On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s^{\text{th}}$ (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.     

现代分析仪器在高中化学教学中的应用*——以“防晒霜的防晒效果实验探究”为例

从高中化学原子、分子等抽象性的知识教学出发,以防晒霜的防晒效果实验探究为例,探讨了现代分析仪器在高中化学教学中的应用。通过利用现代分析仪器紫外可见分光光度计,结合紫外线变色球实验探究防晒霜的防晒效果,分析并认识防晒霜中化学防晒的机理。结果表明,防晒霜的厚度越大、SPF越高,防晒霜的防晒效果越好,并且防晒霜的防晒效果可以持续一段时间,但是不同类型的防晒产品存在一定差异。在化学教学中渗透现代分析仪器的应用,为学生进一步了解化学学科在改变学习方式、拓宽学科知识领域、推动自身全面与可持续发展方面打下坚实的基础。     

On the Relative Gain Array (RGA) with singular and rectangular matrices

This paper identifies a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, it is shown that the conventional use of the Moore–Penrose pseudoinverse is inappropriate because it fails to preserve critical properties that can be assumed in the nonsingular case. It is then shown that such properties can be rigorously preserved using an alternative generalized matrix inverse.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

A branch-and-bound method for the minimum k−enclosing ball problem

The minimum k-enclosing ball problem seeks the ball with smallest radius that contains at least k of m given points. This problem is NP-hard. We present a branch-and-bound algorithm on the tree of the subsets of k points to solve this problem. Our method is able to solve the problem exactly in a short amount of time for small and medium sized datasets.     

Conventional vs. Microwave- or Mechanically-Assisted Synthesis of Dihomooxacalix[4]arene Phthalimides: NMR,X-ray and Photophysical Analysis §

Direct O-alkylation of p-tert-butyldihomooxacalix[4]arene (1) with N-(bromopropyl)- or N-(bromoethyl)phthalimides and K₂CO₃ in acetonitrile was conducted under conventional heating (reflux) and using microwave irradiation and ball milling methodologies. The reactions afforded mono- and mainly distal di-substituted derivatives in the cone conformation, in a total of eight compounds. They were isolated by column chromatography, and their conformations and the substitution patterns were established by NMR spectroscopy (¹H, ¹³C, COSY and NOESY experiments). The X-ray structures of four dihomooxacalix[4]arene phthalimide derivatives (2a, 3a, 3b and 5a) are reported, as well as their photophysical properties. The microwave (MW)-assisted alkylations drastically reduced the reaction times (from days to less than 45 min) and produced higher yields of both 1,3-di-substituted phthalimides (3a and 6a) with higher selectivity. Ball milling did not reveal to be a good method for this kind of reaction.     

Comprehensive performance of a ball-milled La0.5Pr0.5Fe11.4Si1.6B0.2Hy/Al magnetocaloric composite

Due to the hydrogen embrittlement effect, La(Fe,Si)₁₃-based hydrides can only exist in powder form, which limits their practical application. In this work, ductile and thermally conductive Al metal was homogeneously mixed with La_0.5Pr_0.5Fe_11.4Si_1.6B_0.2 using the ball milling method. Then hydrogenation and compactness shaping of the magnetocaloric composites were performed in one step via a sintering process under high hydrogen pressure. As the Al content reached 9 wt.%, the La_0.5Pr_0.5Fe_11.4Si_1.6B_0.2H_y/Al composite showed the mechanical behavior of a ductile material with a yield strength of ~44 MPa and an ultimate strength of 269 MPa accompanied by a pronounced improvement in thermal conductivity. Due to the ease of formation of Fe-Al-Si phases and the several micron and submicron sizes of the composite particles caused by ball milling process, the magnetic entropy change of the composites was substantially reduced to ~1.2 J/kg· K-1.5 J/kg· K at 0 T-1.5 T.     

Ultra-Small-Bandgap Conjugated Polymers Based on an N−B←N Unit

Since the breakthrough of conductive polymers in 1977, scientists have made great efforts to create small band gap (E_g) conjugated polymers. Two general strategies to design small E_g conjugated polymers are quinoid structure and donor-acceptor structure. Ultrasmall E_g conjugated polymers (E_g<1.0 eV) always suffer from poor air stability because of high-lying HOMO energy levels. In this work, we report a new strategy to design ultrasmall E_g conjugated polymers by N−B←N unit, i.e. balanced resonant boron-nitrogen covalent bond (B−N) and boron-nitrogen coordination bond (B←N). The resulting polymer exhibits an E_g of 0.82 eV and an onset absorption wavelength of >1500 nm. Moreover, the polymer exhibits excellent air stability because of its low-lying LUMO/HOMO energy levels. An unprecedented property of this polymer is the selective light absorption in the infrared range (800–1500 nm) and high transparency in the visible range (400–780 nm). Using this property, for the first time, we demonstrate the application of conjugated polymers as transparent thermal-shielding coating layer on glass, which reduces indoor solar irradiation through window and consequently reduces power consumption for cooling of buildings and cars in summer.