一类压缩型映射的不动点定理

在完备的度量空间中,讨论了一类新型的非线性压缩映射ρ(Tx,Ty)≤a(ρ(x,y))ρ(x,Tx)+b(ρ(x,y))ρ(y,Ty)+c(ρ(x,y))ρ(x,y)通过构造迭代序列,指出该映射的不动点的存在性和唯一性,并给出相应的误差估计式,拓展和改进了有关文献的范围.     

改进的PNC方法及其在非线性偏微分方程多解问题中的应用

2017年，李昭祥等提出了一种偏牛顿-校正法（Partial Newton-Correction Method，简记为PNC方法），并利用它成功地计算出了三类非线性偏微分方程的多重不稳定解.本文在PNC方法的基础上，提出并发展了一种改进的PNC方法.首先，利用Nehari流形$\mathcal{N}$与零平凡解的可分离性，建立并证明了$\mathcal{N}$的某特殊子流形$\mathcal{M}$上的全局分离定理及其推广（即局部分离定理）.全局分离定理只跟非线性偏微分算子或相应的非线性泛函本身有关，而与具体的计算方法无关.对一些典型的非线性偏微分方程多解问题（比如，Henon方程问题），该全局分离定理的分离条件，经验证是成立的.另一个方面，通过修改或补充原辅助变换的定义，去掉了原辅助变换的奇异性；接着建立并证明了某些非线性偏微分方程问题的新未知解与该非线性偏微分算子零核空间的密切关系；在证明中，去掉了在原奇异变换下所需的标准收敛（standard convergence）假设.最后，计算实例与数值结果验证了改进的PNC方法的可行性和有效性；同时表明子流形$\mathcal{M}$与已知解的可分离性是PNC方法和本文新方法能成功找到多解的关键.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Synthesis of Highly Functionalized N,N-Diarylamides by an Anodic C,N-Coupling Reaction

We report an innovative, sustainable and straightforward protocol for the synthesis of N,N-diarylamides equipped with nonprotected hydroxyl groups by using electrosynthesis. The concept allows the application of various substrates furnishing diarylamides with yields up to 57 % within a single and direct electrolytic protocol. The method is thereby easy to conduct in an undivided cell with constant current conditions offering a versatile and short-cut alternative to conventional pathways.     

Kinetics and mechanism of trichloroisocyanuric acid/NaNO2-triggered nitration of aromatic compounds under acid-free and Vilsmeier-Haack conditions

Kinetics and mechanism of nitration of aromatic compounds using trichloroisocyanuric acid (TCCA)/NaNO₂, TCCA-N,N-dimethyl formamide (TCCA-DMF)/NaNO₂, and TCCA-N,N-dimethyl acetamide (TCCA-DMA)/NaNO₂ under acid-free and Vilsmeier-Haack conditions. Reactions followed second-order kinetics with a first-order dependence on [Phenol] and [Nitrating agent] ([TCCA], [(TCCA-DMF)], or [(TCCA-DMA)] >> [NaNO₂]). Reaction rates accelerated with the introduction of electron-donating groups and retarded with electron-withdrawing groups, but did not fit well into the Hammett's theory of linear free energy relationship or its modified forms like Brown-Okamoto or Yukawa-Tsuno equations. Rate data were analyzed by Charton's multiple linear regression analysis. Isokinetic temperature (β) values, obtained from Exner's theory for different protocols, are 403.7 K (TCCA-NaNO₂), 365.8 K (TCCA-DMF)/NaNO₂, and 358 K (TCCA-DMA)/NaNO₂. These values are far above the experimental temperature range (303-323 K), indicating that the enthalpy factors are probably more important in controlling the reaction.     

Homogenization of Periodically Heterogeneous Thin Beams

Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions. When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero the authors obtain three different one-dimensional models of beam depending upon the limit of the ratio of these two small parameters.     

Lamb波频域逆散射全聚焦成像方法研究

Lamb波因其传播距离远、衰减小常被用于板状结构的无损检测中,在基于Lamb波损伤检测的诸多成像技术中,全聚焦方法(Total Focus Method,TFM)方法因其成像分辨率高、信噪比高而受青睐。然而Lamb波的频散效应导致时域延时量不能被准确计算,进而影响传统TFM方法对损伤定位及成像的精度;此外,既有的TFM方法仅以回波幅值作为成像指标,忽略了Lamb波与损伤的相互作用,故而不能通过损伤表面的物理参数增强成像质量。针对这两个问题,本文首先在时域TFM基础上发展了频域TFM,在计算中纳入频散关系以规避频散的影响;其次以包含明确的损伤特征参数--反射率为成像指标,结合频域TFM方法建立损伤逆散射模型,以实现对损伤的准确成像。仿真和实验结果表明:频域逆散射TFM成像方法能够实现对铝板结构中的损伤检测,在工程实践中具有应用价值。     

Order-Stability in Complex Biological,Social, and AI-Systems from Quantum Information Theory

This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the ability to preserve its order structure or, in mathematical terms, to prevent increasing of entropy. However, Schrödinger’s analysis shows that the classical theory is not able to adequately describe the order-stability in a biosystem. Schrödinger also appealed to the ambiguous notion of negative entropy. We apply quantum theory. As is well-known, behaviour of the quantum von Neumann entropy crucially differs from behaviour of classical entropy. We consider a complex biosystem S composed of many subsystems, say proteins, cells, or neural networks in the brain, that is, S=(Si). We study the following problem: whether the compound system S can maintain “global order” in the situation of an increase of local disorder and if S can preserve the low entropy while other Si increase their entropies (may be essentially). We show that the entropy of a system as a whole can be constant, while the entropies of its parts rising. For classical systems, this is impossible, because the entropy of S cannot be less than the entropy of its subsystem Si. And if a subsystems’s entropy increases, then a system’s entropy should also increase, by at least the same amount. However, within the quantum information theory, the answer is positive. The significant role is played by the entanglement of a subsystems’ states. In the absence of entanglement, the increasing of local disorder implies an increasing disorder in the compound system S (as in the classical regime). In this note, we proceed within a quantum-like approach to mathematical modeling of information processing by biosystems—respecting the quantum laws need not be based on genuine quantum physical processes in biosystems. Recently, such modeling found numerous applications in molecular biology, genetics, evolution theory, cognition, psychology and decision making. The quantum-like model of order stability can be applied not only in biology, but also in social science and artificial intelligence.     

The self-assemblies of a newly designed star-shaped molecule end-capped with bromine atoms studied by scanning tunneling microscopy

A new star-shaped molecule StOF-Br_3 containing oligofluorenes and halogen atoms(Bromine) has been synthesized and studied by Scanning Tunneling Microscopy(STM) at the highly oriented pyrolytic graphite(HOPG) surface.We have obtained the high-resolution self-assembled STM images,from which the highly ordered and closely packed non-porous arrangements of the StOF-Br_3 molecular selfassemblies at the heptanoic acid/HOPG surface could be observed.The molecular models and selfassembled StOF-Br_3 architectures have been given in the following text.Besides,we have also figured out the surface free energy by the density functional theory(DFT) calculation,which proved that the halogen...halogen interaction was strong enough to stabilize the ordered molecular self-assemblies.This work verifies the existence of bromine...bromine interactions,and meanwhile provides a kind of effective approach for quickly building ordered molecular nanoarchitectures with large areas and different geometries.