分数函数

The research shows that gas concentration shows generally an increasing tendency along the working face airflow direction, from 0.095% to 0.18%, under the effect of gas drainage points, up-and-down changing locally; gas distribution in a transect of working face is unequal, under the effect of gas emission, ventilation and drainage, the gas concentration near coal wall is larger than that near goaf, which is a concave function distribution within a distance of 1876.5 m from the inlet of working face, after greater than 76.5m, which is a decreasing distribution from coal wall to goaf.

研究表明，该工作面瓦斯体积分数沿风流方向总体呈逐渐增加趋势，介于0.095%~0.18%之间，受瓦斯抽放点的影响，局部有起伏变化；工作面横断面瓦斯体积分数分布不均匀，受瓦斯涌出、通风和抽放的影响，靠近煤壁侧瓦斯体积分数要高于采空区侧，在距离进风口约18~76.5 m范围内为凹函数分布，大于76.5 m之后，从煤壁至采空区为单调下降函数分布。

It is proved that the fractional Fourier transform has the cascade feature, and the optical transfer function is the special case of fractional transfer function.

证明了其具有分数傅里叶变换的级联性，指出常规傅里叶变换下的光学传递函数为分数传递函数的特例。

Present a Ambiguity Function based on Fractional Fourier Transform. Its properties are studied. The application is discussed carefully that how to extract cube phase information by the Fractional Ambiguity Function.

给出了基于分数阶傅立叶变换的模糊函数，研究了它们的性质，重点讨论了分数阶模糊函数在三次相位信息提取中的应用。

Based on the normative fractional Fourier transform and the concept of the optical transfer function, using the linear system theory, the mathematic expressions of fractional transfer function in Fresnel diffraction system and Fraunhofer diffraction system are given respectively.

基于规范光学分数傅里叶变换，从光学传递函数概念出发，根据菲涅耳衍射、夫琅和费衍射定义，应用线性系统理论，分别给出了菲涅耳与夫琅和费衍射系统在规范分数傅里叶变换下的光学传递函数数学表达式。

One position control system is introduced as the objective controlled by FOC. Based on Mittag-Leffler function, the fractional order control system with full memory length is adopted as evaluation criterion, and an approximation performance function of discretization methods is proposed. Analyzing the time domain error and approximation performance, the investigation to various discretization methods is carried out comparatively in detail.

引入一个位置控制系统作为分数阶控制对象，将Mittag-Leffler函数的全记忆长度分数阶控制系统作为评估基准，提出了一种离散算法的近似性能函数，通过时域误差分析以及近似性能分析，对比研究了分数阶微积分算子的数字实现方法。

The main results are as follows: the relations between local fractional integrated semigroups and the corresponding Cauchy problem, global fractional integrated semigroups and regularized semigroups are given; introduction of the notion of regularized resolvent families, and the generation theorem and analyticity criterions for regularized resolvent families are obtained; the spectral inclusions between fractional resolvent family and its generator, and the approximation for fractional resolvent families in the cases of generators approximation and fractional orders approximation; elliptic operators with variable coefficients generating fractional resolvent family on L^2 by using numerical range techniques; and the L^p theory for elliptic operators with real coefficients highest order are obtained by Sobolev''s inequalities and the a priori estimates for elliptic operators; and a kind of coercive differential operators generates fractional regularized resolvent family by applying the Fourier multiplier method, functional calculus and some basic properties of Mittag-Leffler functions.

主要结论是：给出了局部分数次积分半群和相应的Cauchy问题的关系以及分数次积分半群和正则半群的关系；引入了正则预解族的概念，并给出了其生成定理和解析生成法则；给出了分数次预解族与其生成元的谱包含关系，并研究了在生成元逼近和分数阶逼近两种情况下相应的预解族的逼近问题；利用数值域方法证明了具变系数的椭圆算子在L^2上生成分数次预解族；利用Sobolev不等式和椭圆算子的先验估计证明了具变系数的椭圆算子在其最高项系数为实数时在L^p上生成分数次预解族；运用Fourier乘子理论、泛函演算和Mittag-Leffler函数证明了一类强制微分算子可以生成分数次正则预解族，并给出了该预解族的范数估计。

Based on the analysis of some aspects, such as the approximative algorithm of FOC, the Tustin transform theory and its generating function formula's character, the convergence guarantee of binomial power function by Maclaurin expanding, and the consideration of the limitation of conventional methods, an improved method is proposed to compute the numerical evalution of FOC using PSE and Tustin transform and is further applied to solving the linear FOS.

以分数阶算子近似方法的分析研究为基础，基于Tustin变换理论及其用于分数阶算子的离散生成函数公式特点，利用二项式幂函数的Maclaurin展开能够保证收敛的特性，考虑常用算法的局限性，提出了一种改进的基于幂级数展开和Tustin变换的分数阶运算方法，并应用于线性分数阶系统的求解，给出了递推算法的详细推导。

School of Information Engineering,Zhengzhou University,Zhengzhou 450052,ChinaA novel chirp-typed watermarking scheme which uses the properties of index additivity and orthonormality of fractional Fourier transform to embed watermark in the different fractional Fourier domain is presented.

提出了一种新的分数阶Fourier域Chirp类数字水印方案，该方案利用分数阶Fourier变换基函数的正交性和旋转相加性，在不同的分数阶Fourier域嵌入Chirp水印，并利用分数阶Fourier域Chirp信号的聚集性进行盲检测。

It is found thatthe fractal dimension D=1.25 corresponds to the lowest criticalcoupling constant αc=1.9,D=1.73 corresponds to the highest criticalratio of dielectric constants ηc=0.163,and when D≤1.145 bipolaronscan not exist at any rate.In chap,4,we will propose a novelapproach to the calculation of the exciton ground-state energy for thestrong-coupling case.Different from all previous methods,the wavefunction of the phonon part is assumed to take a form related to thewave function of the relative motion.We obtain the exciton energy bysolving the derived integrodifferential equation rather than select ahydrogen-like form to minimize the energy expectation.

结果发现，分数维的维数D＝1.25对应最小的临界的电-声耦合常数(αo＝1.9)，D＝1.73对应最大的临界的介电常数比(ηc＝0.163)，当分数维的维数D≤1.145时，双极化子无论如何也不可能存在，在第四章中，我们将提出一种新颖的变分方法来计算强耦合的激子-声子系统的基态能，不同于以前所有的方法，我们取声子的波函数与相对运动波函数有关的形式，而不是假定一个固定的关于相对运动坐标r的函数形式，得到相对运动波函数所满足的非线性的微分积分方程，我们数值求解这个微分积分方程得到系统基态能，而不是选择一个类氢原子的波函数变分使得能量的期待值最小。

Based on the actual conditions of the 5342 fully mechanized caving face of Wulan Coal Mine, the 3-D distribution of gas in the face is tested by measuring points arrangement in the stereo grid. The isograms of gas distribution of the face are drawn. Gas concentration nadir distribution curve along the working face incline direction is drawn by method of seeking function minimum points. The method of calculating gas emission components ratio of the face by means of the curve is brought forward.

针对乌兰煤矿5342综放工作面的实际条件，采用立体网格状的测点布置，对该工作面瓦斯体积分数三维分布进行了测定，绘出了瓦斯分布等值线图，采用求函数极值点的方法，得出了工作面沿倾向瓦斯体积分数最低点分布曲线，提出了基于瓦斯体积分数最低点分布函数的瓦斯涌出分量比例计算方法。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。