对一道积分不等式题的再思考

利用提升维度的方法并结合几何图形直观分析,给出一道一元函数积分均值不等式的新证明,并将原不等式推广至形式较为对称的不等式,使得原不等式成为新不等式的特例.最后证明新不等式与函数单调递减的定义等价.     

一道不等式的多种证法

我们知道数学教学的一个中心内容就是解题教学.解题教学是培养数学思维能力的重要途径.但是当解题教学成为习题解答或题海战术时,数学思维训练也就消失了.因此我们不仅要向学生展现问题的解答方法,还要注意揭示思维的过程.在教学过程中既要重视逻辑训练,也要重视非逻辑的直觉思维能力的培养.克服思维的封闭性,培养思维的广阔性.引导学生多方面、多角度地观察事物,分析问题,揭示规律,建立联系,在丰富的联想中寻找解决问题的方法,从而培养学生的对一个问题的多角度考虑,而一题多解、多证是培养和提高学生思维能力和解题技巧的途径之一.从对一道题目的多种解法或证法,找出知识间的内在联系,就可以收到举一反三、融会贯通的效果,下面举例说明.     

一道考研试题的探讨

通过对一道考研试题的探讨,得到一系列有趣的结果,旨在激发学生学习数学的兴趣.培养学生的创新意识和能力.     

微分学中不等式的证明

针对微分学不等式列出五种常用证明方法,即利用单调性证明法,利用拉格朗日中值定理证明法,利用最值证明法,利用泰勒公式证明法,和利用凹凸性证明法.实例说明每种方法的使用细节,以达到使初学者能尽快掌握微分学不等式证明的目的.     

一个新的Hilbert型积分不等式

Hilbert积分不等式在分析学中有重要的应用.本文通过引入独立参数λ,利用权函数方法和实分析技巧研究了Hilbert型积分不等式,建立了一个具有独立参数的Hilbert型积分不等式∫0∞∫0∞1lnx-lny1f(x)g(x)/xλ+yλ+max{xλ,yλ}dxdy＜A/λ2{∫0∞x1-λf2(x)dx∫0∞x1-λg2(x)dx}1/2,证明了它的常数因子为最佳值.作为应用,给出它的具有最佳常数因子的等价形式∫0∞y-1[∫0∞1lnx-lny1f(x)/xλ+yλ+max{xλ,yλ}dx2＜[A/λ2]2∫0∞x1-λf2(x)dx.本文所得到的不等式是新的不等式.     

分布意义下两独立变量的Gollwitzer型积分不等式

本文得到了含有Stieltjes积分的两独立变量积分不等式,推广了两独立变量的Gollwitzer型积分不等式。     

微分法在证明不等式中的应用

不等式的证明方法灵活多样,本文以微分学中的一些理论来综述不等式证明的若干方法,并通过范例加以说明.     

一型曲线积分一型曲面积分和Stieltjes积分

本文论述了一型曲线积分、一型曲面积分就是Stieltjes积分,并推出了一型曲线积分和一型曲面积分的计算公式,就是Stieltjes积分化为Riemann积分的公式.     

一个反常积分估计式

类比于反常积分∫+∞0sint/tdt=π2,对积分∫+∞χcost/tdt 做出估计.当x>0时,有∫+∞χcost/tdt=ln1/χ-γ+o(χ)其中γ为Euler常数.     

谈运用高等数学证明不等式的方法

通过实例,探讨了一些运用高等数学证明不等式的方法。     

Predicted and Measured Response of an Integral Abutment Bridge

This project examined several uncertainties of integral abutment bridge design and analysis through field-monitoring of an integral abutment bridge and three levels of numerical modeling. Field monitoring data from a Pennsylvania bridge site was used to refine the numerical models that were then used to predict the integral abutment bridge behavior of other Pennsylvania bridges of similar construction. The instrumented bridge was monitored with 64 gages; monitoring pile strains, soil pressure behind abutments, abutment displacement, abutment rotation, girder rotation, and girder strains during construction and continuously thereafter. Three levels of numerical analysis were performed in order to evaluate prediction methods of bridge behavior. The analysis levels included laterally loaded pile models using commercially available software, two-dimensional (2D) single bent models, and 3D finite element models. In addition, a weather station was constructed within the immediate vicinity of the monitored bridge to capture environmental information including ambient air temperature, solar radiation, wind speed and direction, humidity, rainfall, and barometric pressure. Laterally loaded pile models confirmed that inclusion of multilinear soil springs created from p-y curves is a valid approach for modeling soil–pile interaction within a finite element program. The 2D and 3D numerical models verified the field data indicating that primary accommodation of superstructure expansion and contraction is through rotation of the abutment about its base rather than longitudinal translation, as assumed in the original design of this bridge. Girder axial forces were suspected to be influenced by creep and shrinkage effects in the bridge superstructure. Pile strains were found to be well below strains corresponding to pile plastic moment. Overall, the 2D numerical model and the 3D numerical model predicted very similar behavior.     

Risk Assessment as an Integral Aspect of Capacity Evaluations.

Objective: To highlight the need for rehabilitation psychologists' evaluation of potential risks when examining an individual's decision-making capacity (DMC). Design: A literature review of research regarding decision making, predictive accuracy, and rehabilitation outcomes. Conclusions: The perceived level of risk or potential harm entailed in a decision determines the "sliding scale" for DMC. There is much less research on risk assessment than on the cognitive components needed for DMC, meaning that clinicians usually have to rely on clinical judgment. Clinical judgment is often inaccurate in terms of identifying factors that could increase the risk for harm and is susceptible to errors when predicting future risk. It is therefore argued that the process of assessing an individual's DMC must be matched by an equally rigorous analysis of clinicians' ability to quantify situational risk. Only when these 2 components are properly considered can an adequate standard for DMC be established. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Numerical Study of an Integral Abutment Bridge Supported on Drilled Shafts

The majority of integral abutment bridges (IABs) in the United States are supported on steel H-piles to provide the flexibility necessary to minimize the attraction of large lateral loads to the foundation and abutment. In Hawaii, steel H-piles have to be imported, corrosion tends to be severe in the middle of the Pacific Ocean, and the low buckling capacity of steel H-piles in scour-susceptible soils has led to a preference for the use of concrete deep foundations. A drilled shaft-supported IAB was instrumented to study its behavior during and after construction over a 45-month period. This same IAB was studied using the finite-element method (FEM) in both two- (2D) and three dimensional (3D). The 3D FEM yields larger overall pile curvature and moments than 2D because in 3D, the high plasticity soil is able to displace in between the drilled shafts thereby “dragging” the shafts to a more highly curved profile while soil flow is restricted by plane strain beam elements in 2D. Measured drilled shaft axial loads were higher than the FEM values mainly due to differences between the assumed and actual axial stiffness and to a lesser extent on concrete creep in the drilled shafts and uneven distribution of loads among drilled shafts. Numerical simulations of thermal and stream loadings were also performed on this IAB.     

Field Behavior of an Integral Abutment Bridge Supported on Drilled Shafts

The abutments of integral bridges are traditionally supported on a single row of steel-H-piles that are flexible and that are able to accommodate lateral deflections well. In Hawaii, steel-H-piles have to be imported, corrosion tends to be severe in the middle of the Pacific Ocean, and the low buckling capacity of steel-H-piles in scour-susceptible soils has led to a preference for the use of drilled shaft foundations. A drilled shaft-supported integral abutment bridge was monitored from foundation installation to in-service behavior. Strain gauge data indicate that drilled shaft foundations worked well for this integral bridge. After 45 months, the drilled shafts appear to remain uncracked. However, inclinometer readings provide a conflicting viewpoint. Full passive earth pressures never developed behind the abutments as a result of temperature loading because thermal movements were small and the long term movements were dominated by concrete creep and shrinkage of the superstructure that pulled the abutments towards the stream. In the stream, hydrodynamic loading during the wet season had a greater effect on the abutment movements than seasonal temperature cycling. After becoming integral, the upright members of the longitudinal bridge frame were not vertical because the excavation and backfilling process caused deep seated movements of the underlying clay resulting in the drilled shafts bellying out towards the stream. This indicates the importance and need for staged construction analysis in design of integral bridges in highly plastic clays. Also, the drilled shaft axial loads from strain gauges are larger than expected.     

Field-Measured Response of an Integral Abutment Bridge with Short Steel H-Piles

Integral abutment bridges (IABs) with short steel H-pile (HP) supported foundations ( ? 4?m of pile depth) are economical for many environmentally sensitive sites with shallow bedrock. However, such short piles may not develop an assumed, fixed-end support condition at some depth below the pile cap, which is inconsistent with traditional pile design assumptions involving an equivalent length for bending behavior of the pile. In this study, the response of an IAB with short HP-supported foundations and no special pile tip details such as drilling and socketing is investigated. Instrumentation of a single-span IAB with 4-m-long piles at one abutment and 6.2- to 8.7-m-long piles at the second abutment is described. Instrumentation includes pile strain gauging, pile inclinometers, extensometers to measure abutment movement, earth pressure cells, and thermistors. Pile and bridge response during construction, under controlled live load testing, and due to seasonal movements are presented and discussed. Abutment and pile head rotations due to self-weight, live load, and seasonal movements were all found to be significant. Measured abutment movements were likely affected by both temperature changes and deck creep and shrinkage. Based on the field study results presented here, moderately short HPs driven to bedrock without special tip details appear to perform well in IABs and do not experience stresses larger than those seen by longer piles.