When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,…,N}, chosen randomly according to a binomial model with parameter p(N), with N^?1 = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N^?1/2) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N^?1/2 = o(p(N)) then both the sum and difference sets almost surely have size (2N + 1) ? O(p(N)^?2), and so the ratio in question is almost surely very close to one. If p(N) = c · N^?1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_f,g · N^?1/2, for some computable constant c_f,g, such that one form almost surely dominates below the threshold and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Lebesgue type decompositions for nonnegative forms

A nonnegative form on a complex linear space is decomposed with respect to another nonnegative form : it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by which is almost dominated by . The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.     

任意随机序列关于非齐次马氏链的随机和的一类随机偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic sequence, as a measure of dissimilarity between the joint distribution on measure P and the Markov distribution on measure Q, is introduced. A class of random approximation theorems for arbitrary stochastic dominated sequence are obtained by using the tools of generating functions and the tailed-probability generating functions.     

具有主导晶面的BiOIO3/BiOCl异质结制备与光催化性能

分别以乙二醇/去离子水为溶剂,通过溶剂热/水热法分别制备了具有不同主导晶面的BiOIO₃/{110}BiOCl和BiOIO₃/{001}BiOCl异质结。采用X射线衍射、扫描电子显微镜、能量色散谱和紫外可见漫反射光谱对制备的BiOIO₃/BiOCl光催化剂进行了表征。在可见光照射下,通过对罗丹明B和苯酚水溶液的光催化降解,考察了BiOIO₃/BiOCl异质结的光催化活性。结果显示25%BiOIO₃/{110}BiOCl异质结具有最高的光催化效率。BiOIO₃/{110}BiOCl较好的光催化性能是由于其在可见光区较强的光吸收,以及异质结结构和BiOCl所具有的(110)主导晶面有利于光生载流子的分离。超氧自由基(·O₂^-)和空穴(h⁺)是光催化过程中的主要活性物质。此外,根据实验结果探讨了光催化性能增强的机理。     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

On the existence of Pettis integrable functions which are not Birkhoff integrable

Let be a weakly Lindelöf determined Banach space. We prove that if is non-separable, then there exist a complete probability space and a bounded Pettis integrable function that is not Birkhoff integrable; when the density character of is greater than or equal to the continuum, then is defined on with the Lebesgue measure. Moreover, in the particular case (the cardinality of being greater than or equal to the continuum) the function can be taken as the pointwise limit of a uniformly bounded sequence of Birkhoff integrable functions, showing that the analogue of Lebesgue's dominated convergence theorem for the Birkhoff integral does not hold in general.

     

Parametric data-based turbulence modelling for vortex dominated flows

Computational fluid dynamics simulations employing eddy-viscosity turbulence models remain the baseline numerical tool in the aerospace industry, mainly due to their numerical stability and computational efficiency. However, many industrially relevant cases require a level of accuracy that is not routinely achieved by global turbulence models. The simulation of leading-edge vortices shed at low aspect ratio wings is one such class of flows that remains a challenge for turbulence modelling. A local approach is proposed in which a parametrised eddy-viscosity turbulence model is calibrated using experimental results of configurations and flow conditions similar to the one being analysed. In this paper, the Spalart–Allmaras one-equation model is enhanced with additional source terms, which are exclusively active in the vortex field. An automatic optimisation procedure with experimental data as reference is then applied. The resulting optimised model improves the eddy viscosity distribution for a limited but relevant range of configurations and flow conditions.     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A modified Crank-Nicolson scheme with incremental unknowns for convection dominated diffusion equations

A modified Crank-Nicolson scheme based on one-sided difference approximations is proposed for solving time-dependent convection dominated diffusion equations in two-dimensional space. The modified scheme is consistent and unconditionally stable. A priori L² error estimate for the fully discrete modified scheme is derived. With the use of the incremental unknowns preconditioner at each time step, a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified Crank-Nicolson scheme.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.