Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

A theorem of the Dore-Venni type for noncommuting operators

A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations and evolutionary integral equations which are nonautonomous. The domains of the involved operators may depend on , but is required to satisfy a certain smoothness property. The results are then applied to parabolic partial differential and integro-differential equations.

     

Phase transitions in two-dimensional uniformly frustratedXY models. I. Antiferromagnetic model on a triangular lattice

A most popular model in the family of two-dimensional uniformly-frustratedXY models is the antiferromagnetic model on a triangular lattice [AFXY(t) model]. Its ground state is both continuously and twofold discretely degenerated. Different phase transitions possible in such systems are investigated. Relevant topological excitations are analyzed and a new class of such (vortices with a fractional number of circulation quanta) is discovered. Their role in determining the properties of the system proves itself essential. The characteristics of phase transitions related to breaking of discrete and continuous symmetries change. The phase diagram of the generalized AFXY(t) model is constructed. The results obtained are rederived in the representation of the Coulomb gas with half-integer charges, equivalent to the AFXY(t) model with the Berezinskii-Villain interaction.     

Fractionation of aluminum in natural waters by fluorometry based on the competitive complexation

A novel strategy for the speciation of aluminum has been developed by fluorometry using selective analytical reagents under different pH conditions, which is based on the measurement of the complexation ability of Al for analytical reagents and native organic matters. With Eriochrome Blue Black R, labile monomeric Al and total monomeric Al fractions are determined at pH 4.5 and pH 8.0, respectively. With morin only the labile monomeric Al is determined at pH 4.0, while with 8-hydroxyquinoline the total monomeric Al is determined at pH 5.5. So, acid soluble Al fraction is obtained indirectly. This method is successfully applied to the fractionation of Al in natural waters. The results well agree with those obtained by Driscoll’s 8-hydroxyquinoline/cation-exchange protocol. This fluorometric method possesses some advantages such as high sensitivity, easy manipulation, no separation step and less time-consuming.     

Glucofructans from <Emphasis Type="Italic">Cousinii radians</Emphasis> roots

Structures of glucofructans isolated from Cousinii radians roots were studied by methylation, periodate oxidation, and IR and ¹³C NMR spectroscopies. It was found that fructofuranose units were bonded to each other by inulin-type β-(2→1)-gylcoside bonds. __________ Translated from Khimiya Prirodnykh Soedinenii, No. 1, pp. 3–5, January–February, 2007.     

A note on small ball probability of a Gaussian process with stationary increments

Let {X(t), 0t1} be a Gaussian process with mean zero and stationary increments. Let ²(h) =EX ²(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.Research supported by Charles Phelps Taft Post-doctoral Fellowship of the University of Cincinnati and by the Fok Yingtung Education Foundation of China.     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

短脉冲激光加热分数阶导热及其热应力研究

短脉冲激光加热引起材料内部复杂的传热过程及热变形,现有的以Fourier定律或Cattaneo-Vernotte松弛方程结合弹性理论为框架建立起来热应力理论在刻画其热物理过程存在严重缺陷. 本文基于分数阶微积分理论, 以半空间为研究对象, 建立了分数阶Cattaneo热传导方程和相应的热应力方程, 给出了问题的初始条件和边界条件, 采用拉普拉斯变换方法, 给出了非高斯时间分布激光热源辐射下温度场和热应力场的解析解, 研究了短脉冲激光加热的温度场及热应力场的热物理行为. 数值计算中, 首先对理论解进行数值验证, 然后取分数阶变量$p=0.5$研究温度场和热应力场的变化特点及激光参数对温度和热应力的影响,最后数值计算分数阶参数对温度和热应力场的影响. 计算结果表明, 分数阶Cattaneo传热方程和热应力方程描述的温度和热应力任然具有波动特性,与经典的Fourier传热模型和标准的Cattaneo传热模型相比, 分数阶阶次越大, 热波波速越小, 热波波动性越明显; 反之, 则热波波速越大, 热扩散性越强.激光加热和冷却的速度越快, 温度上升和下降的速度越快, 压应力和拉应力交替变化越快, 温度变化幅值越小, 热应力幅值影响不明显.      

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.