Correspondence between continuous variable and discrete quantum systems of arbitrary dimens
泰特猜想的延续 ——四色定理的书面证明
Pure Mathematics 理论数学, 2019, 9(8), 949-960Published Online October 2019 in Hans. /journal/pmhttps:///10.12677/pm.2019.98121Tait’s Conjecture Continue—The Proof of the Four-Color TheoremWenzhen HanJincheng Energy Co. Ltd., Jincheng ShanxiReceived: Sep. 30th, 2019; accepted: Oct. 22nd, 2019; published: Oct. 29th, 2019AbstractThe four-color theorem also known as the four-color conjecture or the four-color problem is one of the world’s three largest mathematical conjecture. Although it has been proved on computer, which owes to its powerful computing ability, after all, it isn’t strictly reasoned mathematically.Lots of math enthusiasts devote themselves to studying the problem around the globe. In this pa-per, the new concepts of two-color dyeable continuous line are put forward. A new method is used to prove that the 3-coloring of 3-regular planar graph lines is equivalent to the 4-coloring of maximal graph points. It is also proved that the 3-coloring of 3-regular planar graph lines is in-evitably possible. Thus, a universal four-color coloring method for vertices of any maximal graph is given.KeywordsFour Colors Enough, Two-Color Dyeable Continuous Line, 3-Regular Plane, Maximum Graph,Even Ring Elimination Method泰特猜想的延续——四色定理的书面证明韩文镇晋城能源有限责任公司，山西晋城收稿日期：2019年9月30日；录用日期：2019年10月22日；发布日期：2019年10月29日摘要四色定理，又称四色猜想、四色问题，是世界三大数学猜想之一。
SPSS统计词汇
英汉统计学常用词汇(SPSS)Aabsolute deviation 绝对离差absolute residuals 绝对残差acceptable hypothesis 可接受假设acceptable region 接受域actual frequency 实际频数adaptive estimator 自适应估计量addition theorem 加法定理additivity 可加性adjusted R square 调整判别系数admissible error 容许误差alphafactorin g α因子提取法alternative hypothesis 备择假设among groups 组间analysis of correlation 相关分析analysis of covariance 协方差分析analysis of regression 回归分析BBayesian estimation Beyes估计bell-shaped curve 钟形曲线best-trim estimator 最好切尾估计量beta distribution β分布between groups 组间的between measures 重复测量间的bivariate 双变量的bivariate correlate 二变量相关biweight interval 双权区间biweight M-estimator 双权M估计量block 区组/配伍组boxplot 箱线图Ccanonical correlation 典型相关case-control study 病例对照研究categorical variable 分类变量Cauchy distribution 柯西分布centering and scaling 中心化和定标central tendency 集中趋势chance statistics 随机统计量chance variable 随机变量chi-square distribution 卡方分布chi-square statistics 卡方统计量chi-square test 卡方检验classified variable 分类变量coefficient of skewness 偏度系数coefficient of variation 变异系数communality variance 共性方差compare means 均值比较分析complete association 完全正相关concomitant variable 伴随变量conditional likelihood 条件似然conditional probability 条件概率confidence limit 置信限consistency check 一致性检验consistent estimate 一致估计contingency tables 列联表continuous variable 连续变量control charts 控制图controlled experiments 对照实验conventional depth 常规深度correction coefficient 校正系数critical point 临界点critical ratio 临界比cumulative probability 累计概率curvature 曲率cyclist 周期性Ddata capacity 数据容量data deficiencies 数据缺乏1data handling 数据处理data reduction 数据简化分析data transformation 数据变换degree of precision 精密度degree of reliability 可靠性程度density function 密度函数density of data points 数据点的密度derivative matrix 导数矩阵description 描述descriptive 描述性的deviation from average 离均差Df. Fit 拟合差值df(degree of freedom)自由度dichotomous variable 二分变量discriminant analysis 判别分析discriminant coefficient 判别系数disproportional 不成比例的dissimilarity 不相似性distribution shape 分布形状disturbance 随机扰动项double logarithmic 双对数Eeffect 实验效应effects of interaction 交互效应efficiency 有效性eigenvector 特征向量enumeration data 计数资料equal size 相等的数量error of estimate 估计误差error type Ⅰ第一类错误error type Ⅱ第二类错误estimation 估计量Euclidean distance 欧氏距离expectation plane 期望平面expectation surface 期望曲面expected value 期望值experimental sampling 试验抽样explanatory variable 解释变量explore Summarize 探索-摘要EXSMOOTH 指数平滑方法extended fit 扩充拟合extra parameter 附加参数extreme observation 末端观测值extreme value 极值Ffactor score 因子得分factorial designs 因子设计factorial experiment 因子试验failure rate 失效率family of estimators 估计量族fatality rate 病死率finite population 有限总体finite-sample 有限样本first derivative 一阶导数first quartile 第一四分位数Fisher information Fisher信息量fitting a curve 曲线拟合fixed model 固定模型fixed variable 固定变量fluctuation 随机起伏fourth 四分点fractional error 相对误差frequency polygon 频数多边图frontier point 界限点F-test F检验function 函数function relationship 泛函关系Ggamma distribution 伽玛分布general census 全面普查geometric mean几何均值Gini’s mean difference 基尼均差goodness-of-fit 拟合优度gross-error sensitivity 大错敏感度group averages 分组平均grouped data 分组资料grouped median 组中值growth curve 生长曲线Hhalf-life 半衰期happenstance 偶然事件harmonic mean 调和均值hazard function 风险均数hazard rate 风险率Hessian array Hessian立体阵Heterogeneity 不同质heterogeneity 不齐性HOMAIS 多重响应分析homogeneity of variance 方差齐性homogeneity test 齐性检验Huber M-estimators Huber M 估计量hyperbola 双曲线hypothesis 假设hypothesis test 假设检验hypothetical universe 假设总体Iimage factoring 典型因子提取法impossible event 不可能事件independent samples 独立样本independent variable 自变量indirect standardization 间接标准化法infinitely great 无穷大information capacity 信息容量interclass correlation 组内相关inter-item correlation 样本内相关interpolation 内插法interquartile range 四分位距interclass correlation 组间相关inverse matrix 逆矩阵item means 样本均值L large sample problem 大样本问题Latin square 拉丁方Latin square design 拉丁方设计Least-square estimation 最小二乘估计L-estimator of location 位置L估计量level of significance 显著性水平leverage value 中心化杠杆值life expectance 预期期望寿命life table 寿命表life table method 生命表法light-tailed distribution 轻尾分布likelihood function 似然函数likelihood ratio 似然比likelihood ratio test 似然比检验linear relation 线性关系linear trend 线性预测值loading 载荷location invariance 位置不变性log rank test 时序检验logarithmic scale 对数尺度logic check 逻辑检查logistic 逻辑的logistic distribution logistic分布logit model logit模型logit transformation logit转换logarithms 对数lost function 损失函数lower limit 下限MMahal Distance 马氏距离main effect 主效应maintainability 可维护度matched data 配对资料matched distribution 匹配分布matrix 矩阵maximum 最大值mean 均值mean difference 均值差值mean square 均方mean sum of square 均方和measure 度量median 中位数median lethal dose 半数致死量median polish 中位数平滑m-estimator M估计midpoint 中值model specification 模型的确定modeling statistics 型统计models for outliers 离群值模型modifying the model 模型的修正Monte Carle method 蒙特卡洛法multiple comparison 多重比较multiple correlation 多元相关系数multiple response 多重响应multiple response sets 多重响应集合multiple solutions 多解multiplication theorem 乘法定理multi-response 多元响应multi-stage sampling 多阶段抽样multivariate 多元的multivariate analysis 多元分析mutual exclusive 互不相容mutual independence 互相独立Nnegative correlation 负相关nominal variable 名义变量nonlinear regression 非线性相关nonlinear regression 非线性回归nonparamtric statistics 非参数统计nonparametric test 非参数检验normal distribution 正态分布normal P-P 正态P-P图normal probability 正态概率normal Q-Q 正态Q-Q图normal value 正常值null hypothesis 零假设Oobjective function 目标函数observation unit 观察单位observed value 观察值one sided test 单侧检验one-sample 单样本one-tailed test 单侧检验one-way classification 单因素分类order statistics 顺序统计量ordered categories 有序分类ordinal 序数ordinal variable 有序变量origin 原点orthogonal 正交的orthogonal design 正交试验设计Ppaired observations 成对观测数据paired design 配对设计paired sample 配对样本parametric statistics 参数统计Pearson curves Pearson曲线P-estimator P估计量pie chart 饼(圆)图Pitman estimator Pitman估计量pivot 枢轴量pivot table 枢轴表polynomial regression 多项式回归population 总体positive correlation 正相关posterior distribution 后验分布preliminary analysis 预备性分析probability 概率probability density 概率密度probability of F F显著性概率probit analysis 概率分析product moment 乘积矩/协方差QQ-Q Plot Q-Q概率图quadratic regression 二次多项式回归quadratic term 二次项quality control charts 质量控制图quantitative analysis 定量分析quartile 四分位数RR square 判别系数random 随机random event 随机事件random number 随机数random sampling 随机取样random variable 随机变量randomization 随机化rank statistic 秩统计量rank sum test 秩和检验rank test 秩检验ranked data 等级资料ratio analysis 比率分析raw data 原始资料Rayleigh's test 雷氏检验reciprocal 倒数reject region 拒绝域rejection point 拒绝点relative dispersion 相对离散度relative number 相对数reliability 可靠性reliability analysis 可靠性分析reliability test 可靠性检验report summaries 报告摘要residual 残差residual sum of square 残差平方和response 响应root mean square 均方根rotation 旋转row effects 行效应run test 游程检验S S. E.mean 均值的标准差S.E.of Kurtosis 峰度的标准差S.E.of Skewness 偏度的标准差sample size 样本容量sample space 样本空间sampling design 抽样设计sampling distribution 抽样分布sampling error 抽样误差sampling inspection 抽样检验scatter diagram 散点图schematic plot 示意图/简图score statistic 得分统计score test 计分检验sensitivity curve 敏感度曲线sequential analysis 贯序分析sequential data set 顺序数据集serial tests 系列试验series mean 系列均值sign test 符号检验signed rank 符号秩significance digits 有效数字significance test 显著性检验significant figure 有效数字similarity 相似性simple regression 简单回归skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题Smirnov test Smirnov检验specific factor 特殊因子specific factor variance 特殊因子方差standard deviation 标准差standard error 标准误standard residual plots 标准化残差图standardize 标准化standardized coefficients标准化系数standardized residual 标准化残差statistics 统计学(量)、统计图表std.predicted value 标准预测值Std.residual 标准残差stem and leaf display 茎叶图step factor 步长因子stochastic models 随机模型stochastic process 随机过程survival 生存分析symmetry 对称systematic error 系统误差systematic sampling 系统抽样Ttest criterion 检验判据test for linearity 线性检验test of goodness of fit 拟和优度检验test of homogeneity 齐性检验test oh independence 独立性检验test rules 检验法则testing function 检验函数testing of hypotheses 假设检验theoretical frequency 理论频数time series 时间序列tolerance 容忍度tolerance interval 容忍区间tolerance limits 容限tolerance lower limit 容忍下限tolerance upper limit 容忍上限total sum of square 总平方和total variation 总变异transfer function 转换函数Uunbiased estimation 无偏估计unbiasedness 无偏性unequal size 不等含量unweight 不加权upper limit 上限upward rank 升秩Vvalidity 有效性value 数值value of estimator 计值variability 变异性variable 变量variance components 方差成分variance ratio 方差比variation 变异various 不同的vector 向量WWeibull distribution 威布尔分布weighted mean square 加权平均方差weighted sum of square 加权平方和weighting coefficient 权重系数weighting method 加权法weighted average 加权平均值ZZ score Z分数Z test Z检验zero correlation 零相关Z-transformation Z变换6。
变动区间上的确界函数是连续函数的证明
㊀㊀第41卷㊀第5期㊀吉首大学学报(自然科学版)V o l.41㊀N o.5㊀㊀㊀㊀2020年9月J o u r n a l o f J i s h o uU n i v e r s i t y(N a t u r a l S c i e n c e sE d i t i o n)S e p t.2020㊀㊀文章编号:10072985(2020)05000104变动区间上的确界函数是连续函数的证明∗邢家省1,2,吴㊀桑1,2,杨义川1,2(1.北京航空航天大学数学科学学院,北京100191;2.数学㊁信息与行为教育部重点实验室,北京100191)㊀㊀摘㊀要:研究了连续函数在变动区间上的确界函数的连续性问题.通过变动区间与单位区间的对应关系,将变动区间上的确界函数表示为单位区间上的确界函数.利用2个函数的上确界相减的不等式,由函数的一致连续性,证明了变动区间上的确界函数是一致连续的.关键词:变动区间;确界函数;连续函数;一致连续中图分类号:O177.2㊀㊀㊀㊀㊀㊀㊀文献标志码:A D O I:10.13438/j.c n k i.j d z k.2020.05.001闭区间上的连续函数在变动区间上的上确界函数和下确界函数是连续函数,这个结果在文献[12]中有完整的论证,其方法是利用确界函数的单调性来证明确界函数的左连续和右连续,从而证明确界函数是连续的.由于该证明过程较长,叙述麻烦,不利于掌握和传播,因此笔者拟在文献[39]的基础上,给出连续函数在变动区间上的确界函数是一致连续的简单证明.1㊀2函数的上确界相减的不等式定理1[34]㊀设f(x),g(x)在I上有定义且有界,则有:(1)s u p xɪI f(x)-s u p xɪI g(x)ɤs u p xɪI f(x)-g(x);(2)i n f xɪI f(x)-i n f xɪI g(x)ɤs u p xɪI f(x)-g(x);(3)s u p xɪI f(x)-s u p xɪI g(x)ɤs u p xɪI f(x)-g(x);(4)i n f xɪI f(x)-i n f xɪI g(x)ɤs u p xɪI f(x)-g(x).证明㊀(1)由f(x)=(f(x)-g(x))+g(x)ɤf(x)-g(x)+g(x)ɤs u p xɪI f(x)-g(x)+s u p xɪI g(x)㊀㊀xɪI,∗收稿日期:20191211基金项目:国家自然科学基金资助项目(11771004);北京航空航天大学校级重大教改项目(北航培育项目201901 202112)作者简介:邢家省(1964 ),男,河南泌阳人,北京航空航天大学数学科学学院副教授,博士,主要从事偏微分方程㊁微分几何和泛函分析研究;杨义川(1970 ),男,甘肃天水人,北京航空航天大学数学科学学院教授,博士,主要从事逻辑代数㊁序代数㊁软计算及其应用研究.可得s u p x ɪIf (x )ɤs u p x ɪIf (x )-g (x )+s u p x ɪIg (x ),从而s u p x ɪIf (x )-s u p x ɪIg (x )ɤs u p x ɪIf (x )-g (x ).由对称性,有s u p x ɪIg (x )-s u p x ɪIf (x )ɤs u p x ɪI g (x )-f (x ),于是s u p x ɪIf (x )-s u p x ɪIg (x )ɤs u p x ɪIf (x )-g (x ).(2)由f (x )=(f (x )-g (x ))+g (x )ɤf (x )-g (x )+g (x )ɤs u p x ɪIf (x )-g (x )+g (x ),可得i n f x ɪIf (x )ɤs u p x ɪIf (x )-g (x )+g (x ),于是i n f x ɪIf (x )ɤi n f x ɪIs u p x ɪIf (x )-g (x )+g (x )()=s u p x ɪIg (x )-f (x )+i n f x ɪIg (x ),从而i n f x ɪIf (x )-i n f x ɪIg (x )ɤs u p x ɪIg (x )-f (x ).由对称性,有i n f x ɪIg (x )-i n f x ɪIf (x )ɤs u p x ɪIg (x )-f (x ),于是i n f x ɪIf (x )-i n f x ɪIg (x )ɤs u p x ɪIf (x )-g (x ).或者注意到i n f x ɪIf (x )=-s u p x ɪI(-f (x )),i n f x ɪIg (x )=-s u p x ɪI(-g (x )),利用(1)可得(2).(3)利用(1)可得,s u p x ɪIf (x )-s u p x ɪIg (x )ɤs u p x ɪIf (x )-g (x )ɤs u p x ɪIf (x )-g (x ).(4)利用(2)可得,i n f x ɪIf (x )-i n f x ɪIg (x )ɤs u p x ɪIf (x )-g (x )ɤs u p x ɪIf (x )-g (x ).定理1在研究一致收敛的函数列的性质方面起着重要作用[59].2㊀连续函数在变动区间上的确界函数的一致连续性定理2[12]㊀设f (x )在[a ,b ]上连续,对于∀x ɪ[a ,b ],记M (x )=s u p a ɤt ɤx f (t ),m (x )=i n f a ɤt ɤxf (t ),则M (x )和m (x )在[a ,b ]上连续.证明㊀(1)M (x )=s u p a ɤt ɤxf (t )=s u p 0ɤu ɤ1f ((1-u )a +u x ),对于∀x ,x 0ɪ[a ,b ],有M (x )-M (x 0)ɤs u p 0ɤu ɤ1f ((1-u )a +u x )-s u p 0ɤu ɤ1f ((1-u )a +u x 0)ɤs u p 0ɤu ɤ1f ((1-u )a +u x )-f ((1-u )a +u x 0).由f (x )在[a ,b ]上连续可知f (x )在[a ,b ]上一致连续.对于∀ε>0,存在δ>0,当x 1,x 2ɪ[a ,b ],x 1-x 2<δ时,有f (x 1)-f (x 2)<ε,于是当x ,x 0ɪ[a ,b ],x -x 0<δ时,((1-u )a +u x )-((1-u )a +u x 0)ɤx -x 0<δ㊀㊀u ɪ[0,1],f ((1-u )a +u x )-f ((1-u )a +u x 0)<ε,从而M (x )-M (x 0)<ε,即M (x )在[a ,b ]上连续.(2)m (x )=i n f a ɤt ɤxf (t )=in f 0ɤu ɤ1f ((1-u )a +u x ),对于∀x ,x 0ɪ[a ,b ],有m (x )-m (x 0)ɤi n f 0ɤu ɤ1f ((1-u )a +u x )-in f 0ɤu ɤ1f ((1-u )a +u x 0)ɤs u p 0ɤu ɤ1f ((1-u )a +u x )-f ((1-u )a +u x 0).由f (x )在[a ,b ]上连续可知f (x )在[a ,b ]上一致连续.对于∀ε>0,存在δ>0,当x 1,x 2ɪ[a ,b ],x 1-x 2<δ时,有f (x 1)-f (x 2)<ε,于是当x ,x 0ɪ[a ,b ],x -x 0<δ时,((1-u )a +u x )-((1-u )a +u x 0)ɤx -x 0<δ㊀u ɪ[0,1],f ((1-u )a +u x )-f ((1-u )a +u x 0)<ε,从而m (x )-m (x 0)<ε,即m (x )在[a ,b ]上连续.或者利用m (x )=i n f a ɤt ɤxf (t )=-s u p a ɤt ɤx(-f (t )),可得m (x )在[a ,b ]上连续.从定理2的证明过程可以发现,其实是直接证明M (x )和m (x )在[a ,b ]上一致连续.由定理2的证明可得如下结果:2吉首大学学报(自然科学版)第41卷定理3㊀设f (x )在[a ,b )上连续,对于∀x ɪ[a ,b ),记M (x )=s u p a ɤt ɤxf (t ),m (x )=i n f a ɤt ɤxf (t ),则M (x )和m (x )在[a ,b )上连续.定理4㊀设f (x )在[a ,+ɕ)上一致连续,对于∀x ɪ[a ,+ɕ),记M (x )=s u p a ɤt ɤxf (t ),m (x )=i n f a ɤt ɤxf (t ),则M (x )和m (x )在[a ,+ɕ)上一致连续.3㊀二元函数在变动区间上的确界函数的一致连续性定理5㊀设f (x ,y )在[a ,b ]ˑ[c ,d ]上连续,对于∀x ɪ[a ,b ],y ɪ[c ,d ],记M (x ,y )=s u p a ɤu ɤx ,c ɤv ɤyf (u ,v ),m (x ,y )=i n f a ɤu ɤx ,c ɤv ɤyf (u ,v ),则M (x ,y )和m (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.证明㊀记x t 1=(1-t 1)a +t 1x ,y t 2=(1-t 2)c +t 2y ,则有M (x ,y )=s u p a ɤu ɤx ,c ɤv ɤyf (u ,v )=s u p 0ɤt 1,t 2ɤ1,f (x t 1,y t 2),m (x ,y )=i n fa ɤu ɤx ,c ɤv ɤyf (u ,v )=i n f 0ɤt 1,t 2ɤ1,f (x t 1,y t 2).由f (x ,y )在[a ,b ]ˑ[c ,d ]上连续,可得f (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.对于∀ε>0,存在δ>0,当x 1,x 2ɪ[a ,b ],y 1,y 2ɪ[c ,d ],x 1-x 2<δ,y 1-y 2<δ时,有f (x 1,y 1)-f (x 2,y 2)<ε,于是|M (x 1,y 1)-M (x 2,y 2)|=|s u p 0ɤt 1,t 2ɤ1,f (x 1t 1,y 1t 2)-s u p 0ɤt 1,t 2ɤ1,f (x 2t 1,y 2t 2)|ɤs u p 0ɤt 1,t 2ɤ1,|f (x 1t 1,y 1t 2)-f (x 2t 1,y 2t 2)|<ε,即M (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.由m (x ,y )=i n fa ɤu ɤx ,c ɤv ɤyf (u ,v )=-s u pa ɤu ɤx ,c ɤv ɤy-f (u ,v ),可得m (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.二元确界函数的连续性是一个新发现的问题,文献[12]中的方法不适用.这里将变动区间上的确界函数转化为固定区间上的确界函数,即将问题一般化,就比较容易解决了.由定理5的证明可得如下结果:定理6㊀设f (x ,y )在[a ,b )ˑ[c ,d )上连续,对于∀x ɪ[a ,b ),y ɪ[c ,d ),记M (x ,y )=s u p a ɤu ɤx ,c ɤv ɤyf (u ,v ),m (x ,y )=i n f a ɤu ɤx ,c ɤv ɤyf (u ,v ),则M (x ,y )和m (x ,y )在[a ,b )ˑ[c ,d )上连续.定理7㊀设f (x ,y )在[a ,+ɕ)ˑ[c ,+ɕ)上一致连续,对于∀x ɪ[a ,+ɕ),y ɪ[c ,+ɕ),记M (x ,y )=s u p a ɤu ɤx ,c ɤv ɤyf (u ,v ),m (x ,y )=i n f a ɤu ɤx ,c ɤv ɤyf (u ,v ),则M (x ,y )和m (x ,y )在[a ,+ɕ)ˑ[c ,+ɕ)上一致连续.4㊀2函数的上确界相减的不等式的应用举例定理8㊀设f (x ,y )在[a ,b ]ˑ[c ,d ]上连续,对于∀x ɪ[a ,b ],记F (x )=s u p c ɤy ɤdf (x ,y ),G (x )=i n f c ɤy ɤdf (x ,y ),则F (x )和G (x )在[a ,b ]上一致连续.证明㊀由f (x ,y )在[a ,b ]ˑ[c ,d ]上连续,可得f (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.对于∀ε>0,存在δ>0,当x 1,x 2ɪ[a ,b ],y 1,y 2ɪ[c ,d ],x 1-x 2<δ,y 1-y 2<δ时,有f (x 1,y 1)-f (x 2,y 2)<ε,于是|F (x 1)-F (x 2)|=|s u p c ɤy ɤdf (x 1,y )-s u p c ɤy ɤdf (x 2,y )|ɤs u p c ɤy ɤd|f (x 1,y )-f (x 2,y )|<ε,即F (x )在[a ,b ]上一致连续.同理可证G (x )在[a ,b ]上一致连续.由定理8的证明可得如下结果:3第5期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀邢家省,等:变动区间上的确界函数是连续函数的证明定理9㊀设f (x ,y ,z )在[a ,b ]ˑ[c ,d ]ˑ[e ,f ]上连续,对于∀x ɪ[a ,b ],记F (x )=s u p c ɤy ɤde ɤz ɤff (x ,y ,z ),G (x ,y )=s u p e ɤz ɤff (x ,y ,z ),则F (x )在[a ,b ]上连续,G (x ,y )在[a ,b ]ˑ[c ,d ]上一致连续.参考文献:[1]费定晖,周学圣.吉米多维奇数学分析习题集题解(一)[M ].济南:山东科学技术出版社,1980:457458.[2]裴礼文.数学分析中的典型问题与方法[M ].北京:高等教育出版社,2002:129130.[3]匡继昌.常用不等式[M ].长沙:湖南教育出版社,1989:8192.[4]徐利治,王兴华.数学分析的方法及例题选讲(修订版)[M ].北京:高等教育出版社,1984:7988.[5]邢家省,杨小远.一类欧拉积分公式与广义菲涅尔积分的计算[J ].吉首大学学报(自然科学版),2018,39(1):16.[6]邢家省,杨义川.分布函数列的一致收敛性[J ].吉首大学学报(自然科学版),2018,39(2):14.[7]邢家省,杨义川.最大模估计和一类J e n s e n 不等式的应用[J ].吉首大学学报(自然科学版),2018,39(4):14.[8]邢家省,杨义川.一些奇异实积分的特殊复围道积分计算方法[J ].吉首大学学报(自然科学版),2018,39(6):15.[9]邢家省,杨义川.最速降线问题解的充分条件的证明[J ].吉首大学学报(自然科学版)2019,40(2):14.M e t h o do fP r o v i n g De f i n i t eB o u n dF u n c t i o no nV a r i a b l e I n t e r v a l a sC o n t i n u o u s F u n c t i o nX I N GJ i a s h e n g 1,2,WUS a n g1,2,Y A N G Y i c h u a n 1,2(1.S c h o o l o fM a t h e m a t i c s a n dS y s t e mS c i e n c e ,B e i h a n g U n i v e r s i t y ,B e i j i n g 100191,C h i n a ;2.L M I Bo f t h eM i n i s t r y o fE d u c a t i o n ,B e i h a n g U n i v e r s i t y ,B e i j i n g 100191,C h i n a )A b s t r a c t :W e s t u d y t h e c o n t i n u i t y of t h ed e f i n i t eb o u n d f u n c t i o no f c o n t i n u o u s f u n c t i o no nt h ev a r i a b l e i n t e r v a l .W i t hc o r r e s p o n d e n c eb e t w e e nt h ev a r i a b l e i n t e r v a la n dt h eu n i t i n t e r v a l ,t h ed e f i n i t eb o u n d f u n c t i o no n t h e c h a ng e i n t e r v a l i s e x pr e s s e d a s t h e d e f i n i t e b o u n d f u n c t i o no n t h e u n i t i n t e r v a l .W i t h t h e s u b t r a c t i o n i n e q u a l i t y o f u p p e r b o u n d s o f t w o f u n c t i o n s a n d t h eu n i f o r mc o n t i n u i t y of t h e f u n c t i o n ,i t i s p r o v e d t h a t t h e d e f i n i t eb o u n d f u n c t i o n i s u n i f o r m l y co n t i n u o u s .K e y wo r d s :v a r i a b l e i n t e r v a l ;d e f i n i t eb o u n d f u n c t i o n ;c o n t i n u o u s f u n c t i o n ;u n i f o r m l y c o n t i n u o u s (责任编辑㊀向阳洁)4吉首大学学报(自然科学版)第41卷。
医院专利备案流程
医院专利备案流程The process of filing for a hospital patent can be a lengthy and complicated procedure that requires careful attention to detail and thorough documentation. 医院专利备案流程可能是一个漫长而复杂的程序，需要仔细的注意细节和充分的文档记录。
From the initial idea or invention to the final submission and approval, each step in the process plays a crucial role in determining the success of the patent application. 从最初的想法或发明到最终的提交和批准，流程中的每一步都对确定专利申请的成功起着至关重要的作用。
It is essential for hospital administrators and inventors to understand the requirements and procedures involved in filing for a patent to protect their intellectual property rights and innovations. 医院管理者和发明者必须了解专利备案的要求和程序，以保护他们的知识产权和创新。
Failure to properly navigate the patent filing process could result in lost opportunities for recognition, financial gain, and legal protection.不正确地进行专利备案程序可能会导致被遗漏的获得认可、财务收益和法律保护的机会。
关于真理的英语作文
关于真理的英语作文The Pursuit of Truth.Truth, a fundamental concept that has fascinated philosophers, scientists, and thinkers alike, is often described as the ultimate goal of human inquiry. It is the bedrock of knowledge, the foundation upon which we build our understanding of the world. Yet, what exactly is truth, and how do we come to know it?At its core, truth is the correspondence between our thoughts, beliefs, or statements and the actual state of affairs. It is the alignment of our mental representations with the external reality. When we say something is true, we imply that it corresponds to facts, evidence, orrealities that are independent of our subjective perceptions.The quest for truth has been a driving force in human history. It has motivated us to explore the unknown, toquestion authority, and to seek answers through scientific experimentation and rational argumentation. The scientific method, in particular, is a systematic approach to discovering truth through observation, hypothesis testing, and replication. It is a rigorous process that aims tofilter out falsehoods and arrive at reliable knowledge.However, the pursuit of truth is not always straightforward. Our understanding of truth is shaped by our cultural backgrounds, values, and biases. What we consider true might differ significantly across cultures and historical periods. For example, the ancient Greeks believed that the sun revolved around the earth, a belief that was considered true for centuries until Copernicus proposed a heliocentric model in the 16th century. This shift in our understanding of the universe illustrates how our perceptions of truth can evolve over time.Moreover, the quest for truth often involves difficult ethical choices. In the pursuit of scientific knowledge,for instance, researchers might face ethical dilemmas about the use of animals in experiments or the impact of theirfindings on society. These choices require a careful balancing of the quest for truth with ethical considerations.Despite these challenges, the pursuit of truth remains a crucial endeavor. It is what drives us to question, to explore, and to discover new knowledge. It is what makes us human – the desire to understand the world and our place within it. As we continue to seek truth, we must remain vigilant against bias, dogma, and misinformation. We must embrace a spirit of inquiry and curiosity, always willing to question our own beliefs and those of others.In conclusion, the pursuit of truth is a lifelong journey that requires dedication, courage, and humility. It is a quest that takes us beyond our own局限 and allows us to connect with the vastness of the universe. As we continue to search for truth, we must remember that it is not a static destination but a dynamic process of continuous learning and discovery.(Note: This essay is a condensed version of therequested 1000-word article, focusing on the main ideas and arguments surrounding the pursuit of truth. It is intended to serve as a starting point for further exploration and elaboration.)。
机器学习介绍(英文版：备注里有中文翻译)
The core target of machine learning is to generalize from known experience. Generalization means a power of which the machine learning system to be learned for known data that could predict the new data.
2020/9/13
Classification step
Raw data
Feature extraction
Feature selection
Model training
Байду номын сангаас
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New data
Classification and prediction
Feature selection（ feature reduction ）
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Unsupervised learning
Input data has no labels. It relates to another learning algorithm, i.e. clustering. The basic definition is a course that divide the gather of physical or abstract object into multiple class which consist of similar objects.
spss统计分析软件词汇中英文
S P S S统计分析软件词汇中英文-CAL-FENGHAI-(2020YEAR-YICAI)_JINGBIANSPSS词汇中英文Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显着差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显着差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显着性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换41。
Vitali’s theorem and WWKL
VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove specific mathematical theorems.In many cases,it is shown that a specific mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite different from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will find that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We define C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise definitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we define a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Definition 2.1(measure of an open set).This definition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we defineµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a finite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We define L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise definitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a finite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all sufficiently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suffices to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,defined by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Define a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N defined by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint finite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is defined and discussed in the next section.)In particular,disjoint finite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no infinite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is finitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no infinite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat different approach to measure the-ory in RCA 0.Note that the definition of µ(U )that we have given (Definition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Definition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Definition 3.4(intensional Lebesgue measure).We make this defini-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is defined by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the definitions,since ∞n =0U n is defined as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Wefirst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is defined within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we define f (x )=lim n →∞f n (x ).Thus we see thatf (x )is defined on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our first approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following definition.Definition 4.4.This definition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as defined in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain specific situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Definition 4.5.The following definition is made within RCA 0.Let S be the Boolean algebra of finite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we define the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric difference of E 1and E 2.Thus d is a pseudometric on S ,and we define S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Defi-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by defining an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1defineχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1defineχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can find E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Definition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can effectively find E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our first notion (Definition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Definition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to find a single definition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a finite sequence of intervals.Then we can find a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,fix >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to find a finite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modification needed is in the proof of the claim.Recall from Definition 4.5that E =lim n →∞E n where each E n is a finite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations offixed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。
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|EPR = dq where |q 1 ⊗ |q 2 denotes a product state of two subsystems of a composite system, maximally violates the CHSH inequality, a question which remained unanswered within the phase-space formalism. This is important because the EPR state - the maximally entangled state of CV systems - is considered as a natural resource of entanglement in CV quantum information processing.
In this paper we establish a mapping between a CV and a discrete system of arbitrary dimension. Mathematically, for an infinite-dimensional Hilbert space we construct the generators of SU(n) algebra for finite n, which build up the structure of a n-dimensional Hilbert space. This allows to consider a CV system as representing a quantum system of any dimension, i.e. a CV system can be used in various quantum information tasks even those which require systems of different dimensions. In particular, the EPR state is always mapped onto the maximally entangled state in any finite dimensional Hilbert space. Thus it can be considered as a universal resource of entanglement.
It is intuitively clear that the potentiality of an infinitedimensional system as a resource for quantum information processing goes beyond that of the qubit system. In particular, as it will be shown below, the CHSH inequality for CV systems can be maximally violated even with non-maximally entangled states. To show the full potential of infinite dimensional systems it will be important to find a mapping between CV and discrete quantum systems of arbitrarily high dimensions. An example of the use of mapping is to check the violation of Bell’s inequalities for higher-dimensional systems [16] by the EPR state. Such a mapping is also necessary if one wants to implement those quantum information tasks developed for discrete systems to CV systems, which exclusively requires higher-dimensional Hilbert spaces. These are, for example, the quantum key distribution based on higher alphabets [17] and the quantum solutions of the coin-flipping problem [18], of the Byzantine agreement problem [19], and of a certain communication complexity problem [20].
Very recently a mapping between CV systems and qubits (two-dimensional systems) was established [11, 12]. This enables to construct a Clauser-Horne-ShimonyHolt (CHSH) inequality [13] for CV systems [11], without relying on the phase-space formalism and to analyze the separability of the infinite-dimensional Werner states [12]. It was shown in Ref. [11] that the EinsteinPodolsky-Rosen (EPR) [14] state
With the exception of two-mode bipartite Gaussian states [9] there are no general criteria to test separability of a general state in infinite-dimensional Hilbert spaces. Similarly, the demonstration of the violation of Bell’s inequalities for CV systems is based predominantly on the phase-space formalism [10] and the generalization to CV systems of various Bell’s inequalities derived for discrete systems and the criteria for their violation remains open. It is thus highly desirable to find mapping between CV and discrete systems. This would open up the possibility the CV systems to be exploited to perform quantum information tasks as if they were qudits, by applying protocols which are already developed for discrete d -dimensional systems. It also would allow to apply all criteria known for discrete systems for the classification of states (e.g. for separability or for violation of Bell’s inequalities) to CV systems.
Correspondence between continuous variable and discrete quantum systems of arbitrary dimensions
Cˇ aslav Brukner1, Myungshik S. Kim2, Jian-Wei Pan1, Anton Zeilinger1 1Institut fu¨r Experimentalphysik, Universit¨at Wien, Boltzmanngasse 5, A–1090 Wien, Austria 2School of Mathematics and Physics, Queen’s University, Belfast, BT7 1NN, United Kingdom
PACS numbers: 3.65 Bz, 3.67 -a, 42.50 Ar
arXiv:quant-ph/0208116v2 20 Nov 2002
Quantum information processing enables performance of communication and computational tasks beyond the limits that are achievable on the basis of laws of classical physics [1]. While most of the quantum information protocols were initially developed for quantum systems with finite dimensions (qudits) they have also been proposed for the quantum systems with continuous variables (CV), such as quantum teleportation [2], entanglement swapping [3], entanglement purification [4], quantum computation [5], quantum error correction [6], quantum dense coding [7], and quantum cloning [8].