数学分析中分区间方法及应用

系统地介绍了数学分析中的"分区间法"。结合数学分析教材中关于一致连续、积分的极限、一致收敛等定义以及性质对分区间法的应用进行了系统探讨,得到一些有用的性质和方法。在此基础上,举出一些有代表性的例子,进一步运用所得到的性质和方法,力求使学生在数学分析的学习中加深印象,提高教学效果。

用均匀设计的分区间优化法研究重油乳化剂

用均匀设计与调优软件进行实验方案的设计 ,并用分区间优化方法进行实验结果的优化。考察了能在界面上形成复合物的乳化剂对乳状液的稳定化作用 ,以及非离子表面活性剂间的协同作用 ,并得到了稳定性较好的乳化剂配方。最佳乳化剂配方为 w (脂肪酸 ) =78.1 % ,w ( Tween 80 ) =1 1 .2 % ,w (三乙醇胺 ) =4 .5% ,w ( Tween 2 0 ) =3 .3 % ,w (苯酚 ) =2 .9%。加入量为乳化油总质量的 0 .0 9%时 ,可使油水质量比为70∶ 3 0的油包水型乳化重油在 85℃下稳定 5d以上 ,常温下稳定 1 a以上。研究结果表明 ,乳液稳定性与重油性质密切相关。

用区间分半法确定冷却水最佳出口温度

本文在确定冷却水最佳出口温度方法中对区间分半法和直接迭代法作了比较和分析 ,提出了最优化方法。

“标根分区间”法的引申及应用

我们知道,解高次不等式的基本思路是通过因式分解,将它化成左边为若干个一次或二次因式的积且右边为0的形式,然后可用"标根分区间"法解之.

带有先验估计的区间逐次分半积分法

为准确高效地计算数值积分,针对复化积分法收敛速度慢的缺点。笔者根据复化梯形公式的积分余项,提出了先验估计积分区间等分数的方法,并与区间逐次分半积分法进行了组合,得到了优化后的区间逐次分半积分法,最后对■和■进行了计算,结果表明:区间逐次分半积分法需要分别对分10次和8次才能收敛到目标精度,而优化后的区间逐次分半积分法都只需对分1次便可达到目标精度。证明基于先验估计区间等分法的区间逐次分半积分法不但提高了数值积分的收敛速度,即让区间逐次分半积分法的时间复杂度从O(nlogn)降到了O(n),而且节省了计算空间。

掌握导数符号判定的三种方法

导数是解决函数单调性的有力工具，导数符号就是判断或求解函数单调性问题的关键．课本例题中，给出了三种常见的方法：观察法、不等式法、根分区间法．老师要指导学生进行总结归纳，从而提高学生的解题能力．

利用Mathematica求解方程的近似根

本文分别按牛顿迭代法原理和对分区间法原理,利用Mathematica编程计算了同一个函数在误差允许范围内的所有近似根,通过比较例题的结果,得出牛顿法收敛速度快于对分区间法,且误差较小。

Adaptive Spatial-division Split-step Fourier Migration Method

This paper presents a new depth migration method, adaptive spatial-division split-step Fourier （ASDSSF） migration. In this method we introduce the idea of a stratified phase shift migration into the split-step Fourier （SSF） migration to make an accurate and efficient wave field image when sharp discontinuities appear in the velocity field. In principle, the ASDSSF migration is a multi-reference slowness （reciprocal of velocity） （MRS） migration. Compared to previous MRS migration methods, this method uses fewer reference slowness values without accuracy loss. The reference slowness is determined in this paper according to an error-control parameter of the perturbation term in the SSF operator and the variation of the complet velocity field. The velocity corresponded to reference slowness can define a spatial division. Each division can also be divided into several discontinuous spatial subdivisions to effectively reduce the number of reference slowness values needed. The choice of reference slowness, including the number of reference slowness values needed and how to construct the spatial divisions, is adaptive and reasonable at each extrapolation step （depth step）. A simple and economical smoothing filter in the wave number-frequency domain is designed to avoid artifacts in the wave field extrapolation due to the presence of sharp discontinuities in the velocity field. For comparable conditions the present approach to migration is expected to be computationally more efficient and accurate than other MRS migration methods. The performance of the method is demonstrated on a simple 2D prestack model and the prestack SEG/EAEG salt dataset.

An Urban Open Space Extraction Method: Combining Spectral and Geometric Characteristics

This paper introduces an advanced method based on remote sensing and Geographic Information System for urban open space extraction combining spectral and geometric characteristics. From both semantic and remote sensing perspectives, a hybrid hierarchy structure and class organization of open space are issues and mapped from one to another. Based on per-pixel and segmentation mechanism separately, two classification approaches are performed. Owing to prior of spatial aggregation and spectral contribution, the segmentation-based classification exhibits its superiority over a pixel-based classification. Finally a GIS-based post procedure is hired to eliminate some unsuitable open space components in both spatial and numerical constraints on the one hand, and separate open space some fabrics from fused remote sensing classes by defining their Shape Index on the other hand. The case study of Beer Sheva based on ASTER data proves this method is a feasible way for open space extraction.

Combining TOPSIS and GRA for supplier selection problem with interval numbers

Supplier selection can be regarded as a typical multiple attribute decision-making problem. In real-world situation, the values of the alternative attributes and their weights are always being nondeterministic, and as a result of this, the values are considered interval numbers. In addition, the common approach to measure the similarity between alternatives through their distance suffers from some minor shortcomings. To address these problems, this study develops a novel hybrid decision-making method by combining the technique for order preference by similarity to an ideal solution （TOPSIS） with grey relational analysis （GRA） for supplier selection with interval numbers. By introducing the intervals theory, the extensions of Euclidean distance and grey relational grade are defined. And then a new comprehensive closeness coefficient is constituted for supplier alternatives evaluation based on the interval Euclidean distance and the interval grey relational grade, which could indicate the distance-based similarity and the shape-based similarity simultaneously. A mtmerical example is taken to validate the flexibility of the proposed method, and result shows that this method can tackle the uncertainty in real-world supplier selection and also help decision makers to effectively select optimal suppliers.

Application of Integration of Spatial Statistical Analysis with GIS to Regional Economic Analysis

This paper summarizes a few spatial statistical analysis methods for to measuring spatial autocorrelation and spatial association, discusses the criteria for the identification of spatial association by the use of global Moran Coefficient, Local Moran and Local Geary. Furthermore, a user-friendly statistical module, combining spatial statistical analysis methods with GIS visual techniques, is developed in Arcview using Avenue. An example is also given to show the usefulness of this module in identifying and quantifying the underlying spatial association patterns between economic units.

L_1 adaptive controller of nonlinear reference system in presence of unmatched uncertainties

An extension of L_1 adaptive control is proposed for the unmatched uncertain nonlinear system with the nonlinear reference system that defines the performance specifications. The control law adapts fast and tracks the reference system with the guaranteed robustness and transient performance in the presence of unmatched uncertainties. The interval analysis is used to build the quasi-linear parameter-varying model of unmatched nonlinear system, and the robust stability of the proposed controller is addressed by sum of squares programming. The transient performance analysis shows that within the limit of hardware a large adaption gain can improve the asymptotic tracking performance. Simulation results are provided to demonstrate the theoretical findings of the proposed controller.

Two-Stage Resource Allocation Scheme for Three-Tier Ultra-Dense Network

In 5 G Ultra-dense Network(UDN), resource allocation is an efficient method to manage inter-small-cell interference. In this paper, a two-stage resource allocation scheme is proposed to supervise interference and resource allocation while establishing a realistic scenario of three-tier heterogeneous network architecture. The scheme consists of two stages: in stage I, a two-level sub-channel allocation algorithm and a power control method based on the logarithmic function are applied to allocate resource for Macrocell and Picocells, guaranteeing the minimum system capacity by considering the power limitation and interference coordination; in stage II, an interference management approach based on K-means clustering is introduced to divide Femtocells into different clusters. Then, a prior sub-channel allocation algorithm is employed for Femtocells in diverse clusters to mitigate the interference and promote system performance. Simulation results show that the proposed scheme contributes to the enhancement of system throughput and spectrum efficiency while ensuring the system energy efficiency.

Schedule algorithm for mitigating inter-cell interference based on orthogonal complement space

In order to avoid severe performance degradation led by the inter-cell interference (ICI) in orthogonal frequency division multiple access (OFDMA) systems with a frequency reused factor (FRF) of 1,distributed schedule algorithm (DS-OCS) and distributed proportional fairness schedule algorithm (DPFS-OCS) based on orthogonal complement space (OCS) were proposed. The first right and left singular vectors of the channel that the user experienced were selected as the transmitting and receiving beamforming vectors. An interference space was spanned by the left singular vectors of the entire interference users in the same channel. The most suitable user lay in the OCS of the interference space was scheduled to avoid suffering interference from neighboring cells based on the criterion of system capacity maximizing and proportional fairness. The simulation results show that the average system capacity can be improved by 2%-4% compared with the DS-OCS algorithm with the Max C/I algorithm,by 6%-10% compared with the DPFS-OCS algorithm with the PF algorithm.

A collocation interval analysis method for interval structural parameters and stochastic excitation

Uncertainty propagation, one of the structural engineering problems, is receiving increasing attention owing to the fact that most significant loads are random in nature and structural parameters are typically subject to variation. In the study, the collocation interval analysis method based on the first class Chebyshev polynomial approximation is presented to investigate the least favorable responses and the most favorable responses of interval-parameter structures under random excitations. Compared with the interval analysis method based on the first order Taylor expansion, in which only information including the function value and derivative at midpoint is used, the collocation interval analysis method is a non-gradient algorithm using several collocation points which improve the precision of results owing to better approximation of a response function. The pseudo excitation method is introduced to the solving procedure to transform the random problem into a deterministic problem. To validate the procedure, we present numerical results concerning a building under seismic ground motion and aerofoil under continuous atmosphere turbulence to show the effectiveness of the collocation interval analysis method.

Method to avoid the structural interference of the thrust system of a shield tunneling machine

Several malfunctions of the shield tunneling machine （STM） caused by structural interference have been encountered in actual tunnel excavation. This paper is focusing on providing an effective method to avoid the structural interference based on making the reachable and the required workspaces of the thrust system match each other. The main structure of the thrust mechanism is analyzed, and coordinate systems are built up to describe the pose and workspace of the thrust mechanism. Constraint conditions are derived and the formulation of each constraint condition is carried out to facilitate the analysis of the reachable workspace of the thrust mechanism. Meanwhile, a reachable workspace determination algorithm is introduced based on interval analysis method. The mathematical model for determining the required workspace of the thrust mechanism is presented based on the analysis of the process when the STM excavates along a specific tunnel axis. Two applications are included to show how to avoid these problems by choosing reasonable parameters of the designed tunnel axis and the key structural parameters of the thrust mechanism based on workspace matching.

An iterative interval analysis method based on Kriging-HDMR for uncertainty problems

In recent years,growing attention has been paid to the interval investigation of uncertainty problems.However,the contradiction between accuracy and efficiency always exists.In this paper,an iterative interval analysis method based on Kriging-HDMR(IIAMKH)is proposed to obtain the lower and upper bounds of uncertainty problems considering interval variables.Firstly,Kriging-HDMR method is adopted to establish the meta-model of the response function.Then,the Genetic Algorithm&Sequential Quadratic Programing(GA&SQP)hybrid optimization method is applied to search for the minimum/maximum values of the meta-model,and thus the corresponding uncertain parameters can be obtained.By substituting them into the response function,we can acquire the predicted interval.Finally,an iterative process is developed to improve the accuracy and stability of the proposed method.Several numerical examples are investigated to demonstrate the effectiveness of the proposed method.Simulation results indicate that the presented IIAMKH can obtain more accurate results with fewer samples.

A novel method of Newton iteration-based interval analysis for multidisciplinary systems

A Newton iteration-based interval uncertainty analysis method(NI-IUAM) is proposed to analyze the propagating effect of interval uncertainty in multidisciplinary systems. NI-IUAM decomposes one multidisciplinary system into single disciplines and utilizes a Newton iteration equation to obtain the upper and lower bounds of coupled state variables at each iterative step.NI-IUAM only needs to determine the bounds of uncertain parameters and does not require specific distribution formats. In this way, NI-IUAM may greatly reduce the necessity for raw data. In addition, NI-IUAM can accelerate the convergence process as a result of the super-linear convergence of Newton iteration. The applicability of the proposed method is discussed, in particular that solutions obtained in each discipline must be compatible in multidisciplinary systems. The validity and efficiency of NI-IUAM is demonstrated by both numerical and engineering examples.

Two-Level Additive Schwarz Methods Using Rough Polyharmonic Splines-Based Coarse Spaces

This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines （RPS for short）. As an approximation space of the eUiptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.