Statistical properties of shocks in Burgers turbulence

We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, , with Gaussian whitenoise initial data. This system was originally proposed by Burgers^[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..^[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)s ^1/2,s1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1–P(s)exp{–Cs ³} fors1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.Inadvertently the sequel to this article, Statistical Properties of Shocks in Burgers Turbulence, II. Tail Probabilities for Velocities, Shock-Strengths and Rarefaction Intervals has already appeared in an earlier issue of Commun. Math. Phys. (Commun. Math. Phys.169, 45–59 (1995).     

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

     

改进片铸与烧结技术制备高性能钕铁硼磁体的研究

通过优化合金成分设计与改进速凝片铸技术、烧结技术,应用国内通用的工业生产烧结钕铁硼磁体的各类原材料,在工业生产线上实现了45UH高性能烧结钕铁硼磁体的批量生产。SEM观察和XRD分析结果表明:磁体具有比较高的取向度;其显微组织致密、精细而均匀,平均晶粒尺寸约为5μm。45UH烧结钕铁硼磁体的典型磁性能为Br=1.363 T,Hcb=1060 kA.m-1,Hcj=2140 kA.m-1,Hk=1625 kA.m-1,(BH)max=366.0 kJ.m-3;其Hcj/79.6 kA.m-1+(BH)max/7.96 kJ.m-3=72.8。在295～453 K温度区间,其剩磁与内禀矫顽力的温度系数分别为-0.108%.K-1和-0.486%.K-1。当L/D=0.7时,在473 K保持2 h磁体开路磁通不可逆损失为4.1%左右。批量生产的45UH烧结钕铁硼磁体,其常温磁性能优异,温度稳定性良好。     

Rational Synthesis of Amorphous Iron-Nickel Phosphonates for Highly Efficient Photocatalytic Water Oxidation with Almost 100 % Yield

A simple solvent ligation effect was successfully used to disrupt the growth of a model compound, Fe[(OH)(O₃P(CH₂)₂CO₂H)]⋅H₂O (MIL-37), into an extended 2D structure by replacing water with dimethylformamide (DMF) as the solvent during the synthesis. Owing to the lack of −OH group, which provides the corner-sharing (binding) oxygen atoms for the octahedra, an amorphous and porous structure is formed. When Fe³⁺ is partially replaced by Ni²⁺, the amorphous structure remains and the resultant binary metal catalyst displays excellent photocatalytic oxygen evolution activity with almost 100 % yield achieved under visible light irradiation using [Ru(bpy)₃]²⁺ as the photosensitizer. This study opens up new possibilities of using the simple solvent effect to synthesize high surface area metal phosphonates for catalytic and other applications.     

[1,2,4]三唑并[3,4-b][1,3,4]噻二嗪类衍生物的合成及生物活性研究进展

[1,2,4]三唑并[3,4-b][1,3,4]噻二嗪作为极具前景的活性基团,在抗菌、抗肿瘤等方面具有多种药理活性,在药物和化学领域受到了广泛的关注。本文以含三唑并噻二嗪结构化合物的合成及其在抗肿瘤、抗菌等方面的生物活性为依据,对近几年国内外关于[1,2,4]三唑并[3,4-b][1,3,4]噻二嗪类化合物的研究进行分析总结,以期为后续该类新型抗肿瘤化合物的设计合成提供参考。     

Rational Synthesis of Amorphous Iron‐Nickel Phosphonates for Highly Efficient Photocatalytic Water Oxidation with Almost 100 % Yield

A simple solvent ligation effect was successfully used to disrupt the growth of a model compound, Fe[(OH)(O₃P(CH₂)₂CO₂H)]?H₂O (MIL‐37), into an extended 2D structure by replacing water with dimethylformamide (DMF) as the solvent during the synthesis. Owing to the lack of ?OH group, which provides the corner‐sharing (binding) oxygen atoms for the octahedra, an amorphous and porous structure is formed. When Fe³⁺ is partially replaced by Ni²⁺, the amorphous structure remains and the resultant binary metal catalyst displays excellent photocatalytic oxygen evolution activity with almost 100 % yield achieved under visible light irradiation using [Ru(bpy)₃]²⁺ as the photosensitizer. This study opens up new possibilities of using the simple solvent effect to synthesize high surface area metal phosphonates for catalytic and other applications.     

Poly(2‐Oxazoline)‐Based Functional Peptide Mimics: Eradicating MRSA Infections and Persisters while Alleviating Antimicrobial Resistance

Peptides have important biological functions. However, their susceptibility to proteolysis limits their applications. We demonstrated here for the first time, that poly(2‐oxazoline) (POX) can work as a functional mimic of peptides. POX‐based glycine pseudopeptides, a host defense peptide mimic, had potent activities against methicillin‐resistant S. aureus, which causes formidable infections. The POX mimic showed potent activity against persisters that are highly resistant to antibiotics. S. aureus did not develop resistance to POX owning to the reactive oxygen species related antimicrobial mechanism. POX‐treated S. aureus is sensitive to common antibiotics, demonstrating no observable antimicrobial pressure or cross‐resistance in using antimicrobial POX. This study highlights POX as a new type of functional mimic of peptides and opens new avenues in designing and exploring peptide mimetics for biological functions and applications.     

Onsager's conjecture on the energy conservation for solutions of Euler's equation

We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.     

态|Ψ_n~((2))〉_q的广义非线性等阶N次方H压缩效应

构造了一种新型的多模叠加态 |Ψ(2 )n 〉q=C(R)n |{ -i Z*j }〉q+C(0 )n |{ 0 j}〉q;并首次详细地研究了此量子态的等阶 N次方 H压缩特性 .大量的计算和分析表明 :态 |Ψ(2 )n 〉q是一种多模典型的非经典光场 ;还发现了“相似压缩”等现象     

多模虚共轭相干态与多模真空态组成的叠加态光场的等阶N次方H压缩

构造了由多模虚共轭相干态|{iZ_j^*}〉_q与多模真空态|{0_j}〉_q两者的线性叠加所组成的两态叠加多模叠加态光场|{Ψ_q⁽²⁾}〉_q.利用多模压缩态理论,首次对该态的等阶N次方H压缩特性进行了详细研究.结果发现:态|{Ψ_q⁽²⁾}〉_q是一种典型的多模非经典光场,它可呈现出周期性变化的、任意阶的等阶N次方H压缩效应.