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Daniel Conus Mathew Joseph Davar Khoshnevisan Shang-Yuan Shiu 《Probability Theory and Related Fields》2013,156(3-4):483-533
Consider the stochastic heat equation ${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$ , where the solution u := u t (x) is indexed by ${(t, x) \in (0, \infty) \times {\bf R}^d}$ , and ${\dot{F}}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- ${\|x\|}$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is ${f(x)\propto \|x\|^{-\alpha}}$ , then the “fluctuation exponents” of the solution are ${\psi}$ for the spatial variable and ${2\psi-1}$ for the time variable, where ${\psi:=2/(4-\alpha)}$ . Moreover, these exponent relations hold as long as ${\alpha \in (0, d \wedge 2)}$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987]. 相似文献
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以水热法制备的高磁饱和强度Fe3O4纳米颗粒为核,正硅酸乙酯(TEOS)为前驱体,采用改进的Stöber法,制备介孔SiO2包覆Fe3O4磁性核壳复合微球。利用XRD、SEM、TEM、N2吸-脱附、FTIR和VSM对制备样品的物相结构、形貌和磁性能进行了测试表征。研究结果表明,制备的复合材料呈球形,粒径分布均一,材料的比表面积和磁饱和强度分别为413 m2·g-1和68.93 emu·g-1。研究了TEOS的添加量对复合微球形貌的影响,随着TEOS添加量的增加,SiO2壳层增厚,复合粒子形貌均匀,饱和磁化强度有所下降,仍具有良好的超顺磁性。在此基础上,通过接枝法在复合微球的表面接枝-NH2,制备了一种新型磁性纳米吸附剂(Fe3O4@SiO2@mSiO2-NH2),进而研究了其对水中重金属离子Cr(Ⅳ)的吸附性能。通过动力学拟合,Fe3O4@SiO2@mSiO2-NH2对Cr(Ⅳ)的吸附过程是准二级动力学模型占主导地位。探究了该材料对Cr(Ⅳ)的吸附过程和吸附机理。结果表明,其吸附机理及吸附容量与Cr(Ⅳ)的离子形态及-NH2有关,并通过吸附剂与吸附质之间的电子共用或静电吸附实现。 相似文献
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Mixed-Valence CuI/CuIII Metal–Organic Frameworks with Non-innocent Ligand for Multielectron Transfer
Shang-Yuan Fu Cheng-Han Chang Dr. Alexander S. Ivanov Dr. Ilja Popovs Dr. Jeng-Lung Chen Dr. Yen-Fa Liao Dr. Hsin-Kuan Liu Dr. Suman Chirra Prof. Dr. Yun-Wei Chiang Jui-Chin Lee Wei-Ling Liu Prof. Dr. Watchareeya Kaveevivitchai Prof. Dr. Teng-Hao Chen 《Angewandte Chemie (International ed. in English)》2023,62(50):e202312494
We report two novel three-dimensional copper-benzoquinoid metal–organic frameworks (MOFs), [Cu4L3]n and [Cu4L3 ⋅ Cu(iq)3]n (LH4=1,4-dicyano-2,3,5,6-tetrahydroxybenzene, iq=isoquinoline). Spectroscopic techniques and computational studies reveal the unprecedented mixed valency in MOFs, formal Cu(I)/Cu(III). This is the first time that formally Cu(III) species are witnessed in metal–organic extended solids. The coordination between the mixed-valence metal and redox-non-innocent ligand L, which promotes through-bond charge transfer between Cu metal sites, allows better metal-ligand orbital overlap of the d-π conjugation, leading to strong long-range delocalization and semiconducting behavior. Our findings highlight the significance of the unique mixed valency between formal Cu(I) and highly-covalent Cu(III), non-innocent ligand, and pore environments of these bench stable Cu(III)-containing frameworks on multielectron transfer and electrochemical properties. 相似文献