Differentiable spaces that are subcartesian

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.     

人教版高中化学教科书“研究与实践”栏目分析与实施策略*

对人教版高中化学教科书新设计的“研究与实践”栏目的主题内容、功能价值进行分析,就如何充分发挥该栏目的教学功能和价值,发展学生的化学学科核心素养,提出了创设教学情境线索、开发为研究性学习课题、开发为校本选修课程、开发为STEM课程等实施策略。     

The full basis theorem does not imply analytic wellordering

A finite support product of ${\omega }_1$ clones of Jensen's minimal ${\Pi }_2^1$ singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals.     

Subset representations and eigenvalues of the universal intertwining matrix

We solve a combinatorial question concerning eigenvalues of the universal intertwining endomorphism of a subset representation.     

A unified theory for brittle and ductile shear mode fracture

A unified theory captures both brittle and ductile fracture. The fracture toughness is proportional to the applied stress squared and the length of the crack. For purely brittle solids, this criterion is equivalent to Griffith's theory. In other cases, it provides a theoretical basis for the Irwin-Orowan formula. For purely ductile solids, the theory makes direct contact with the Bilby-Cottrell-Swinden model. The toughness is highest in ductile materials because the shielding dislocations in the plastic zone provide additional resistance to crack growth. This resistance is the force opposing dislocation motion, and the Peach-Koehler force overcomes it. A dislocation-free zone separates the plastic zone from and the tip of the crack. The dislocation-free zone is finite because molecular forces responsible for the cohesion of the surfaces near the crack tip are not negligible. At the point of crack growth, the length of the dislocation-free zone is constant and the shielding dislocations advance in concert. As in Griffith's theory, the crack is in unstable equilibrium. The theory shows that a dimensionless variable controls the elastoplastic behaviour. A relationship for the size of the dislocation-free zone is derived in terms of the macroscopic and microscopic parameters that govern the fracture.     

Preparation of Highly Stable and Photoluminescent Cadmium-Free InP/GaP/ZnS Core/Shell Quantum Dots and Application to Quantitative Immunoassay

Indium phosphide (InP) quantum dots (QDs) are ideal substitutes for widely used cadmium-based QDs and have great application prospects in biological fields due to their environmentally benign properties and human safety. However, the synthesis of InP core/shell QDs with biocompatibility, high quantum yield (QY), uniform particle size, and high stability is still a challenging subject. Herein, high quality (QY up to 72%) thick shell InP/GaP/ZnS core/shell QDs (12.8 ± 1.4 nm) are synthesized using multiple injections of shell precursor and extension of shell growth time, with GaP serving as the intermediate layer and 1-octanethiol acting as the new S source. The thick shell InP/GaP/ZnS core/shell QDs still keep high QY and photostability after transfer into water. InP/GaP/ZnS core/shell QDs as fluorescence labels to establish QD-based fluorescence-linked immunosorbent assay (QD-FLISA) for quantitative detection of C-reactive protein (CRP), and a calibration curve is established between fluorescence intensity and CRP concentrations (range: 1–800 ng mL⁻¹, correlation coefficient: R² = 0.9992). The limit of detection is 2.9 ng mL⁻¹, which increases twofold compared to previously reported cadmium-free QD-based immunoassays. Thus, InP/GaP/ZnS core/shell QDs as a great promise fluorescence labeling material, provide a new route for cadmium-free sensitive and specific immunoassays in biomedical fields.     

Corners of self-conjugate (s,s + 1)-cores and (s‾,s+1‾)-cores

In this paper, we describe a result on self-conjugate $(s,s+1)$-core partitions with the fixed number of corners. We also define shifted corners of a distinct partition and find formulas for the number of $(\stackrel{￣}{s},\stackrel{￣}{s+1})$-core partitions and the number of $(s,s+1)$-core shifted Young diagrams with the fixed number of shifted corners.     

Pure tetrahedral quadruple systems with index two

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order $n$ with index $\lambda$, denoted by ${\text{TQS}}_{\lambda }\left(n\right)$, is a pair $\left(X,\mathrm{?}\right)$, where $X$ is an $n$‐set and $\mathrm{?}$ is a set of oriented tetrahedra (blocks) such that every cyclic triple on $X$ is contained in exactly $\lambda$ members of $\mathrm{?}$. A ${\text{TQS}}_{\lambda }\left(n\right)$ is pure if there do not exist two blocks with the same vertex set. When $\lambda =1$, the spectrum of a pure TQS $\left(n\right)$ has been completely determined by Ji. In this paper, we show that there exists a pure ${\text{TQS}}_2\left(n\right)$ if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$. A corollary is that a simple ${\text{QS}}_4\left(n\right)$ also exists if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$.     

Synthesis of diisocyanides with phenolic groups and their polymerization to helically chiral poly(quinoxaline‐2,3‐diyl)s

The development of synthetic routes which lead to five new diisocyanide monomers with one or two phenolic groups is described. Their polymerization behavior is studied with Pd‐ and Ni‐based initiators, as well as under microwave irradiation. The polymerizability is mainly dominated by steric effects as is concluded from experiments using different protecting groups. Chiroptical properties of these new polymers are studied by CD‐spectroscopy. After deprotection, helically chiral poly(quinoxalin‐2,3‐diyl)s are obtained which display a Brønsted function attached to a stereolabile biaryl axis whose configuration should be influenced by the chiral polymer backbone. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 1320–1329