrast imaging – investigation of defects and defect arrangements in metallic materials

In 1967, Coates discovered the electron channelling contrast of backscattered electrons (BSEs) in scanning electron microscopy, and by this the possibility to investigate arrangements of lattice defects in deformed microstructures of materials. Since that time, a straightforward development of the scanning electron microscopes as well as of the electron channelling contrast technique took place. Nowadays, the performance of scanning electron microscopes is high enough that the resolution of electron channelling contrast imaging (ECCI) micrographs is comparable with conventional bright field transmission electron microscopy (TEM) micrographs. In the first part of the present paper, a historical review on the development of the ECCI technique starting from its discovery more than 45 years ago up to the combination with other advanced methods of scanning electron microscopy like electron backscatter diffraction or high-resolution selected area channelling patterning in the last few years is given. Major important investigations using this technique for the visualization of individual lattice defects like stacking faults (SFs) and dislocations or dislocation arrangements are chronologically summarized. The second part demonstrates that nowadays, ECCI micrographs taken in high-resolution scanning electron microscopes can be called high-resolution ECCI (HR-ECCI). It is shown that the resolution of individual SFs and dislocations in the HR-ECCI micrographs is comparable to that of conventional TEM (about 15 nm defect image width). Furthermore, the paper is demonstrating that HR-ECCI micrographs can be obtained for various types of materials after different mechanical loadings and different grain sizes ranging from large grain size of 500 μm (cast steel) down to less than 2 μm (γ-TiAl).     

Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.     

Flexible working arrangements in context: An empirical investigation through self-organizing maps

This is one of the first studies to utilize Kohonen’s self-organizing maps on flexible work arrangements (FWAs), employee turnover and absenteeism within different national contexts and an array of organizational factors. While the majority of FWAs did not reduce significantly employee turnover or absenteeism, country and industry were significant contextual variables in FWA use: we deciphered six main country regions, where service and manufacturing organizations were important to FWA preferences. We found a curvilinear relationship between turnover and shift-work among manufacturing firms regardless of country: turnover decreases at low levels and increases at high levels of shift-work. We also found strong positive relationships between weekend work and turnover among manufacturing firms regardless of country and firms in the region comprising of Germany, Austria, Sweden, Finland, Denmark, Czech Republic and Belgium. Finally, we found consistently high concentration of organizations with low absenteeism throughout certain industries and countries: noteworthy are service organizations in the Netherlands and manufacturing organizations in Australia. The results demonstrate the contextuality of FWA use across countries and industries, and the usefulness of SOMs for research within human resource management.     

某些对角方程在有限域上的解数

主要运用Gauss和以及Jacobi和的相关性质给出两类对角方程在有限域上的解数公式,分别是形如s∑(i=1) a_ix_i~(m_i)=c的对角方程,其中a_i,c∈F_q~2~*,(m_i,m_j)=1,m_i|(q+1),m_i为奇数或(q+1)/(m_i)为偶数,i=1,2,…,s,以及形如s∑(i=1) x_i~m=c的对角方程,其中c∈F_q~*,m|(q+1),m为奇数或(q+1)/m为偶数.     

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.     

Benzo[f]isoindole derivatives from cycloaddition reaction of 2,4-dimethylpyrimido[2,1-a]isoindole and maleimides

In contrast to the previously reported results in the reaction of maleimides with pyrido[2,1-a]isoindole and 1,2-bis substituted isoindoles, the reaction between 2,4-dimethylpyrimido[2,1-a]isoindole and maleimides leads to the formation of unusual products. Their structure is assessed by NMR and mass spectrometry. An original reaction pathway is proposed. The high quantum yields observed in fluorescence opens the route to applications as biological probes. To cite this article: Z.V. Voitenko et al., C. R. Chimie 9 (2006).     

Neutron scattering study of transverse magnetism in the metamagnet FeBr2

In order to clarify the nature of the additional phase transition at H ₁ ( T ) < H _c ( T ) of the layered antiferromagnetic (AF) insulator FeBr ₂ as found by Aruga Katori et al. (1996) we measured the intensity of different Bragg-peaks in different scattering geometries. Transverse AF ordering is observed in both AF phases, AF I and AF II. Its order parameter exhibits a peak at T ₁ = T ( H ₁ ) in temperature scans and does not vanish in zero field. Possible origins of the step-like increase of the transverse ferromagnetic ordering induced by a weak in-plane field component when entering AF I below T₁ are discussed. Received 27 September 1999 and Received in final form 6 December 1999     

Magnetic properties of ErFe6Fa6 studied by magnetization and neutron diffraction

Neutron-diffraction experiments reveal that ErFe₆Ga₆ forms in the tetragonal ThMn₁₂-type of structure (space group I₄/mmm). The Fe sublattice orders ferromagnetically below K. The Er moments order antiparallel to the Fe moments which, below about 250 K, leads to a decrease of the total magnetization. The easy magnetization direction of ErFe₆Ga₆ is perpendicular to the c-axis in the whole temperature range. Refinement at 2 K shows that ErFe₆Ga₆ orders ferrimagnetically with Er moments of 8.5 (2) and Fe moments at the 8(j) site of 1.9 (1) and at the 8(f) site of 1.7 (1) , respectively. At room temperature, ErFe₆Ga₆ exhibits the same type of magnetic order, however with substantially lower Er moments of 1.0 (4) and Fe moments at the 8(f) site of 1.2 (2) . The Fe moments at the 8(j) site amount to 1.9 (5) /Fe. Received 24 November 1999 and Received in final form 27 March 2000