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While the above two papers mainly focus on how to model the spatial relations of variables on a field, IGM targets at learning the distribution of data variables that are independently and identically distributed.

上面这两类模型研究的范畴属于空间统计邻域，而IGM是一种用来学习随机向量分布的模型。

If the initial value function of an initial value problem is in the form of separated variables, then its exact solution is also in the form of separated variables.

若给定分离变量形式的初值函数，则初值问题的精确解也是分离变量形式的。

In allusion to Funar Conjecture :"If a random triangle lies in a closed unit square,then its inscribed circle s radius,r≤(5-1)/4",an equivalent minimum problem about a function of 2-variables is studied;the stagnation point and its value,value on the boundary of the function of 2-variables are studied,the equivalent problem is proved correct,so the Funar Conjecture is proved correct.

针对Funar猜想："设任意三角形位于闭单位正方形内，则该三角形的内切圆半径，r≤(5-1)/4"，研究了与其等价的某二元函数的最小值问题；利用对此二元函数驻点及其取值、边界取值讨论，证明了等价问题成立，进而此Funar猜想得证。

Unlike other solution methods, integer variables are easier to accommodate in GA than continuous variables.

不同于其他解答方法，整型变量比连续的可变物是容易容纳在GA。

This thesis firstly defines the uniform integrability and absolute continuity of fuzzy variables, discusses their relationships, and provides an approach to checking the uniform integrability of fuzzy variables.

本文首先定义模糊变量序列的一致可积和绝对连续，讨论它们之间的关系并且给出判别一致可积的方法。

A series of spatial syntactic variables is discussed in the paper. Taking Wuhan's traffic axial map as a case, spatial syntactic variables of multi-scale spatial extent based on GIS are analyzed. A conclusion is that the accessibility is the key factor that weighs an urban spatial morphology, and the integration degree is a direct variable describing accessibility.

讨论了空间句法的系列变量，并以武汉市为例，提取武汉市的交通网络轴线地图，在GIS环境下讨论了不同尺度空间范围内的空间句法变量，得出空间可达性是评价城市空间形态的重要因子，而集成度是描述空间可达性的直接变量。

This project also obtained several limit theorems for some important dependent random variables and stochastic processes, such as the Strassen law of the iterated logarithm for negatively dependent random variables, strong limit theorems for mixing random vectors in Banach spaces, sample path properties for two-parameter fractional Wiener processes, and so on.

随机环境中的随机变量与随机过程的研究在国内外相当活跃，本项目主要研究它们的极限性质，着重研究了随机风景中随机变量与随机过程的极限性质，主要取得了以下几个结果：首先对简单对称的Kesten-Spitzer随机游动在低阶矩的条件下给出了强逼近，大大减弱了前人要求任意阶矩的条件，然后对独立风景中的一般随机变量给出了强逼近的一般性结果，由此导出在风景和随机变量都只具有低阶矩的条件下的独立但不同分布、混合相依变量的强逼近，在只有弱高于二阶矩的条件下得到了重相对数律和弱收敛；给出了连续时间参数的Brown风景中Brown运动和稳定风景中稳定过程的滞后增量和连续模等精确样本轨道性质；同时给出了一些重要的相依随机变量和过程的若干极限定理，如负相关随机变量的Strassen重对数律、抽象空间上混合相依变量的一些强极限定理成立的充分必要条件、两参数分数Wiener过程的样本轨道性质等。

Under suitable assumptions, the relationship is established between the unconstrained minimization of the Hestenes-Powell augmented Lagrangian function on the space of problem variables and the solution of the original constrained problem, and a relationship is also presented between the unconstrained minimization of the Hestenes-Powell augmented Lagrangian function on the product space of problem variables and multipliers and the solution of the original constrained problem.

在适当的条件下，我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系，并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系。

The design of the 3-dimensional trajectory in horizontal wells is an optimal control system where the state variables are azimuth, inclination and the control variables are radius, tool-face angle and length of curve.

水平井三维轨道设计是以方位角、井斜角为状态变量，以曲线半径、装置角、曲线段长为控制变量，以入靶精度等为性能指标的非线性多阶段最优控制系统。

Based on the approach of turning some of the relevant extreme-value-type and logarithmic normal distribution variables into independent and normal random variables, the shear-slipping failure probability of a high arch dam is worked out to be of the order of magnitude of 10^(-5) by means of second-order moment method.

在将部分相关的极值型与对数正态分布随机变量转换为独立、正态随机变量的基础上，利用二阶矩法计算得到某高拱坝的剪滑失效概率为10^(-5)量级。

In contrast to the ubiquitous rising-sun-with-rays military flag of the Japanese, Chinese banners and ensigns feature a range of designs.

与遍地都是的太阳军旗不同，中国人的旗帜和徽章设计得各式各样。

From their small corner of Feng's Guangzhou headquarters -- a jumble of pink leashes, squeezable rubber steaks, and plastic doggy Santas for Fido's stocking -- Soleil's designers come up with at least five new products a month.

从Feng 设在广州总部的产品展示柜台上可以看到，Soleil的设计师每月至少设计出5件新产品。