Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Dynamics of laser-induced cavitation bubble during expansion over sharp-edge geometry submerged in liquid – an inside view by diffuse illumination

Laser ablation in liquids is growing in popularity for various applications including nanoparticle production, breakdown spectroscopy, and surface functionalization. When laser pulse ablates the solid target submerged in liquid, a cavitation bubble develops. In case of “finite” geometries of ablated solids, liquid dynamical phenomena can occur inside the bubble when the bubble overflows the surface edge. To observe this dynamics, we use diffuse illumination of a flashlamp in combination with a high-speed videography by exposure times down to 250 ns. The developed theoretical modelling and its comparison with the experimental observations clearly prove that this approach widens the observable area inside the bubble. We thereby use it to study the dynamics of laser-induced cavitation bubble during its expansion over a sharp-edge (“cliff-like” 90°) geometry submerged in water, ethanol, and polyethylene glycol 300. The samples are 17 mm wide stainless steel plates with thickness in the range of 0.025–2 mm. Bubbles are induced on the samples by 1064-nm laser pulses with pulse durations of 7–60 ns and pulse energies of 10–55 mJ. We observe formation of a fixed-type secondary cavity behind the edge where low-pressure area develops due to bubble-driven flow of the liquid. This occurs when the velocity of liquid overflow exceeds ~20 m s⁻¹. A re-entrant liquid injection with up to ~40 m s⁻¹ velocity may occur inside the bubble when the bubble overflows the edge of the sample. Formation and characteristics of the jet evidently depend on the relation between the breakdown-edge offset and the bubble energy, as well as the properties of the surrounding liquid. Higher viscosity of the liquid prevents the generation of the jet.     

关于负超几何分布期望与方差算法的注记

研究了非还原取样模型中负超几何随机变量的联合分布,得到了若干有用的推论.据此给出了负超几何分布的期望和方差的一种分解算法.     

基于2,5-二溴对苯二甲酸配体的锌/钴配合物合成、晶体结构及性质

将有机物2,5-二溴对苯二甲酸(H₂L₁)和2,2′-联吡啶(L₂)作为双配体,使用溶剂热法和七水合硫酸锌(ZnSO₄·7H₂O)、六水合硝酸钴(Co(NO₃)₂·6H₂O)分别反应,得到配合物[Zn(L₁)(L₂)(H₂O)]_n(1)和配合物[Co(L₁)(L₂)(H₂O)]_n(2)。采用单晶X射线衍射、元素分析、红外光谱、紫外光谱、荧光光谱、热重分析等测试方法对这两种物质进行分析研究。单晶测试结果表明配合物1是单斜晶系,以Zn²⁺配位连接L^2-₁与L₂形成一维链状结构,各条链在分子间氢键和π…π共轭作用下有规律地堆叠形成三维网络结构。配合物2是三斜晶系,Co1离子和Co1ⁱ离子由H₂L₁上的羧酸氧原子O4和O4ⁱ连接,形成双齿螯合的配位结构单元,以Co²⁺配位连接 L^2-₁和L₂形成二维网格结构,各层在O—H…O分子间氢键和范德瓦耳斯力作用下有规律的堆叠形成三维网络结构。配合物1和2均含有芳香杂环、羧基杂环和氮杂环,具有良好的荧光性质和热稳定性,最大发射波长分别为345 nm和333 nm。     

Triterpenes and limonoids of Cedrela: Distribution,biosynthesis, and 1H and 13C NMR data

Cedrela genus, a member of the Meliaceae family, presents both chemical characteristics associated with and those that distinguish it from the rest of its members. The presence of triterpenes and limonoids is the characteristic of the Meliaceae family, but the class and type of these chemical constituents are distinctive for each genus. Cedrela includes cycloartane, ursane, oleanane, tirucallane, butyrospermane, and apotirucallane triterpenes, and its limonoids belongs to six class and nine types, known as class Ia-type havanensines, class Ib-type delevoyin, class II-type gedunin, class IIIb-type andirobin, class IIIg-type mexicanolide, class IVa-type evoludone, class Va-type obacunol, class V-type limonin, and class VIII. Each of these structural arrangements includes specific traits, defined by their biosynthetic origin, which can be established by means of structural elucidation techniques, particularly ¹H and ¹³C NMR, which assisted by 2D NMR techniques, allowing to deduce their structures unequivocally. The constant presence of these skeletal arrangements in Cedrela ensures that they are its chemophenetic markers and their recurrence is an important criterion for their identity. This review is a compilation of the occurrence of triterpenes and limonoids in Cedrela genus, detailing their biosynthetic association and collecting and organizing their NMR data, with the purpose of facilitating its location, analysis, and use in the phytochemical study of species from this genus.     

从知识传授到育人:高职“生活中的化学”教学实践*

深挖高职院校公选课“生活中的化学”的育人元素,精心设计教学内容,运用课堂主题辩论、趣味生活实验的设计与展示、探秘生活中的化学等理论知识传授与课内外实践相结合的教学方式,实现全过程育人,充分展现了公选课在立德树人方面的价值。     

The value function of a transportation problem

We investigate the value of an optimal transportation problem with the maximization objective as a function of costs and vectors of production and consumption. The value is concave in production. For generic costs, the numbers of linearity domains and peak points are independent of costs and consumption. The peak points are determined by an auxiliary assignment problem. The volumes of the linearity domains are independent of costs while their dependence on consumption can be expressed via the multinomial distribution.     

基于快速神经网络架构搜索的鲁棒图像水印网络算法

为解决深度学习在图像水印算法中计算量大且模型冗余的问题，提高图像水印算法在抵抗噪声、旋转和剪裁等攻击时的鲁棒性，提出基于快速神经网络架构搜索（neural architecture search，NAS）的鲁棒图像水印网络算法。通过多项式分布学习快速神经网络架构搜索算法，在预设的搜索空间中搜索最优网络结构，进行图像水印的高效嵌入与鲁棒提取。首先，将子网络中线性连接的全卷积层设置为独立的神经单元结构，并参数化表示结构单元内节点的连接，预先设定结构单元内每个神经元操作的搜索空间；其次，在完成一个批次的数据集训练后，依据神经元操作中的被采样次数和平均损失函数值动态更新概率；最后，重新训练搜索完成的网络。水印网络模型的参数量较原始网络模型缩减了92%以上，大大缩短了模型训练时间。由于搜索得到的网络结构更为紧凑，本文算法具有较高的时间性能和较好的实验效果，在隐藏图像时，对空域信息的依赖比原始网络更少。对改进前后的2个网络进行了大量鲁棒性实验，对比发现，本文算法在CIFAR-10数据集上对抵抗椒盐噪声和旋转、移除像素行（列）等攻击优势显著；在ImageNet数据集上对抵抗椒盐高斯噪声、旋转、中值滤波、高斯滤波、JPEG压缩、裁剪等攻击优势显著，特别是对随机移除行（列）和椒盐噪声有较强的鲁棒性。