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《Expositiones Mathematicae》2020,38(3):285-302
The unit theorem that forms the subject of the present article, is a theorem from algebra that has a combinatorial flavour, and that originated in fact from algebraic combinatorics. Beyond a proof, we also address applications, one of which is a proof of the normal basis theorem from Galois theory. 相似文献
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This paper examines how three eighth grade students coordinated lower and higher dimensional units (e.g., composite units and pairs) in the context of constructing a formula for evaluating sums of consecutive whole numbers while solving combinatorics problems (e.g., 1 + 2 + ⋯ + 15 = (16 × 15)/2). The data is drawn from the beginning of an 8-month teaching experiment. The findings from the study include: (1) a framework for understanding how students coordinate lower and higher dimensional units; (2) identification of key learning that occurred as students made the transition between solving two kinds of combinatorics problems; and (3) identification of the links between the way students’ coordinated lower and higher dimensional units and their evaluation of sums of consecutive whole numbers. Implications for research and teaching are considered. 相似文献
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Haotian Tian Chensheng Lin Yuqiao Zhou Xin Zhao Huixin Fan Tao Yan Prof. Dr. Ning Ye Dr. Min Luo 《Angewandte Chemie (International ed. in English)》2023,62(29):e202304858
Ionic organic crystals containing organic planar π-conjugated units has become one of the hot spots as nonlinear optical (NLO) materials. However, although this type of ionic organic NLO crystals commonly have remarkable second harmonic generation (SHG) responses, they also suffer from overlarge birefringences and relatively small band gaps that be hardly beyond 6.2 eV. Herein, a flexible π-conjugated [C3H(CH3)O4]2− unit was theoretically revealed, showing great potential for designing NLO crystals with balanced optical properties. Accordingly, through the reasonable NLO-favourable layered design, a new ionic organic material, NH4[LiC3H(CH3)O4], was successfully obtained. As expected, it achieves not only a large SHG effect (4×KDP), but also a suitable birefringence (0.06@546 nm) and an ultrawide band gap (>6.5 eV). This study provides a new flexible π-conjugated NLO-active unit, contributing to design more ionic organic NLO materials with excellent balanced optical properties. 相似文献
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模糊条件下的决策单元相对有效性评价 总被引：5，自引：0，他引：5
研究了模糊条件下决策单元的相对有效性评价问题。首先分析了模糊性因素对决策单元相对有效性的影响;然后根据模糊规划取截集方法和DEA评价的经济含义,给出了模糊DEA模型的求解方法;最后定义了决策单元的模糊DEA有效性以及进行有效性排序的平均置信有效性。文末是一个模糊DEA应用的例子。 相似文献
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Dr. Jie-Yu Yue Li-Ping Song Yan-Fei Fan Zi-Xian Pan Dr. Peng Yang Dr. Yu Ma Dr. Qing Xu Dr. Bo Tang 《Angewandte Chemie (International ed. in English)》2023,62(38):e202309624
H2O2 is a significant chemical widely utilized in the environmental and industrial fields, with growing global demand. Without sacrificial agents, simultaneous photocatalyzed H2O2 synthesis through the oxygen reduction reaction (ORR) and water oxidation reaction (WOR) dual channels from seawater is green and sustainable but still challenging. Herein, two novel thiophene-containing covalent organic frameworks (TD-COF and TT-COF) were first constructed and served as catalysts for H2O2 synthesis via indirect 2e− ORR and direct 2e− WOR channels. The photocatalytic H2O2 production performance can be regulated by adjusting the N-heterocycle modules (pyridine and triazine) in COFs. Notably, with no sacrificial agents, just using air and water as raw materials, TD-COF exhibited high H2O2 production yields of 4060 μmol h−1 g−1 and 3364 μmol h−1 g−1 in deionized water and natural seawater, respectively. Further computational mechanism studies revealed that the thiophene was the primary photoreduction unit for ORR, while the benzene ring (linked to the thiophene by the imine bond) was the central photooxidation unit for WOR. The current work exploits thiophene-containing COFs for overall photocatalytic H2O2 synthesis via ORR and WOR dual channels and provides fresh insight into creating innovative catalysts for photocatalyzing H2O2 synthesis. 相似文献
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The new polyoxomolybdate 1 was prepared from an aqueous solution by the reaction of Na2MoO4·H2O, NH2NH2·2HCl and HCOONa, and characterized by IR spectroscopy, elemental analysis and single-crystal X-ray structural analysis. Crystal data: C4H72ClMo18Na13O94, Mr = 3685.86, triclinic, space group P, a = 11.922(5), b = 15.003(5), c = 15.171(5) , α = 116.073(5), β = 112.593(5), γ = 93.342(5)o, V = 2164.3(14) 3, Z = 1, Dc = 2.828 g/cm3, F(000) = 1764, μ = 2.741 mm-1, R = 0.0349 and wR = 0.0875 (I > 2σ(I)). X-ray crystal structure analysis shows that 1 is constructed by Mo18 polyoxoanions linked by Na+ cations to form a fascinating 3D open framework with “guest” Na+ cations and formyl groups residing in the channels. 相似文献
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In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings
R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.
相似文献
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The International System of Units’ (SI) base unit of the quantity “amount of substance” is the mole (symbol: mol). After the revision of the SI to be implemented in 2019, when all SI units will be based solely on constants, the mole will be defined through a fixed value of the Avogadro constant NA. One mole contains exactly 6.02214076 × 1023 elementary entities, meaning the mole will no longer be linked to the kilogram. Currently, the mole is defined via the mass of exactly 0.012 kg of the 12C isotope which links it to the kilogram prototype. The history, changes, and implications of the revised definition of the mole are discussed here from the chemist's point of view. The ability to count entities such as atoms or molecules (precisely enough to enable a revision of the SI and preserve consistency of previous and future measurements) is crucial. This is achieved with the realization (Mise en Pratique) based on the X‐ray‐crystal density (XRCD) method (counting the atoms in a silicon sphere). The determination of NA, focusing on the measurement of the molar mass of silicon highly enriched in the 28Si isotope, with the lowest uncertainty so far, is presented. 相似文献