有限群共轭类长度的一点注记

假设G=AB是子群A和B的互相置换积。通过A?B中元素的共轭类长度给出了群G的结构，推广了一些最近的结论。     

Temporal freedom in mathematical thought: A philosophical–empirical enquiry

This paper seeks to introduce and elaborate on the notion of ‘temporal freedom’ in the context of mathematical thinking.The intention is to merge a philosophical and empirical discussion on this topic, and findings will be presented from a qualitative study in which 29 high-school students were invited to explore the symmetry group of a square during clinical interviews. The data indicate how the interviewees – who were given the opportunity to freely explore an unseen mathematical system – repeatedly brought past and future together in a tangible present during their investigations. In doing so they were exceptionally successful in engaging with advanced abstract algebra, far beyond what might have been expected at the outset.By using empirical data to exemplify an otherwise highly philosophical discussion, I hope to draw explicit attention to the temporal aspect of the human mind in mathematical action which instructional methods frequently fail to recognise, value and utilise.     

End group polarity and block symmetry effects on cloud point and hydrodynamic diameter of thermoresponsive block copolymers

Thermoresponsive block copolymers are of interest for delivery vehicles in the body. Often an interior domain is designed for the active agent and the exterior domain provides stability in the bloodstream, and may carry a targeting ligand. There is still much to learn about how block sequence and chain end identity affect micelle structure, size, and cloud points. Here, hydrophilic oligo(ethylene glycol) methyl ether acrylate and more hydrophobic di(ethylene glycol) methyl ether methacrylate monomers were polymerized to give amphiphilic block copolymers with amphiphilic chain ends. The block sequence and chain end identity were both controlled by appropriate choice of RAFT chain transfer agents to study the effect of ‘matched’ and ‘mismatched’ chain end polarity with amphiphilic block sequence. The affect of matching or mismatching chain end polarity and block sequence was studied on the hydrodynamic diameter, cloud point, and temperature range of the chain collapse on linear di‐ and triblock copolymers and star diblock polymers. The affects of matching or mismatching chain end polarity were significant with linear diblock copolymers but more complex with triblock and star copolymers. Explanations of these results may help guide others in designing thermoresponsive block copolymers. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2838–2848     

Generic applications of 13C‐detected NMR diffusion to formulated systems with suppression of thermal convection induced by proton decoupling

Fast and effective structural/compositional analysis on formulated systems represents one of the major challenges encountered in analytical science. ¹³C‐detected diffusion represents a promising tool to tackle the aforementioned challenges, particularly in industry. Toward exploring the generic applications of ¹³C‐detected diffusion, thermal convection induced by ¹H decoupling has been identified as a key factor that resulted in significantly reduced resolution in the diffusion dimension. Optimization of experimental parameters and utilization of double‐stimulated echo‐based pulse sequence both can effectively suppress the thermal convection caused by the ¹H decoupling, the success of which allows robust and generic applications of ¹³C‐detected diffusion to systems from mixtures of small molecules, polymer blends, and copolymers to actual complex formulated systems. The method is particularly powerful in differentiating small molecules from polymers, polymer blends from copolymers, and end‐group analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Thioetherphenolate group 4 metal complexes in the ring opening polymerization of rac‐β‐Butyrolactone

Group 4 complexes 1 – 3 [ 1 = (t‐BuOS)₂Ti(O‐i‐Pr)₂; 2 = (t‐BuOS)₂Zr(O‐t‐Bu)₂; 3 = (t‐BuOS)₂Hf(O‐t‐Bu)₂] supported by two phenolate bidentate ligands (t‐BuOS‐H = 4,6‐di‐tert‐butyl‐2‐phenylsulfanylphenol) promote the well‐controlled ring opening polymerization of rac‐β‐butyrolactone. In presence of isopropanol, low dispersities and molecular weights proportional to the equivalents of isopropanol are achieved. Moreover, the zirconium complex is effective in the copolymerization of rac‐β‐butyrolactone with rac‐lactide. The 13 C nuclear magnetic resonance analysis revealed that the obtained copolymers have a tapered diblock microstructure consisting of an initial block composed of lactide sequences and a terminal block composed of butyrolactone sequences. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 3132–3139     

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.