查询词典 integral of a function

In this paper,a proof is made of the equivalence in three definitions of the integral of bounded function in finite set measure.

关于Lebesgue积分，文献有不同的定义，本文给出了测度有限集上有界函数 Lebesgue积分三种不同定义的等价性的一种证明。

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational, basic, differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式，给出了这几个不等式的一些应用实例，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational ,basic , differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality and Minkowski inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式和闵可夫斯基不等式以及这几个不等式的应用，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

Remove row and column' and 'put a big number' have been used for the first boundary;a continuous Regional Identification Function is used on the basis of non-variant nodal virtual flux method for the free surface boundary;Gauss formula is used to make the curved surface integral of the free surface boundary transformed as the difference between volumn integral and other curved surface integral to calculate the integral item on boundaries, which avoiding finding the position of free surface, while the continuous Regional Identification Function is applied to calculate volumn integral and surface integral;Crout and PCG method is used for solusion.

对于已知水头边界，采用"去行去列法"和"置大数法"进行处理；对于自由面穿过的单元，在固定网格节点虚流量法的基础上，引入连续的区域识别函数；对于非稳定渗流中自由面边界积分项，采用高斯公式将求自由面的面积分转化为求体积分与其他面积分之差，避免了求自由面的具体位置，同时在计算体积分和面积分时采用连续的区域识别函数；在解法上，采用直接解法和PCG法。

By means of traditional criterions of generalized integral's convergence and divergence, this paper, from the analysis of integrated function's nature, discovers a series of new criterions, which are briefer and more suitable: on the aspect of generalized integral for functions of a single variable, through the investigation of integrated function, together with the inner relationship between positive series and generalized integral in infinite interval under the condition of positive function, it gives several criterions of generalized integral's convergence and divergence, which are similar to positive series' criterions of convergence and divergence.

本文从分析被积函数本身所具的性质出发，借助传统的广义积分敛散性判别方法，发现1系列更简捷适用的新判别方法：单变量函数广义积分方面，通过考察被积函数，结合正项级数与正函数情形下无穷区间上广义积分的内在联系，给出了几个与正项级数敛散性判别法相类似的广义积分敛散性判别方法；多变量函数广义积分方面，着重讨论了广义2重积分和广义3重积分，结合被积函数的特点，运用比较判别法和柯西判别法，本文给出了判别广义2重积分收敛的1种新方法。

A series of theoretical and experimental studies on the microstrip patch antennas with dielectric superstrates is carried out systematically. Firstly, a comprehensive survey of the microstrip antenna technology and its theories are presented. Then, two analysis techniques, named as the spectral domain Green's function method and the spectral domain integral equation method are developed. The SDGF method is a simplified analysis technique, which takes advantage of the spectral domain immittance approach to find the dyadic Green's function for the stratified media, and obtains the complete closed-form formulas for radiation patterns, directivity, efficiency, gain etc. Some interesting calculated results are presented. The SDIE method is capable of determining resonant frequency, radiation characteristics, input impedance etc, for the covered microstrip patch antennas. Comparing with other full-wave analyses it reduced computation and mathematical labour remarkably. The singularities near the pole location of the surface-wave modes and the difficulty of the integral computation over infinite range of oscillating integrands are dealt with actively and documented well. Results predicted by the SDIE method agree extremely well with our experimental results for resonant frequencies of various superstrates. The superstrate effects on impedance and radiation characteristics also have been investigated experimentally so that the gap due to lack of measured data and computed results for multi-layered superstrate has been filled to some degrees and two useful observations have been achieved for weakening the resonant frequency shift and optimizing the microstrip antenna gain. As a meaningful application, the design and analysis of a 4×4 patch array are introduced.

实用微带天线普遍加介质覆盖层来防护热，物理损伤和环境影响，也可能在飞行或严酷气候条件下自然形成覆盖层本论文对带介质覆盖层的微带贴片天线系统地作了一系列理论和实验研究首先，对微带天线技术和理论作了全面的综述然后，发展了两种分析方法，称为谱域格林函数法和谱域积分方程法SDGF法是一种简化的分析技术，它利用谱域导抗法来求得分层媒质的并矢格林函数，并得出方向图、方向性系数、增益等参数的完整的闭式公式，给出了一些令人感兴趣的计算结果SDIE法能确定介质覆盖微带天线的谐振频率、辐射特性和输入阻抗等与其它全波分析方法相比，它明显减小了计算量和数学工作量已有效地处理了表面波模极点附近的奇异性和对振荡的被积函数的无限区间积分计算的困难对于不同覆盖层情况下的谐振频率，由SDIE法所预示的结果与我们的实验结果吻合得很好对于覆盖层对阻抗和辐射特性的影响也已作了实验研究这在某种程度上填补了缺少实验数据和计算结果之间的间隙，并得出了缓解谐振频率偏移和优化天线增益的两点有用的结论作为有意义的应用，介绍了一个4×4元矩形贴片天线阵的设计和分析。

Numerical Integration Single variable f - Calculates the definite integral of a single variable function f(x using numerical methods: The Trapezoidal, the Midpoint and the Simpson's method.

计算一个单精度变量的整数部分的函数f 使用的数学方法：梯形法、中间点中和辛普森方法。

It was suggested that aging did not only influence diastolic filling period, but also influence STI. 3. With increasing age, whether at rest or during exercice, fractional shortening of minor semiaxis, mean velocity of circumferential fiber shortening, cardiac index and ejection fraction did not change, but immediately post pacing, PEP prolonged and PEP/ LVET increased in elderly group and did not change in young group. It was suggested that STI was sensitivity in detecting systolic function of LV. 4. There was a good correlation between aging and decreased compliance and diastolic function of left ventricle which was manifested by a reduction of early diastolic peak folw velocity, time-velocity integral of early diastole, 1/3 filling fraction (1/3FF)(r=-0.958,-0.875 and -0.937, P＜0.01～0.001) and a significant compansatory increase in peak flow velocity of atrial contraction, time-velocity integral of atrial contraction, the ratio between APFV and EPFV and the ratio of ATVI and Total TVI (r=0.958, 0.956, 0.966 and 0.986, P＜0.001) and the prolongation of Dec T and RFP with increasing age. It was particularly true for the subgroups above the age of 50 years.

静息状态下收缩时间间期与年龄无相关性，负荷后即刻，老年组射血前期延长(P＜0.05)，射血前期与射血期比值加大(P＜0.05)，青年组无变化，提示年龄因素亦影响STI，但由于静息状态下得到代偿，STI变化往往表现不明显。3、静息及负荷状态左室短轴缩短率、平均周径缩短率、心指数及射血分数均不随年龄增长而变化，而负荷后即刻老年组PEF延长，PEP/LVET加大，提示STI是检测左室收缩功能的敏感指标。4、随年龄增长，心房收缩期充盈峰值流速、流速积分、充盈分数及心房收缩期与舒张早期充盈峰值流速之比值显著增高(r＝0.958、0.956、0.986及0.966，P均＜0.001)，舒张早期充盈峰值流速及流速积分、1/3充盈分数(1/3FF)下降(r＝-0.958、-0.875及-0.937，P＜0.01～0.001)。

Results: The changes following hydrocortisone intervention of rats could be identified by PCA, and the kidney deficiency syndrome rats entered a state of "hyperfunction", involving a series of changes in metabolic pathway and metabolic network. The response integral area of lactic acid (δ 1.37) increased, indicating the accumulation of metabolites of lactic acid. The amount of dimethylamine (δ 2.72) also increased, indicating that the disorder of kidney function was related to kidney-deficiency syndrome. The response of creatinine (δ 3.87), asparaginic acid (δ 2.83), taurine (δ 3.44, 3.28), and hippurate (δ 7.84, 7.56, 7.64, 3.97) all decreased significantly, indicating that the hydrocortisone might cause adrenal cortex excretion injury. The response integral area of succinate (δ 2.41) and citric acid (δ 2.53, 2.68) decreased, which was usually caused by disorder of the mitochondrial function.

结果：PCA方法处理肾虚证和空白对照大鼠尿液数据显示，两组大鼠的数据可以在得分图实现分类，与对照组比较肾虚组大鼠由于氢化可的松诱导，大鼠机体进入一个过消耗后衰弱的状态，相关代谢发生显著变化，乳酸(δ 1.37)代谢发生堆积；二甲胺(δ 2.72)的含量增加，提示肾虚与肾功能异常是密切相关的；天冬氨酸(δ 2.83)，牛磺酸(δ 3.44， 3.25)，马尿酸(δ 7.84， 7.56， 7.64， 3.97)，肌氨酸(δ 3.87)等的相对积分面积明显下降，这预示糖皮质激素可能引起了肾上腺皮质分泌功能的损害；琥珀酸(δ 2.41)和柠檬酸(δ 2.53， 2.68)是三羧酸循环能量代谢及糖酵解的中间产物，其含量的降低，通常是由于线粒体功能紊乱所引起。

Remove row and column' and 'put a big number' have been used for the first boundary;a continuous Regional Identification Function is used on the basis of non-variant nodal virtual flux method for the free surface boundary;Gauss formula is used to make the curved surface integral of the free surface boundary transformed as the difference between volumn integral and other curved surface integral to calculate the integral item on boundaries, which avoiding finding the position of free surface, while the continuous Regional Identification Function is applied to calculate volumn integral and surface integral;Crout and PCG method is used for solusion.

对于已知水头边界，采用&去行去列法&和&置大数法&进行处理；对于自由面穿过的单元，在固定网格节点虚流量法的基础上，引入连续的区域识别函数；对于非稳定渗流中自由面边界积分项，采用高斯公式将求自由面的面积分转化为求体积分与其他面积分之差，避免了求自由面的具体位置，同时在计算体积分和面积分时采用连续的区域识别函数；在解法上，采用直接解法和PCG法。

Plunder melds and run with this jewel!

掠夺melds和运行与此宝石！

My dream is to be a crazy growing tree and extend at the edge between the city and the forest.

此刻，也许正是在通往天国的路上，我体验着这白色的晕旋。