Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

GTSP重钙工艺减少反应料浆含水量的初步研究

1 引言 我国耕地严重缺磷的面积所占比例高达2/3,而目前施肥的磷/氮比例又远远低于国际平均水平,如此的比例失调将严重影响农业增产.在当前施用的磷肥中,普钙和重钙占了很大的比例.由于重钙是用磷酸而非硫酸酸解磷矿,所以不含硫酸钙杂质,有效五氧化二磷的含量也由16～21%提高到45～52%,肥效比普钙提高两到三倍.为此国家"八五"重点项目引进了Hy dro GTSP工艺2×40万吨/年重钙装置.该装置于1999年3月试车以来难以达到满负荷生产. 作者通过对生产工艺各环节的分析,指出反应料浆含水量偏高是影响装置生产负荷的主要因素.料浆中水分的重要来源之一是二次球磨.     

Microchemical and Microstructural Characterization of the Early Stages of the Discontinuous Precipitation Reaction in Al-22 at.% Zn Alloy

High-spatial energy dispersive X-ray (EDX) microanalysis has been applied in order to reveal changes of chemical composition accompanying the formation of the lamellar structure due to discontinuous precipitation reaction in Al-22at.% Zn alloy. The line-scans performed in the reaction zone have shown that the region left behind the grain boundary (GB) after its displacement from the original position is solute depleted, similar to the de-alloying case of diffusion induced grain boundary migration. This enables the allotriomorphs to be transformed into solute-rich lamellae and development of lamellar + structure, especially if the GB moves in the direction perpendicular to its original location. If the direction of movement is strongly inclined to the original location of the GB, the growth usually stops after a short period of time, mainly due to formation of the precipitation free zone (PFZ) ahead of the reaction front. The detailed EDX examination revealed that there is a sharp change in solute content at the lamella/PFZ and PFZ/matrix alloy interfaces. The solute content in the PFZ is lower than in the neighbouring lamella and it corresponds to the equilibrium state. Therefore, one can suppose that the PFZ is formed due to backward movement of the reaction front in order to consume the excess of solute atoms stored in the lamellae in comparison with the equilibrium state. As soon as the equilibrium state is attained within the PFZ, the growth of the colony is completed.     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Diffusion Along Migrating Grain Boundaries

The paper summarizes recent experiments on diffusion at migrating grain boundaries (GBs) occurring during discontinuous reactions, like discontinuous precipitation (DP) and diffusion induced grain boundary migration. Analytical electron microscopy was used for measurements of the solute concentration across individual solute-depleted lamellae. These data combined with information on the growth velocity and the thickness of an individual lamella allowed the determination of the local values of the diffusivities of the moving reaction front of the DP cell in Al–Zn, Ni–Sn, Cu–In and Co–Al alloys. The obtained diffusivities and activation energies are very similar to the relevant parameters of stationary GBs. This allows us to conclude that there is no significant difference in the rates of diffusion along migrating and stationary GBs in the systems investigated. It is therefore believed that the diffusivity values of the moving reaction front of the DP reaction can be a source of reliable information on interfacial diffusion characteristics, especially in systems and/or at temperatures where radiotracer data are not readily available.     

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Process

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ ^d space. This process has an infinitesimal generator of the form mI–(m ^2/α I–Δ) ^α/2, where 0<α<2, m>0, and reduces to the isotropic α-stable process for m=0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in Byczkowski et al. (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint). ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough. Research supported by KBN Grants.     

抢渡长江最优路径的讨论

给出了2003年全国大学生数学建模竞赛D题的一种求解方法.运用拉格朗日乘数法与变分法建立了水速变化时最优路径适合的方程.特别针对问题4,当水流速度线性变化时,导出了求解最优解的函数方程.     

Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ ^N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.