“多角度欣赏面心立方晶胞”教学设计*

尝试利用“火花学院”App等在线交互资源,引导学生从立体几何及密置层堆积模型视角,观察、描述、绘制面心立方晶胞的正四面体填隙、正八面体填隙、三视图和沿体对角线投影图像,加深学生对以面心立方晶胞结构为基础的CaF₂→金刚石晶胞、NaCl→钙钛矿晶胞结构的理解;再结合新型材料透明镁铝尖晶石晶体结构的探讨,了解晶体制备中的掺杂改性等常用方法,发展学生的微观探析能力和科学创新意识。     

Mathematical attitudes and semantic differentials

The semantic differential—one approach to attitude measurement— basically records a combination of a person's associations with a particular concept with a scaling procedure. This paper considers the implications of such a device for teachers of mathematics.     

Blind separation of fluorescence spectra using sparse non‐negative matrix factorization on right hand factor

Sparse non‐negative matrix factorization on right side factor (SNMF/R) has better performance in feature extraction than non‐negative matrix factorization. In this work, SNMF/R was first used to separate the overlapped three‐dimensional fluorescence spectra of polycyclic aromatic hydrocarbons mixtures in pure water, lake water, and river water, respectively. It is found that the similarity coefficients between the acquired three‐dimensional spectra and the corresponding reference spectra with random initials are all above 0.80; the recognition rate of SNMF/R is higher than that of PARAFAC and non‐negative matrix factorization algorithms, especially in the case of lake water and river water samples. In addition, SNMF/R does not need any initialization scheme designing during spectra separation. These results demonstrate that SNMF/R is an appropriate algorithm to separate the overlapped fluorescence spectra of polycyclic aromatic hydrocarbons in aquatic environment accurately and effectively. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit–implicit mapping approach to nonlinear blind separation of sparse nonnegative dependent sources from a single mixture: pure component extraction from nonlinear mixture mass spectra

The nonlinear, nonnegative single‐mixture blind source separation problem consists of decomposing observed nonlinearly mixed multicomponent signal into nonnegative dependent component (source) signals. The problem is difficult and is a special case of the underdetermined blind source separation problem. However, it is practically relevant for the contemporary metabolic profiling of biological samples when only one sample is available for acquiring mass spectra; afterwards, the pure components are extracted. Herein, we present a method for the blind separation of nonnegative dependent sources from a single, nonlinear mixture. First, an explicit feature map is used to map a single mixture into a pseudo multi‐mixture. Second, an empirical kernel map is used for implicit mapping of a pseudo multi‐mixture into a high‐dimensional reproducible kernel Hilbert space. Under sparse probabilistic conditions that were previously imposed on sources, the single‐mixture nonlinear problem is converted into an equivalent linear, multiple‐mixture problem that consists of the original sources and their higher‐order monomials. These monomials are suppressed by robust principal component analysis and hard, soft, and trimmed thresholding. Sparseness‐constrained nonnegative matrix factorizations in reproducible kernel Hilbert space yield sets of separated components. Afterwards, separated components are annotated with the pure components from the library using the maximal correlation criterion. The proposed method is depicted with a numerical example that is related to the extraction of eight dependent components from one nonlinear mixture. The method is further demonstrated on three nonlinear chemical reactions of peptide synthesis in which 25, 19, and 28 dependent analytes are extracted from one nonlinear mixture mass spectra. The goal application of the proposed method is, in combination with other separation techniques, mass spectrometry‐based non‐targeted metabolic profiling, such as biomarker identification studies. Copyright © 2015 John Wiley & Sons, Ltd.     

The Order of Automorphisms of Quasigroups

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.     

Perfect 1‐Factorizations of a Family of Cayley Graphs

A 1‐factorization of a graph G is a decomposition of G into edge‐disjoint 1‐factors (perfect matchings), and a perfect 1‐factorization is a 1‐factorization in which the union of any two of the 1‐factors is a Hamilton cycle. We consider the problem of the existence of perfect 1‐factorizations of even order 4‐regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted , which are Cayley graphs if and only if k is even or . By solving the perfect 1‐factorization problem for a large class of graphs, we obtain a new class of 4‐regular bipartite circulant graphs that do not have a perfect 1‐factorization, answering a problem posed in 7 . With further study of graphs, we prove the nonexistence of P1Fs in a class of 4‐regular non‐bipartite circulant graphs, which is further support for a conjecture made in 7 .     

Factorizations of complete graphs into caterpillars of diameter 5

We examine factorizations of complete graphs K_2n into caterpillars of diameter 5. First we present a construction generalizing some previously known methods. Then we use the new method along with some previous partial results to give a complete characterization of caterpillars of diameter 5, which factorize the complete graph K_2n.     

Accurate computation of the smallest eigenvalue of a diagonally dominant -matrix

If each off-diagonal entry and the sum of each row of a diagonally dominant -matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.

     

Polynomial factorization over

We describe algorithms for polynomial factorization over the binary field , and their implementation. They allow polynomials of degree up to to be factored in about one day of CPU time, distributing the work on two processors.