Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

Reactivity of a N-Coordinated Germylene-Borane Complex: From Ge→B to Ge→Ga Coordination

Reactivity studies of the Ge^II→B complex L(Cl)Ge⋅BH₃ ( 1 ; L=2-Et₂NCH₂-4,6-tBu₂-C₆H₂) were performed to determine the effect on the Ge^II→B donation. N-coordinated compounds L(OtBu)Ge⋅BH₃ ( 2 ) and [LGe⋅BH₃]₂ ( 3 ) were prepared. The possible tuning of the Ge^II→B interaction was proved experimentally, yielding compounds 1-PPh₂-8-(LGe)-C₁₀H₆ ( 4 ) and L(Cl)Ge⋅GaCl₃ ( 5 ) without a Ge^II→B interaction. In 5 , an unprecedented Ge^II→Ga coordination was revealed. The experimental results were complemented by a theoretical study focusing on the bonding in 1 − 5 . The different strength of the Ge^II→E (E=B, Ga) donation was evaluated by using energy decomposition analysis. The basicity of different L(X)Ge groups through proton affinity is also assessed.     

Chemistry of Pentafluorosulfanyl Derivatives and Related Analogs: From Synthesis to Applications

Pentafluorosulfanyl (SF₅)-containing compounds and corresponding analogs are a highly valuable class of fluorine-containing building blocks owing to their unique properties. The reason for that is the set of peculiar and tremendously beneficial characteristics they can impart on molecules once introduced onto them. Despite this, their application in distinct scientific fields remains modest, given the extremely harsh reaction conditions needed to access such compounds. The recent synthetic approaches via S−F, and C−SF₅ bond formation as well as the use of SF₅-containing building blocks embody a “stairway-to-heaven” loophole in the synthesis of otherwise-inaccessible chemical scaffolds only a few years ago. Herein, we report and evaluate the properties of the SF₅ group and analogs, by summarizing synthetic methodologies available to access them as well as following applications in material science and medicinal chemistry since 2015.     

Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.     

Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups

We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear. © 2005 Wiley Periodicals, Inc. J Graph Theory