可列值随机变量序列的一个小偏差定理

本文得到了可列值随机变量序列的用不等式表示的强极限定理,即小偏差定理,包含了［１］的结果．［１］中研究的是相对于齐次Ｍａｒｋｏｖ 链的偏差,而本文允许非齐次的情形．     

Nonnormality and stochastic differential equations

A highly nonnormal Jacobian may give rise to large transients. This behaviour has been shown to have implications for (a) the relevance of linearising a nonlinear system and (b) the timestep restrictions required to keep a numerical method stable. Here, we show that nonnormality also manifests itself for stochastic differential equations. We give an example of a family of systems that is stable without noise, but can be made exponentially unstable in mean-square by a noise perturbation that shrinks to zero as the nonnormality increases. We then show via finite-time convergence theory that an Euler approximation shares the same property, giving a discrete analogue of the result. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65C30, 34F05     

大型海湾水交换计算中随机游动方法的应用研究

水交换矩阵是反映大型海湾不同子区域之间水交换关系的有效工具,该文对随机游动方法在水交换矩阵计算中的应用进行了研究．通过与对流扩散模型的对比,指出用随机游动方法计算水交换矩阵比用对流扩散方程求解更快更灵活．同时,为了对不同区域之间的水交换特性进行长期快速预测,采用了Thompson提出的预测矩阵方法．通过理论分析得出,影响该矩阵预测效果的因素主要有:粒子数目、分区数和预测矩阵取值时间．以一个矩形海湾为例,将数值模拟结果和预测矩阵预测结果进行对比,分析了以上各因素对预测效果的影响．结果表明,粒子数目越多,预测矩阵取值时间越长,预测矩阵的预测效果越好．在粒子数目足够多的情形下,分区数越多,预测效果越好．相反,粒子数目如果较少,过多的分区数将可能导致预测矩阵效果太差而不可用．     

Distinguishing between Decaffeinated and Regular Coffee by HS-SPME-GC×GC-TOFMS,Chemometrics, and Machine Learning

Coffee, one of the most popular beverages in the world, attracts consumers by its rich aroma and the stimulating effect of caffeine. Increasing consumers prefer decaffeinated coffee to regular coffee due to health concerns. There are some main decaffeination methods commonly used by commercial coffee producers for decades. However, a certain amount of the aroma precursors can be removed together with caffeine, which could cause a thin taste of decaffeinated coffee. To understand the difference between regular and decaffeinated coffee from the volatile composition point of view, headspace solid-phase microextraction two-dimensional gas chromatography time-of-flight mass spectrometry (HS-SPME-GC×GC-TOFMS) was employed to examine the headspace volatiles of eight pairs of regular and decaffeinated coffees in this study. Using the key aroma-related volatiles, decaffeinated coffee was significantly separated from regular coffee by principal component analysis (PCA). Using feature-selection tools (univariate analysis: t-test and multivariate analysis: partial least squares-discriminant analysis (PLS-DA)), a group of pyrazines was observed to be significantly different between regular coffee and decaffeinated coffee. Pyrazines were more enriched in the regular coffee, which was due to the reduction of sucrose during the decaffeination process. The reduction of pyrazines led to a lack of nutty, roasted, chocolate, earthy, and musty aroma in the decaffeinated coffee. For the non-targeted analysis, the random forest (RF) classification algorithm was used to select the most important features that could enable a distinct classification between the two coffee types. In total, 20 discriminatory features were identified. The results suggested that pyrazine-derived compounds were a strong marker for the regular coffee group whereas furan-derived compounds were a strong marker for the decaffeinated coffee samples.     

On the imaginary parts of chromatic roots

While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (ie, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We also show that for any fixed $\text{◂+▸}p\in \left(0,1\right)$, almost every random graph $G$ in the Erdös-Rényi model has a nonreal root.     

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

Discrete Green's Function for Lattice Random Walk

A lattice random walk model based on particles scattering on discrete lattice of homogenous space is introduced. The discrete Green's function (DFG) for two-dimensional and three-dimensional lattice random walk of photon is found and proved by mathematical induction. The convolution theorem of photon lattice random walk is presented. They can be used with the method of images to calculate the photon density distribution in semi-infinite and finite slab homogenous turbid media such as tissue.     

Collision problems of random walks in two-dimensional time

The collision problems of two-parameter random walks are studied. That is, some criteria have been established in terms of the characteristic functions of two or more mutually independent random walks in order to determine if they meet infinitly often in certain restricted time sets.     

Dissipation, decoherence and preparation effects in the spin-boson system

The dynamics of the reduced density matrix of the driven dissipative two-state system is studied for a general diagonal/off-diagonal initial state. We derive exact formal series expressions for the populations and coherences and show that they can be cast into the form of coupled nonconvolutive exact master equations and integral relations. We show that neither the asymptotic distributions, nor the transition temperature between coherent and incoherent motion, nor the dephasing rate and relaxation rate towards the equilibrium state depend on the particular initial state chosen. However, in the underdamped regime, effects of the particular initial preparation, e.g. in an off-diagonal state of the density matrix, strongly affect the transient dynamics. We find that an appropriately tuned external ac-field can slow down decoherence and thus allow preparation effects to persist for longer times than in the absence of driving. Received 23 October 1998 and Received in final form 26 February 1999     

The existence of phase V in the Mandelbrot percolation process

Dekking and Meester defined six phases for a subclass of random Cantor sets consisting of those generated by Bernoulli random substitutions. They proved that the random Sierpinski carpet passed through all these phases asp tended from 0 to 1, but the were not able to prove the existencne of phase V in the Mandelbrot percolation process. In this paper, we accomplish the proof by improving their methods.Research supported by the Chinese Natural Science Foundation.