Upper and lower bounds of Borel–Cantelli Lemma in a general measure space

It is known that the Borel–Cantelli Lemma plays an important role in probability theory. Many attempts were made to generalize its second part. In this article, we investigate the upper and lower bounds of Borel–Cantelli Lemma for the nonnegative functions in a general measure space. Our results extend the corresponding results obtained in a probability space. Some examples including dependent random variables are illustrated to our results.     

On the average‐case complexity of Shellsort

We prove a lower bound expressed in the increment sequence on the average‐case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked. A special case of this lower bound yields the general Jiang‐Li‐Vitányi lower bound. We obtain new results, for example, determining the average‐case complexity precisely in the Yao‐Janson‐Knuth 3‐pass case.     

Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution     

含参强向量均衡问题的适定性

研究实Hausdorff拓扑线性空间中含参强向量均衡问题的适定性。证明了在适当条件下,由近似网定义的含参适定性等价于近似解映射的上半连续性,并给出了所研究问题两种适定性的充分性条件。     

甲烷/石松子粉尘混合体系爆炸下限的变化规律

基于标准20 L球形爆炸装置,在相同测试条件下, 分别测量了石松子粉尘、甲烷和不同浓度配比的甲烷/石松子粉尘混合体系爆炸下限,并将测试结果与Le Chatelier’s law、Bartknecht curve、Jiang method等混合体系爆炸下限预测结果进行了对比。结果表明:低于爆炸下限的甲烷和低于爆炸下限的石松子粉尘混合后仍具有爆炸危险性。石松子粉尘爆炸下限随混合体系中甲烷体积分数的增高而减小。Le Chatelier’s law、Bartknecht curve、Jiang method均不能准确预测甲烷/石松子粉尘混合体系爆炸下限。Le Chatelier’s law对甲烷体积分数φ与甲烷爆炸下限φ_L之比φ/φ_L＜0.5的混合体系爆炸下限的预测值偏小,而对φ/φ_L＞0.5的混合体系预测值偏大;Bartknecht curve在预测φ/φ_L＞0.5的混合体系爆炸下限时适用性较好,而对于φ/φ_L＜0.5的混合体系预测值偏小;Jiang method不适用于预测甲烷/石松子粉尘混合体系爆炸下限。     

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.     

非负不可约矩阵Perron根新的下界序列

Estimate bounds for the Perron root of a nonnegative matrix are important in theory of nonnegative matrices. It is more practical when the bounds are expressed as an easily calculated function in elements of matrices. For the Perron root of nonnegative irreducible matrices, three sequences of lower bounds are presented by means of constructing shifted matrices, whose convergence is studied. The comparisons of the sequences with known ones are supplemented with a numerical example.     

一类三阶常微分方程非线性三点边值问题解的存在性

In this paper, existence of solutions of third-order differential equation
y′″（t）=f（t,y（t）,y′（t）,y″（t））
with nonlinear three-point boundary condition
{g（y（a）,y′（a）,y″（a））=0,
h（y（b）,y′（b））=0,
I（y（c）,y′（c）,y″（c））=0
is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a, b, c∈ R,a〈 b〈 c; f ： [a,c]×R^3→R,g：R^3→R,h：R^2→R and I：R^3→R are continuous functions. The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem. As an application, an example with an explicit solution is given to demonstrate the validity of the results in this paper.     

Modeling of the three-phase equilibrium in systems of the type water + nonionic surfactant + alkane

Liquid–liquid equilibria of systems water (A) + C_iE_j surfactant (B) + n-alkane (C) have been modeled by a mass-action law model previously developed and so far successfully applied to a series of binary water + C_iE_j systems and to the ternary system water + C₄E₁ + n-dodecane. These calculations provide the basis for the presented modeling. The aqueous systems give information about the association constants and the χ_AB-parameter of the Flory–Huggins theory and the ternary C₄E₁-system provides universal temperature functions for the χ_AC- and the χ_BC-parameter. The three-phase equilibrium for seven ternary C_iE_j systems (i = 6–12, j = 3–6) has been calculated by fitting one additional parameter for each of both temperature functions to the characteristic “fish-tail” point. The agreement with the experimental data is reasonably well. For systems with very small three-phase areas the results can considerably be improved by individual temperature functions that incorporate the experimental temperature maximum of the “fish” into the parameter fit. Based on the parameters of the system water + C₈E₄ + n-C₈H₁₈ the “fish-shaped” phase diagram of the system water + C₈E₄ + n-C₁₄H₃₀ was predicted reasonably well.     

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

New lower bounds for three- and four-level designs under the centered -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.