Free Groups from Fields

In the nineteenth century, field theory brilliantly resolveda number of questions that had taxed mathematicians for centuries;for example, ‘The circle cannot be squared’ by straightedge and compass, and solving polynomial equations by radicalsis not always possible. These successes have continued to beheld up as superb examples of the power of mathematical thought,and are demonstrated at an undergraduate level. The purposeof this article is to provide another such natural example whichleads to a concrete realisation of the free group on 2 generators.     

基于变芯径光纤的光学系统在光造型中的应用

光学系统采用变芯径光纤，可大幅度提高光的传输效率，增加光学子系统柔性。描述了光造型工作原理及其理论依据，在介绍所研制的光的成型系统光学子系统的基础上，阐述了紫外光源的选择与设计，及变芯径光纤的设计、传输、耦合与聚焦等技术。该套基于变芯径光纤的光学子系统成本低、体积小、效率高，巳成功应用于所研制的台式低成本快速成型系统中，为该系统的进一步推广奠定了技术基础。     

Second-order relativistic electron capture

Starting with the impulse approximation, we analyse second-order effects in relativistic electron capture. The relation of this model with relativistic distorted-wave approximations is clarified. In particular it is shown that the second-order spin-coupling terms in the RCDW theory are consistent with the correct form given by perturbation theory. In the semirelativistic limit, the RCDW results are shown to accord with the formulae of Moiseiwitsch for flip and nonflip transitions in the ultra-relativistic limit. This confirms that the continuum-distorted-wave model generalises to relativistic spinors, and highlights the defects of scalar models. We also present a new symmetric eikonal theory which gives reliable results for capture without change of spin, but leads to a divergent total cross section for spin-flip transitions in the second-order term. This effect, which is quite distinct from the spurious spin-flip amplitudes of the scalar symmetric eikonal theory, is taken as further evidence that the eikonal approximation is not valid for magnetic transitions.     

Applications of maximum likelihood principal component analysis: incomplete data sets and calibration transfer

The application of a new method to the multivariate analysis of incomplete data sets is described. The new method, called maximum likelihood principal component analysis (MLPCA), is analogous to conventional principal component analysis (PCA), but incorporates measurement error variance information in the decomposition of multivariate data. Missing measurements can be handled in a reliable and simple manner by assigning large measurement uncertainties to them. The problem of missing data is pervasive in chemistry, and MLPCA is applied to three sets of experimental data to illustrate its utility. For exploratory data analysis, a data set from the analysis of archeological artifacts is used to show that the principal components extracted by MLPCA retain much of the original information even when a significant number of measurements are missing. Maximum likelihood projections of censored data can often preserve original clusters among the samples and can, through the propagation of error, indicate which samples are likely to be projected erroneously. To demonstrate its utility in modeling applications, MLPCA is also applied in the development of a model for chromatographic retention based on a data set which is only 80% complete. MLPCA can predict missing values and assign error estimates to these points. Finally, the problem of calibration transfer between instruments can be regarded as a missing data problem in which entire spectra are missing on the ‘slave’ instrument. Using NIR spectra obtained from two instruments, it is shown that spectra on the slave instrument can be predicted from a small subset of calibration transfer samples even if a different wavelength range is employed. Concentration prediction errors obtained by this approach were comparable to cross-validation errors obtained for the slave instrument when all spectra were available.     

Effect of molecular oxygen on the variable-temperature 29Si MAS NMR spectra of zeolite-sorbate complexes

Structure determinations of siliceous zeolite-sorbate host-guest complexes by solid-state NMR require highly resolved 29Si MAS NMR spectra. As the temperature is lowered, the 29Si MAS NMR spectra of many zeolite-sorbate complexes become broadened such that the resolution of the individual 29Si peaks is lost, limiting the application of solid-state NMR for structure determination. It is shown that the 29Si peak widths are related to the 29Si T2 relaxation times and that the source of the 29Si relaxation and the line broadening is paramagnetic molecular oxygen in the channels of the zeolite. Removal of the oxygen by purging the sample with nitrogen gas leads to a dramatic increase in the resolution of the 29Si MAS NMR spectrum of the p-dibromobenzene/ZSM-5 complex. An analysis of the individual 29Si T1 relaxation times reveals that the oxygen molecules are localized mainly in the zigzag channels of ZSM-5, suggesting that the p-dibromobenzene molecules are located in the channel intersections.     

Rapid synthesis of pomalidomide-conjugates for the development of protein degrader libraries

Current methods for the preparation of heterobifunctional pomalidomide-conjugates rely on methods that are often low yielding and produce intractable byproducts. Herein we describe our strategy for the reliable and succinct preparation of pomalidomide-linkers which is essential to the formation of these conjugates. We present the preparation of 18 pomalidomide-linkers in high yield compared to current literature methods. Our findings show that secondary amines consistently afford greater yields than their primary counterparts, a trend that we were able to exploit in the synthesis of several new pomalidomide homo-dimers in enhanced yields compared to similar literature syntheses. This trend was further utilised to develop the first one-pot synthesis of JQ1-pomalidomide conjugates in yields up to 62%, providing a method that is suited to rapid preparation of conjugate libraries as is frequently required for the development of new protein degraders.

Current methods for the preparation of heterobifunctional pomalidomide-conjugates rely on methods that are often low yielding and produce intractable byproducts. Herein we describe our strategy for the succinct preparation of pomalidomide-linkers.