On the Kernel of Restriction of Characters*

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.     

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.     

The distribution of 4-full numbers

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = \sum\limits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = \mathop {res}\limits_{s = 1/j} (F(s)/s), F(s) = \mathop \prod \limits_P \left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)$$ and R(x) is the remainder. This paper proves that $$R(x) \ll x^{3626/35461 + \varepsilon }$$ where ε is any positive number.     

Lewis-acid assisted cross metathesis of acrylonitrile with functionalized olefins catalyzed by phosphine-free ruthenium carbene complex

The exchange of the PPh3 ligand in the complex [1,3-bis(2,6-dimethylphenyl)4,5-dihydroimidazol-2-ylidene](PPh3)(Cl)2Ru=CHPh (7) for a pyridine ligand at ambient temperature leads to the formation of the stable phosphine-free carbene ruthenium complex [1,3-bis(2,6-dimethylphenyl)4,5-dihydroimidazol-2-ylidene](C5H5N)2(Cl)2 Ru=CHPh (8). The resulted ruthenium complex exhibits highly catalytic activity for the cross metathesis of acrylonitrile with various functionalized olefins under mild conditions, and its activity can be further improved by the addition of a Lewis acid such as Ti(OiPr)4. In the mixture products, the Z-isomer predominates.     

High performance organic field-effect transistors based on amphiphilic tris(phthalocyaninato) rare earth triple-decker complexes

LB films of three amphiphilic tris(phthalocyaninato) rare earth triple-decker complexes with crown-ethers as hydrophilic heads and long alkyl chains as hydrophobic tails have been prepared and found to display very well ordered layer structures, as proved by pi-A isotherms, UV-vis and polarized absorption spectra, X-ray diffraction experiments, and microscopic morphology characterization. These LB films have been fabricated into field-effect transistor (FET) devices, which show carrier mobilities as high as 0.24-0.60 cm2 V-1 s-1, among the highest mobilities achieved thus far for all LB film-based OFETs.