SU(1,1) Random Polynomials
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MATLAB稀疏Cholesky分解1. 介绍MATLAB是一种常用的数学软件,其在矩阵运算和线性代数方面有着强大的功能。
稀疏矩阵是指大部分元素为零的矩阵,而Cholesky分解是一种用于解决对称正定矩阵的线性方程组的方法。
本文将探讨MATLAB中稀疏Cholesky分解的原理、使用方法以及其在实际应用中的意义。
2. 稀疏矩阵与Cholesky分解稀疏矩阵是指矩阵中大部分元素为零,只有少数非零元素。
在实际问题中,许多矩阵具有这种特性,比如网络数据传输矩阵、有限元法中的刚度矩阵等。
对于这种稀疏矩阵,传统的直接方法(如高斯消去法)效率较低,因此需要使用特殊的方法进行计算。
Cholesky分解是一种有效的方法,特别适用于对称正定矩阵。
对于一个对称正定矩阵A,Cholesky分解将该矩阵表示为A=LL^T,其中L为下三角矩阵。
与传统的LU分解相比,Cholesky分解能够减少一半的计算量,因此在求解线性方程组时具有更高的效率和稳定性。
3. MATLAB中的稀疏Cholesky分解在MATLAB中,稀疏矩阵可以使用sparse函数进行定义。
而Cholesky分解则可以通过chol函数进行求解。
对于稀疏矩阵A,可以使用[ch, p] = chol(A, 'lower')来进行Cholesky分解,其中ch为下三角矩阵,p为置换矩阵。
通过Cholesky分解后,可以得到A=ch*ch^T。
MATLAB中对稀疏矩阵进行Cholesky分解的函数使用非常方便,能够高效地处理大规模稀疏矩阵的计算问题。
MATLAB还提供了一系列的稀疏矩阵运算函数,如sparse乘法、转置、求逆等,为稀疏矩阵的计算提供了强大的支持。
4. 实际应用稀疏矩阵和Cholesky分解在实际应用中有着广泛的意义。
以金融衍生品定价为例,通常会涉及大规模的稀疏矩阵和线性方程组的求解。
Cholesky分解能够极大地提高计算效率,为复杂金融问题的求解提供了重要支持。
用su(2)李代数的
SU(2)李代数是数学上的一个概念,与群论和李代数理论相关。
SU(2)是特殊幺正群(Special Unitary Group)的简写,是由幺正矩阵组成的群,其行列式为1。
SU(2)李代数的生成元可以表示为泡利矩阵,常见的三个泡利矩阵分别是σ₁,σ₂,和σ₃。
这些矩阵是:
\[
\sigma_1=\begin{bmatrix}0&1\\1&0\end{bmatrix},\
\sigma_2=\begin{bmatrix}0&-i\\i&0\end{bmatrix},\
\sigma_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}
\]
SU(2)李代数的基本关系是:
\[[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k\]
其中\(\epsilon_{ijk}\)是三维Levi-Civita符号。
这个关系反映了SU(2)李代数的非交换性。
SU(2)李代数在量子力学和粒子物理学中有广泛的应用,特别是在描述自旋和角动量的量子态时。
这些代数结构在研究基本粒子的性质和相互作用中发挥着重要作用。
机器学习--⽀持向量机(SVM)算法的原理及优缺点⼀、⽀持向量机(SVM)算法的原理 ⽀持向量机(Support Vector Machine,常简称为SVM)是⼀种监督式学习的⽅法,可⼴泛地应⽤于统计分类以及回归分析。
它是将向量映射到⼀个更⾼维的空间⾥,在这个空间⾥建⽴有⼀个最⼤间隔超平⾯。
在分开数据的超平⾯的两边建有两个互相平⾏的超平⾯,分隔超平⾯使两个平⾏超平⾯的距离最⼤化。
假定平⾏超平⾯间的距离或差距越⼤,分类器的总误差越⼩。
1.⽀持向量机的基本思想 对于线性可分的任务,找到⼀个具有最⼤间隔超平⾯,如图所⽰, (1)⽀持向量机的基本型为: (2)软间隔的优化⽬标: 其中,0-1函数为错分样本的个数。
(3)核⽅法: 其中为特征映射函数。
2、实验⼀般步骤: (1)导⼊数据; (2)数据归⼀化; (3)执⾏svm寻找最优的超平⾯; (4)绘制分类超平⾯核⽀持向量; (5)利⽤多项式特征在⾼维空间中执⾏线性svm (6)选择合适的核函数,执⾏⾮线性svm; 3、算法优缺点: 算法优点: (1)使⽤核函数可以向⾼维空间进⾏映射 (2)使⽤核函数可以解决⾮线性的分类 (3)分类思想很简单,就是将样本与决策⾯的间隔最⼤化 (4)分类效果较好 算法缺点: (1)SVM算法对⼤规模训练样本难以实施 (2)⽤SVM解决多分类问题存在困难 (3)对缺失数据敏感,对参数和核函数的选择敏感 ⼆、数学推导过程 对于线性可分的⽀持向量机求解问题实际上可转化为⼀个带约束条件的最优化求解问题: 推理过程: 结果: 对于线性不可分的⽀持向量机求解问题实际上可转化为⼀个带约束条件的soft-margin最优化求解问题:三、代码实现1、线性svmimport numpy as npfrom sklearn.datasets import load_irisimport matplotlib.pyplot as pltfrom sklearn.preprocessing import StandardScalerfrom sklearn.svm import LinearSVCfrom matplotlib.colors import ListedColormapimport warningsdef plot_decision_boundary(model,axis):x0,x1=np.meshgrid(np.linspace(axis[0],axis[1],int((axis[1]-axis[0])*100)).reshape(-1,1),np.linspace(axis[2],axis[3],int((axis[3]-axis[2])*100)).reshape(-1,1))x_new=np.c_[x0.ravel(),x1.ravel()]y_predict=model.predict(x_new)zz=y_predict.reshape(x0.shape)custom_cmap=ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])plt.contourf(x0,x1,zz,linewidth=5,cmap=custom_cmap)w = model.coef_[0]b = model.intercept_[0]plot_x = np.linspace(axis[0],axis[1],200)up_y = -w[0]/w[1]*plot_x - b/w[1] + 1/w[1]down_y = -w[0]/w[1]*plot_x - b/w[1] - 1/w[1]up_index = (up_y>=axis[2]) & (up_y<=axis[3])down_index = (down_y>=axis[2]) & (down_y<=axis[3])plt.plot(plot_x[up_index],up_y[up_index],c='black')plt.plot(plot_x[down_index],down_y[down_index],c='black')warnings.filterwarnings("ignore")data = load_iris()x = data.datay = data.targetx = x[y<2,:2]y = y[y<2]scaler = StandardScaler()scaler.fit(x)x = scaler.transform(x)svc = LinearSVC(C=1e9)svc.fit(x,y)plot_decision_boundary(svc,axis=[-3,3,-3,3])plt.scatter(x[y==0,0],x[y==0,1],c='r')plt.scatter(x[y==1,0],x[y==1,1],c='b')plt.show()输出结果:2、⾮线性-多项式特征import numpy as npfrom sklearn import datasetsimport matplotlib.pyplot as pltfrom sklearn.preprocessing import PolynomialFeatures,StandardScaler from sklearn.svm import LinearSVCfrom sklearn.pipeline import Pipelinefrom matplotlib.colors import ListedColormapimport warningsdef plot_decision_boundary(model,axis):x0,x1=np.meshgrid(np.linspace(axis[0],axis[1],int((axis[1]-axis[0])*100)).reshape(-1,1), np.linspace(axis[2],axis[3],int((axis[3]-axis[2])*100)).reshape(-1,1) )x_new=np.c_[x0.ravel(),x1.ravel()]y_predict=model.predict(x_new)zz=y_predict.reshape(x0.shape)custom_cmap=ListedColormap(['#EF9A9A','#FFF59D','#90CAF9']) plt.contourf(x0,x1,zz,linewidth=5,cmap=custom_cmap)def PolynomialSVC(degree,C=1.0):return Pipeline([('poly',PolynomialFeatures(degree=degree)),('std_scaler',StandardScaler()),('linearSVC',LinearSVC(C=1e9))])warnings.filterwarnings("ignore")poly_svc = PolynomialSVC(degree=3)X,y = datasets.make_moons(noise=0.15,random_state=666)poly_svc.fit(X,y)plot_decision_boundary(poly_svc,axis=[-1.5,2.5,-1.0,1.5])plt.scatter(X[y==0,0],X[y==0,1],c='red')plt.scatter(X[y==1,0],X[y==1,1],c='blue')plt.show()输出结果:3、⾮线性-核⽅法from sklearn.preprocessing import StandardScalerfrom sklearn.svm import SVCfrom sklearn.pipeline import Pipelinefrom sklearn import datasetsfrom matplotlib.colors import ListedColormapimport numpy as npimport matplotlib.pyplot as pltimport warningsdef plot_decision_boundary(model,axis):x0,x1=np.meshgrid(np.linspace(axis[0],axis[1],int((axis[1]-axis[0])*100)).reshape(-1,1), np.linspace(axis[2],axis[3],int((axis[3]-axis[2])*100)).reshape(-1,1) )x_new=np.c_[x0.ravel(),x1.ravel()]y_predict=model.predict(x_new)zz=y_predict.reshape(x0.shape)custom_cmap=ListedColormap(['#EF9A9A','#FFF59D','#90CAF9']) plt.contourf(x0,x1,zz,linewidth=5,cmap=custom_cmap)def RBFKernelSVC(gamma=1.0):return Pipeline([('std_scaler',StandardScaler()),('svc',SVC(kernel='rbf',gamma=gamma))])warnings.filterwarnings("ignore")X,y = datasets.make_moons(noise=0.15,random_state=666)svc = RBFKernelSVC(gamma=100)svc.fit(X,y)plot_decision_boundary(svc,axis=[-1.5,2.5,-1.0,1.5])plt.scatter(X[y==0,0],X[y==0,1],c='red')plt.scatter(X[y==1,0],X[y==1,1],c='blue')plt.show()输出结果:。
附录A连续型随机变量构成的Askey 正交多项式系包含如下4种正交多项式。
1) 勒让德多项式(Legendre Polynomials ) 三项递推关系为:1121()()11n n n n nP xP x P x n n +-+-++=(A1)正交性关系为:112()()21n m mn P x P x dx n δ-=+⎰ (A2)显然,该正交关系的权重函数为常数1,即ω(x )=1。
该形式的ω(x )可与平均分布的概率密度函数相联系。
前三项勒让德多项式分别为:0(x)1P =,1(x)P x =,2231(x)22Px =-,… 2) 埃尔米特多项式(Hermite Polynomials )三项递推关系为:11()(),0n n n H xH x nH x n +-=-> (A3)正交性关系为:()()()!m n mn H x H x x dx n ωδ∞-∞=⎰(A4)其中权重函数为:2/2()x x ω-=该形式的ω(x )可与正态分布的概率密度函数相联系。
前三项埃尔米特多项式分别为:0()1H x =,1()H x x =,22()1H x x =-,…3) 拉盖尔多项式(Laguerre Polynomials ) 三项递推关系为:()()()11(1)()(21)()()()n n n n L x x n L x n L x ααααα+-+=-+++-+(A5)正交性关系为:()(),0(1)()()()!m n m n n L x L x x dx n αααωδ∞Γ++=⎰(A6) 其中权重函数为:()e x x x αω-=该形式的ω(x )可与Gamma 分布的概率密度函数相联系。
前三项拉盖尔多项式分别为:0()1L x =,1()1L x x =-+,221()(42)2L x x x =-+,… 4) 雅可比多项式(Jacobi Polynomials ) 三项递推关系为:(,)(,)122(,)(,)12(1)(1)()()(21)(22)()(2)(22)2()()()(2)(21)n n n n n n xJ x J x n n J x n n n n J x n n αβαβαβαβαβαβαββααβαβαβαβαβ+-++++=++++++-++++++++++++++ (A7)正交性关系为:1()()11,()()()2(1)(1)21(1)!mn m nJ x J x x dx n n n n n αααβωαβδαβαβ-++=Γ++Γ+++++Γ+++⎰ (A8)其中权重函数为:()(1)(1)x x x αβω=-+该形式的ω(x )可与Beta 分布的概率密度函数相联系。
absolute value 绝对值acceptable region 接受域additivity 可加性alternative hypothesis 对立假设analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption checking 假设检验availability 有效度band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的binomial distribution 二项分布binomial test 二项检验center of gravity 重心central tendency 中心趋势hi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性components 构成,分量compound 复合的confidence interval 置信区间consistency 一致性continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值cross tab 列联表cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计default 默认的deleted residual 剔除残差density function 密度函数dependent variable 因变量design of experiment 试验设计df.(degree of freedom) 自由度diagnostic 诊断discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析D-optimal design D-优化设计effects of interaction 交互效应eigenvalue 特征值equal size 等含量estimation of parameters 参数估计estimations 估计量exact value 精确值expected value 期望值exponential指数的exponential distribution 指数分布extreme value 极值factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fitted line 拟合线fitted value 拟合值fixed variable 固定变量fractional factorial design 部分析因设计F-test F检验full factorial design 完全析因设计gamma distribution 伽玛分布geometric mean 几何均值harmonic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis test 假设检验independence独立independent variable 自变量independent-samples 独立样本index of correlation 相关指数interclass correlation 组内相关interval estimate 区间估计inverse 倒数的iterate 迭代kurtosis 峰度large sample problem 大样本问题least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level of significance 显著性水平leverage value 中心化杠杆值life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数lost function 损失函数main effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数Monte Carle method 蒙特卡罗法moving average移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析negative nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数one-sample 单样本one-tailed test 单侧检验one-way ANOV A 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值paired observations 成对观测数据paired-sample 成对样本parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数quadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的,度量quartiles 四分位数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线rejection region 拒绝域residual 残差residual sum of squares 剩余平方和risk function 风险函数robustness 稳健性root mean square 标准差row 行run test 游程检验sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线sets 集合sign test 符号检验significance level 显著性水平significance testing 显著性检验significant digits 有效数字skewed distribution 偏态分布small sample problem 小样本问题sort 排序sources of variation 方差来源ion 标准离差standard error of mean 均值的标准误差statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum of squares 平方和t-distribution t分布test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布value of estimator 估计值variance 方差variance components 方差分量variance ratio 方差比weighted average 加权平均值within groups 组内的Z score Z分数active constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子bandwidth 带宽be equivalent to 等价于best-fit 最佳拟合coefficient 系数complex-value 复数值component 分量constrained 有约束的constraint function 约束函数converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合data-fitting 数据拟diagonal 对角的direct search method 直接搜索法direction of search 搜索方向eigenvalue 特征值empty matrix 空矩阵exceeded 溢出的feasible solution 可行解finite-difference 有限差分first-order 一阶Gauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient method 梯度法handle 句柄Hessian matrix 海色矩阵independent variables 独立变量inequality 不等式infeasibility 不可行性initial feasible solution 初始可行解initialize 初始化invoke 激活iteration 迭代Jacobian 雅可比矩阵Lagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method 列文伯格-马夸尔特法line search 一维搜索linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解medium-scale 中型的mixed quadratic and cubic polynomial interpolation and extrapolationmethod 混合二次、三次多项式内插、外插法multi objective 多目标的norm 范数observed data 测量数据optimization routine 优化过程optimizer 求解器over-determined system 超定系统partial derivatives 偏导数polynomial interpolation method 多项式插值法quadrati二次的quadratic interpolation method 二次内插法quadratic programming 二次规划real-value 实数值residuals 残差robust 稳健的robustness 稳健性,鲁棒性scalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method 序列二次规划法simplex search method 单纯形法sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust region method 子空间置信域法symmetric matrix 对称矩阵termination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置unconstrained 无约束的under-determined system 负定系统weighting matrix 加权矩阵approximation 逼近a spline in b-form/b-spline b样条 a spline of polynomial piece /ppform spline 分段多项式样条bivariate spline function 二元样条函数break/breaks 断点coefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation 三次样条插值/三次样条内插curve 曲线degree of freedom 自由度end conditions 约束条件input argument 输入参数interpolation 插值/内插interval 取值区间knot/knots 节点least-squares approximation 最小二乘拟合multiplicity 重次multivariate function 多元函数optional argument 可选参数output argument 输出参数point/points 数据点rational spline 有理样条rounding error 舍入误差(相对误差)sequence 数列(数组spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条tolerance 允许精度univariate function 一元函数absolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格boundary condition 边界条件contour plot 等值线图coordinate 坐标系decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditions 边界条件eigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解generalized Neumann boundary condition 推广的Neumann边界条件geometry description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface(GUI)图形用户界面hyperbolic 双曲线的initial mesh 初始网格jiggle 微调Lagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值machine precision 机器精度mixed boundary condition 混合边界条件Neuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量Parabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stress 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定refined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual norm 残差范数singular 奇异的postulate假定, 基本条件, 基本原理,要求, 假定,要求conic, conical圆锥的;圆锥形的ellipse椭圆, 椭圆形ellipt hyperbolic 双曲线的parabolic用寓言表达的: 抛物线的,像抛物线的algebraic代数的, 关于代数学的mineralogy 矿物学axiom公理collinear在同一直线上的同线的convex 凸出的;凸面的triangle三角形, 三人一组, 三角关系parallelogram平行四边形straight angle平角right angle 直角acute angle锐角obtuse angle钝角reflex angle优角rectilinear直线的;由直线组成的;循直线进行的isosceles triangle等腰三角形equilateral triangle等边三角形right triangle n. 直角三角形obtuse triangle钝角三角形acute triangle锐角三角形equiangular triangle正三角形,等角三角形hypotenuse(直角三角形的)斜边infinitesimal 无穷小的, 极小的, 无限小的calculus 微积分学, 结石inscribe 记下polygon多角形, 多边形curvilinear曲线的, 由曲线组成的intuition 直觉, 直觉的知识integral积分, 完整, 部分defective有缺陷的, (智商或行为有)欠缺的differential coefficient 微分系数irrational numbers无理数domain 定义域contradiction 矛盾continuous variable 连续变量;[连续变数]variation 变分, 变化independent variable 自变量dependent variable 应变量rectangular coordinate 直角坐标abscissa〈数〉横坐标ordinate纵线, 纵座标differential 微分的,微分(differentiation)Integral 积分, 完整, 部分(integration) trigonometry 三角法exponential 指数的, 幂数的logarithm 对数derivative导数;微商tangent 切线正切definite integral 定积分culminate 达到顶点differential equation 微分方程extreme value 极值multiple integral 多重积分functional analysis 泛函分析cardinal number 基数(如:1, 2, 3, ... 有别于序数)denumerable可数的aggregate 合计, 总计, 集合体,合计的, 集合的, 聚合的,聚集, 集合, 合计purport主旨,声称superior 长者, 高手, 上级,较高的, 上级的, 上好的, 出众的, 高傲的cumbersome 讨厌的, 麻烦的, 笨重的drastically 激烈地, 彻底地conservation 守衡律quadrature求积, 求积分interpolation插值extrapolation外推法, 推断internal point 内点generalized solution 广义解hydrodynamics 流体力学,水动力学divergence 发散(性),梯度,发散integro-interpolation method 积分插值法Variational method 变分方法comparatively 比较地, 相当地self-adjoint (nonself-adjoint) 自治的,自伴的,自共轭的finite element method 有限元法spline approximation 样条逼近Particles-in-the-Cell 网格质点法herald 使者, 传令官, 通报者, 先驱, 预兆,预报, 宣布, 传达, 欢呼advection水平对流fluctuation波动, 起伏mean-square 均方dispersion离差, 差量nterpolation 插值divisible可分的dice, die 骰子pitfall 缺陷celestial天上的macroscopic肉眼可见的, 巨观的classical field theory 经典场理论rigit 刚硬的, 刚性的, 严格的quantum量, 额, [物] 量子, 量子论inception 起初, 获得学位pertain 适合, 属于encompass 包围, 环绕, 包含或包括某事物ingredient 成分, 因素acquainted有知识的, 知晓的synonymous同义的configuration 构造, 结构, 配置, 外形inertia 惯性, 惯量attribute 特性momentum动量designate 指明projectile 射弹,发射的ballistics 弹道学, 发射学intractable 难处理的furnish 供应, 提供, 装备, 布置torque n. 扭矩, 转矩moment 力矩的dissipation 消散, 分散, 挥霍, 浪费, 消遣, 放荡, 狂饮constitutive构成的, 制定的continuum mechanics 连续介质力学superposition重叠, 重合, 叠合reckon 计算, 总计, 估计, 猜想,数, 计算, 估计, 依赖, 料想strength 强度load 载荷empirical 以经验为依据的insofar 在……范围cohesive 内聚性的stiffness 硬度furnish 供给turbulent 湍流laminar 层流isothermal 等温isotropic 各向同性eddy 旋涡viscosity 粘性、粘度adiabatic 绝热的reversible 可逆的isentropic 等熵的stream tube 流管tangential 切线的incompressible 不可压缩的similitude 相似性hydraulic 水力的,水力学的spillway (河或水坝的)放水道,泄洪道prototype 原型,样板vibratory 振动的,摆动的propagation 传播acoustic 听觉的,声学的damp 阻尼,衰减restore 复职,归还neutral 平衡exciting force 激励力resonant共振的,谐振的stiffness 刚度,刚性magnitude 数值,大小substantially实质上的perturb 干扰,扰乱Fourier series 傅里叶级数shredder 切菜器metropolitan 大都市的at-grade 在同一水平面上elevated 高架的guide way 导轨rigid body 刚体medium 介质aging 老化polymeric聚合(物)的consolidate 把…联合为一体,统一radically 根本地,本质上deliberate 从容不迫的,深思熟虑Attribute赋予medieval 中世纪的etch 蚀刻,蚀镂fingernail 指甲bar chart 直方图joystick 游戏杆trial-error 试制, 试生产junction n. 连接, 接合, 交叉点, 汇合处contrive v. 发明, 设计, 图谋snooker (=snooker pool)彩色台球, 桌球****公理axiom 命题proposition 被加数augend , summand 加数addend 被减数minuend 减数subtrahend 差remainder 被乘数multiplicand, faciend 乘数multiplicator 积product 被除数dividend 除数divisor 商quotient 大于等于is equal or greater than 小于等于is equal or lesser than 运算符operator 算术平均数geometric mean n个数之积的n次方根(reciprocal)x的倒数为1/x 有理数rational number 无理数irrational number 整数integer小数点decimal point分数fraction 分子numerator 分母denominator 比ratio 十进制decimal system 二进制binary system 十六进制hexadecimal system 权weight, significance 截尾truncation 四舍五入round 下舍入round down 上舍入round up 有效数字significant digit 无效数字insignificant digit 代数algebra 单项式monomial 多项式polynomial, multinomial 系数coefficient 未知数unknown, x-factor, y-factor, z-factor 等式,方程式equation 一次方程simple equation 二次方程quadratic equation 三次方程cubic equation 四次方程quartic equation 阶乘factorial 对数logarithm 指数,幂exponent 乘方power 二次方,平方square 三次方,立方cube 四次方the power of four, the fourth power n次方the power of n, the nth power 开方evolution, extraction 二次方根,平方根square root 三次方根,立方根cube root 四次方根the root of four, the fourth root n次方根the root of n, the nth root 坐标系coordinates 坐标轴x-axis, y-axis, z-axis 横坐标x-coordinate 纵坐标y-coordinate 原点origin 象限quadrant 截距(有正负之分)intercede (方程的)解solution 线段segment 射线radial 平行parallel 相交intersect 角度degree 弧度radian 钝角obtuse angle 平角straight angle 周角perigon 底base 锐角三角形acute triangle 直角边leg 斜边hypotenuse 勾股定理Pythagorean theorem 钝角三角形obtuse triangle 不等边三角形scalene triangle 等腰三角形isosceles triangle 等边三角形equilateral triangle 四边形quadrilateral 平行四边形parallelogram 周长perimeter 全等congruent 三角trigonometry 正弦sine 余弦cosine 正切tangent 余切cotangent 正割secant 余割cosecant 反正弦arc sine 反余弦arc cosine 反正切arc tangent 反余切arc cotangent 反正割arc secant 反余割arc cosecant 集合aggregate 空集void 子集subset 交集intersection 并集union 补集complement 映射mapping 定义域domain, field of definition 值域range 单调性monotonicity 图象image 数列,级数series 导数derivative 无穷小infinitesimal 复数complex number 矩阵matrix 行列式determinant 半圆semicircle 扇形sector 环ring 椭圆ellipse 圆周circumference 轨迹locus, loca(pl.) 平行六面体parallelepiped 立方体cube 七面体heptahedron 八面体octahedron 九面体enneahedron 十面体decahedron 十一面体hendecahedron 十二面体dodecahedron 二十面体icosahedron 多面体polyhedron 四面体tetrahedron 五面体pentahedron 六面体hexahedron 菱形rhomb, rhombus, rhombi(pl.), diamond 正方形square 梯形trapezoid 直角梯形right trapezoid 等腰梯形isosceles trapezoid 五边形pentagon 六边形hexagon 七边形heptagon 八边形octagon 九边形enneagon 十边形decagon 十一边形hendecagon 十二边形dodecagon 多边形polygon 正多边形equilateral polygon 相位phase 振幅amplitude 内心incentre(BrE), incenter(AmE) 外心excentre(BrE), excenter(AmE) 旁心escentre(BrE), escenter(AmE) 垂心orthocentre(BrE), orthocenter(AmE) 重心barycentre(BrE), barycenter(AmE) 内切圆inscribed circle 外切圆circumcircle 方差variance 标准差root-mean-square deviation, standard deviation 百分点percentage 百分位数percentile 排列permutation 分布distribution 正态分布normal distribution 非正态分布abnormal distribution 条形统计图bar graph 柱形统计图histogram 折线统计图broken line graph 曲线统计图curve diagram 扇形统计图pie diagram**** mutually disjoint events 互不相交事件mutually disjoint subsets 互不相交子集mutually independent events 互相独立事件myria 万myriad 无数的multiplicity 重数mid square method 平方取中法midperpendicular 中垂线minor 子式minor arc 劣弧mixed number 带分数regular convergence 正则收敛relative discriminant 相对判别式relative error 相对误差relative extremum 局部极值ricci equatoin 李奇恒等式ricci identity 李奇恒等式riemann function 黎曼函数riemann integral 黎曼积分right direct product 右直积right endpoint 右端点right inner product 右内积ring of integers 整数环ring of matrices 矩阵环root mean square error 均方根差root of equation 方程式的根rotation of axes 坐标轴的旋转rotation of co ordinate system 坐标轴的旋转round off error 舍入规则round up error 舍入规则runge kutta method 龙格库塔法n disk n维圆盘nth member 第n项nth partial quotient 第n偏商nth power operation n次幂运算nth root n次根nth term 第n项n times continuously differentiable n次连续可微的natural injection 自然单射natural isomorphism 自然等necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量neutral element 零元素neutral line 中线nonhomogeneous linear boundary value problem 非齐次线性边值问题nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组interval algebra 区间代数interval analysis 区间分析interval closed at the right 右闭区间interval estimation 区域估计interval function 区间函数interval graph 区间图interval of convergence 收敛区间interval of definition 定义区间interval topology 区间拓扑irreducible set 不可约集irreducible r module 不可约r模periodical decimal fraction 循环十进小数pentad 拼五小组pentadecagon 十五边形pentagon 五角形pentagonal number 五角数pentagonal pyramid 五角锥pentagram 五角星pentahedron 五面体pentaspherical coordinates 五球坐标penalty method 补偿法pascal distribution 帕斯卡分布partition function 分折函数partial differential equation of elliptic type 椭圆型偏微分方程partial differential equation of first order 一阶偏微分方程partial differential equation of hyperbolic type 双曲型偏微分方程partial differential equation of mixed type 混合型偏微分方程partial differential equation of parabolic type 抛物型偏微分方程partial differential operator 偏微分算子parametric test 参数检验particular solution 特解parallelogram axiom 平行四边形公理orthogonality relation 正交关系ordinary differential equation 常微分方程optimal value function 最优值函数opposite angles 对角opposite category 对偶范畴one to one mapping 一一映射onto mapping 满射open mapping theorem 开映射定理one to many mapping 一对多映射one sided limit 单侧极限numerical solution of linear equations 线性方程组的数值解法null set 空集null solution 零解third boundary condition 第三边界条件two sided neighborhood 双侧邻域unbiased estimating equation 无偏估计方程unbounded function 无界函数unbounded quantifier 无界量词uncertainty principle 测不准原理uncorrelated random variables 不相关随机变量undetermined coefficient 末定系数velocity distribution 速度分布velocity optimal 速度最优的weak approximation theorem 弱逼近定理weak completeness 弱完备性weak continuity 弱连续性weak convergence 弱收敛wiener measure 维纳测度word group 自由群sample correlation coefficient 样本相关系数sample covariance 样本协方差schwarz inequality 施瓦尔兹不等式second boundary condition 诺伊曼边界条件second comparison test 第二比较检验second limit theorem 第二极限定理self adjoint differential equation 自伴微分方程semimajor axis 半长轴semiminor axis 半短轴sentential calculus 命题演算set of measure zero 零测度集set topology 集论拓扑simple connectedness 单连通性slope function 斜率函数solution curve 积分曲线solution domain 解域solution set of equation 方程的解集spatial co ordinate 空间坐标specific address 绝对地址spherical bessel function 球贝塞耳函数spherical cap 球冠spherical coordinates 球极坐标spherical curvature 球面曲率spherical shell 球壳spherical zone 球带spline function 样条函数spline interpolation 样条内插stability conditions 稳定条件statistical hypothesis testing 统计假设检验strict inequality 严格不等式strict isotonicity 严格保序性strict isotony 严格保序性strict increasing 严格递增system of partial differential equations 偏微分方程组system of ordinary differential equations 常微分方程组system of linear homogeneous equations 线性齐次方程组system of linear inhomogeneous equations 线性非齐次方程组system of inequalities 联立不等式system of polar coordinates 极坐标系system of variational equations 变分方程组system with concentrated parameters 集中参数系统system with distributed parameters 分布参数系统t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间tangent cone 切线锥面telegraph equation 电报方程theorem for damping 阻尼定理****充分条件sufficient condition必要条件necessary condition 充要条件sufficient and necessary condition……的充要条件是………if and only if …****abscissa 横坐标alternating series 交错级数angle of the sector 扇形角arbitrary constant 任意常数augmented matrix 增广矩阵axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴binomial series 二项级数binomial theorem 二项式定理binomial distribution 二项分布bisection method 分半法;分半方法bounded above 有上界的;上有界的bounded below 有下界的;下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号Cartesian coordinates 笛卡儿坐标certain event 必然事件circumcentre 外心;外接圆心circumcircle 外接圆classical theory of probability 古典概率论cofactor 余因子; 余因式common denominator 同分母;公分母common difference 公差common divisor 公约数;公约common logarithm 常用对数common multiple 公位数;公倍common ratio 公比commutative law 交换律compasses 圆规Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理centripedal acceleration 向心加速度concave downward 凹向下的concurrent 共点concyclic 共圆concyclic points 共圆点Euclidean geometry 欧几里德几何Euler's formula 尤拉公式;欧拉公式even function 偶函数even number 偶数game (1)对策;(2)博奕Gaussian distribution 高斯分布greatest term 最大项greatest value 最大值harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数higher order derivative 高阶导数improper fraction 假分数improper integral 广义积分; 非正常积分implicit function 隐函数incircle 内切圆inclined plane 斜included angle 夹角indefinite integral 不定积分initial condition 原始条件;初值条件initial-value problem 初值问题interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代Lagrange interpolating polynomial 拉格朗日插值多项代Laplace expansion 拉普拉斯展式lemniscate 双纽线left hand limit 左方极限limiting case 极限情况limiting position 极限位置line of best-fit 最佳拟合line segment 线段logarithmic equation 对数方程mathematical analysis 数学分析mathematical induction 数学归纳法monotonic decreasing function 单调递减函数monotonic convergence 单调收敛性monotonic increasing function 单调递增函数multiple-angle formula 倍角公式multiple root 多重根mutually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直numerical method 计算方法;数值法oblique cone 斜圆锥orthogonal circles 正交圆orthogonality 正交性oscillatory convergence 振动收敛性ordinary differential equation 常微分方程pairwise mutually exclusive events 两两互斥事件place holder 补位数字point of inflection (inflexion) 拐点; 转折点Pisson distribution 泊松分布point-slope form 点斜式polar coordinate plane 极坐标平面polynomial equation 多项式方程posterior probability 后验概率; 事后概率premultiply 前乘; 自左乘prime factor 质因子;质因素prime number 素数;质数principal angle 主角principal axis 主轴principal value 主值prior probability 先验概率; 事先概率probability density function 概率密度函数product and sum formula 和积互变公式product sample space 积样本空间product to sum formula 积化和差公式proof by contradiction 反证法; 归谬法proper fraction 真分数proper integral 正常积分proper subset 真子集propositional calculus 命题演算propositional inference 命题推演protractor 量角器Pythagoras' theorem 勾股定理Pythagorean triplet 毕氏三元数组quadratic convergence 二阶收敛性quadrature 求积法quotient set 商集radial component 沿径分量radical axis 根轴range 值域;区域;范围;极差;分布域rationalization 有理化raw data 原始数据rectifiable 可求长的reciprocal 倒数rectangular coordinate plane 直角坐标平面recurrence formula 递推公式reducibility 可约性; 可化简性reflexive relation 自反关系reference angle 参考reference line 基准线reflex angle 优角;反角region of acceptance 接受区域region of convergency 收敛区域region of rejection 否定区域right circular cone 直立圆锥(体)resolution of vector 向量分解; 矢量分解right hand limit 右方极限right prism 直立棱柱;直立角柱(体) right pyramid 直立棱锥;直立角锥(体) right-angled triangle 直角二角形scalene triangle 不等边三角形;不规则三角形scatter diagram 散点图scientific notation 科学记数法semi-conjugate axis 半共轭轴semi-transverse axis 半贯轴semi-vertical angle 半顶角separable differential equation 可分微分方程septic equation 七次方程set square 三角尺;三角板shaded portion 有阴影部分significance level 显著性水平significant figure 有效数字similar triangles 相似三角形simple iteration method 简单迭代法simple pendulum 单摆Simpson's integral 森逊积分standard deviation 标准差;标准偏离standard normal distribution 标准正态分布; 标准常态分布stationary point 平稳点; 逗留点; 驻点strictly monotonic 严格单调statistical chart 统计分析submultiple angle formula 半角公式subsidiary angle 辅助角substitution 代入; 代入法successive approximation 逐次逼近法successive derivative 逐次导数successive differentiation 逐次微分法suffix 下标sum to infinity 无限项之和sum to product formula 和化积公式superimposing 迭合supplementary angle 补角surjection 满射symmetric relation 对称关系tautology 恒真命题;恒真式Taylor’s expansion 泰勒展开式Taylor’s series 泰勒级数Taylor’s theorem 泰勒定理test criterion 检验标准test of significance 显著性检验to the nearest 至最接近之torque 转矩torus 环面transcendental function 超越函数transformation of variable 变数转换transitive 可传递的transpose of matrix 倒置矩阵;转置矩阵transversal 截;横截的triangle law of addition 三角形加法travel graph 行程图tree diagram 树形图trapezoidal integral 梯形积分truncated Taylor’s series 截断泰勒级数two-tailed test 双尾检验;只端检验type I error I型误差type II error II型误差unbiased estimator 无偏估计量undetermined coefficient 待定系数unique solution 唯一解vertical asymptote 垂直渐近线vertically opposite angles 对顶角without loss of generality 不失一般性****分子Numerator 分母Denominator 阿拉伯数字Hindu-Arabic numeral假分数Improper fraction 最大公因子Highest Common Factor (H.C.F.) 最小公倍数Lowest Common Multiple (L.C.M.) 行列式determinant****interval closed at the right 右闭区间interval of convergence 收敛区间interval of definition 定义区间invariance theorem 不变性定理invariant of an equation 方程的不变量inverse circular function 反三角函数inverse hyperbolic function 反双曲函数inversion formula 反演公式isotonic injective mapping 保序单射映射jacobi identity 雅可比恒等式jump point 跳跃点law of double negation 双重否定律law of inertia 惯性律law of large numbers 大数定律leading ideal 猪想liouville theorem 刘维尔定理lipschitz condition 李普希茨条件markov transform 马尔可夫变换mathematical approximation 数学近似法mathematical model 数学模型maximum condition 极大条件maximum deviation 最大偏差mean square deviation 方差mean square of error 误差的均方meromorphic function 亚纯函数柱形统计图histogram 折线统计图broken line graph 曲线统计图curve diagram 扇形统计图pie diagram排列permutatio内切圆inscribed circle 外切圆circumcircle正多边形equilateral polygon metric space 度量空间metric subspace 度量子空间method of runge kutta type 朗格库塔型的方法method of steepest ascent 最速上升法method of steepest descent 最速下降法method of finite elements 有限元法method of fractional steps 分步法method of exhaustion 穷竭法method of approximation 近似法method of artificial variables 人工变量法method of balayage 扫除法method of characteristic curves 特者法method of comparison 比较法method of conjugate gradients 共轭梯度法lateral area 侧面积last multiplier 最后乘子large sample test 大样本检验lattice constant 点阵常数lattice design 格子设计method of difference 差分法method of elimination 消元法method of estimation 估计法meromorphic differential 亚纯微分median 中位数measuring rule 量尺mean term 内项mean term 内项mean term 内项irreducibility criterion 不可约性判别准则irreducible polynomial 不可约多项式irreducible generating set 不可约生成集irregular divisor class 非正则因子类irregular point 非正则点irregular singular point 非正则奇点isometric circle 等距圆isometric embedding 等距嵌入isomorphic field 同构域isomorphic graph 同构图isomorphic group 同构群isomorphic image 同构象isothermal parameter 等温参数iterated function 叠函数iterated integral 累积分joint distribution 联合分布jordan algebra 约当代数kernel of an integral equation 积分方程的核l'hospital rule 洛必达规则laboratory system of coordinates 实验室坐标系labyrinth 迷宫lacation principle 介值定理lag correlation coefficient 滞后相关系数lag regression 落后回归laguerre differential equation 拉盖尔微分方程lame equation 拉梅方程language of formula 公式语言laplace beltrami operator 拉普拉斯贝尔特拉米算子lateral area 侧面积last multiplier 最后乘子large sample test 大样本检验lattice constant 点阵常数lattice design 格子设计left adjoint 左伴随的left derivative 左导数left differential 左微分left direct product 左直积left end point 左端点left length 左长left limit value 左极限值left multiplication ring 左乘环length of curve 曲线的长length of normal 法线的长levi decomposition 列维分解limes inferior 下极限limes superior 上极限limit circle 极限圆limit circle type 极限圆型logarithm to the base 10 常用对数logarithmic normal distribution 对数正态分布logic of relations 关系逻辑magic circle 幻圆magic cube 幻立方manifold without boundary 无边廖many valued mapping 多值映射marginal distribution density function 边缘分布密度函数marginal distribution function 边缘分布函数mathematical programming 数学规划mathematical random sample 数学随机样本mathematical statistics 数理统计maximum likelihood estimating function 极大似然估计量independent variable 自变量dependent variable 应变量equiangular triangle 正三角形,等角三角形命题proposition 差remainder 积product 除数divisor 商quotient 截尾truncation 未知数unknown, x-factor, y-factor, z-factor 阶乘factorial 集合aggregate 空集void 子集subset 交集intersection 并集union 补集complement 映射mapping 勾股定理Pythagorean theorem 菱形rhomb, rhombus, rhombi(pl.), diamond 双曲线hyperbola 抛物线parabola topology of bounded convergence 有界收敛拓扑toroid 超环面toroidal coordinates 圆环坐标trace of dyadic 并向量的迹transcendental integral function 超越整函数transformation formulas of the coordinates 坐标的变换公式transformation to principal axes 轴变换transversal lines 截线trapezoid method 梯形公式trefoil knot 三叶形纽结truth function 真值函项two sided test 双侧检定two sided neighborhood 双侧邻域two sided surface 双侧曲面two termed expression 二项式ultrahyperbolic equation 超双曲型方程a priori bound 先验界限a priori distribution 先验分布a priori probability 先验概率a summable a可和的abacus 算盘abbreviate 略abbreviation 简化abel equation 阿贝耳方程abel identity 阿贝耳恒等式abel inequality 阿贝耳不等式abel summation method 阿贝耳求和法abelian algebra 阿贝耳代数abelian differential 阿贝耳微分abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分abelian summation 阿贝耳求和法abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号abscissa 横坐标abscissa of absolute convergence 绝对收敛坐标abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute 绝对形absolute address 绝对地址absolute class field 绝对类域absolute coding 绝对编码absolute cohomology 绝对上同调absolute conic 绝对二次曲线absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率。
matlabrandom函数Matlab是一款功能强大的数学软件,其内置了许多实用的函数,其中之一便是random函数。
本文将围绕这一函数展开,介绍random 函数的用法及其在实际应用中的作用。
我们来了解一下random函数的基本用法。
在Matlab中,random函数用于生成随机数。
它可以接受不同的参数,从而生成不同类型的随机数。
例如,random('unif',a,b)可以生成[a,b]范围内的均匀分布随机数,而random('norm',mu,sigma)则可以生成满足指定均值mu和标准差sigma的正态分布随机数。
除了这些常见的分布,random函数还可以生成其他类型的随机数,如泊松分布、指数分布等。
在实际应用中,random函数可以发挥重要作用。
例如,在模拟实验中,我们经常需要生成服从某种分布的随机数。
通过使用random函数,我们可以方便地生成满足特定要求的随机数序列,从而进行各种模拟实验。
这对于评估系统性能、优化算法等方面具有重要意义。
random函数还可以用于数据分析和统计推断。
在统计学中,我们经常需要从样本中抽取随机样本,并对其进行分析。
通过使用random 函数,我们可以方便地生成具有相同分布特征的随机样本,从而进行统计推断和模型拟合。
除了生成随机数,random函数还可以用于随机排列和随机抽样。
在实际应用中,我们经常需要对数据进行随机排序,例如在数据挖掘中,我们可以使用随机排列来打乱数据集,以避免模型过拟合。
而对于大规模数据集,我们通常需要进行随机抽样。
通过使用random 函数,我们可以方便地实现这些功能,为数据处理和分析提供便利。
除了上述功能,random函数还可以用于生成随机矩阵、随机向量等。
这在矩阵计算和线性代数中非常有用。
通过使用random函数,我们可以生成具有特定分布特征的随机矩阵,从而进行各种矩阵运算和模型求解。
random函数是Matlab中一个非常实用的函数,它可以用于生成各种类型的随机数,并在模拟实验、数据分析和统计推断等方面发挥重要作用。
symmetric random variables的一些概念-回复什么是对称随机变量?对称随机变量是指在其概率密度函数或概率质量函数关于某个中心对称轴对称的随机变量。
换句话说,对称随机变量的取值在概率分布上关于某个点对称。
关于对称随机变量的概念,有一些相关的概念和特性需要我们探讨。
首先,我们来讨论对称随机变量的中心。
中心:对称随机变量的中心通常是指其概率分布函数的中心对称轴所在的值。
中心轴将概率分布函数分成两个对称的部分。
对称随机变量的中心可以是一个特定的值,也可以是一个区间。
通常来说,对称随机变量的中心是其期望值(即均值),也就是其概率分布函数的中心位置。
对称轴的性质:对于对称随机变量来说,其对称轴具有一些特殊的性质。
首先,对称轴是随机变量的一个固定点或一个固定区间,无论概率密度函数如何变化,对称轴仍然不变。
其次,任意两个关于对称轴对称的点的概率相等。
也就是说,如果x是对称随机变量的一个取值,那么与x关于对称轴对称的另一个取值的概率与x取值的概率相等。
对称随机变量的特征:对称随机变量的特性可以通过其概率密度函数或概率质量函数来推导。
对于连续对称随机变量,其概率密度函数关于对称轴对称,且概率密度函数在对称轴上的值最高。
对于离散对称随机变量,其概率质量函数关于对称轴对称,且概率质量函数在对称轴上的值最高。
对称随机变量的例子:下面是一些常见的对称随机变量的例子:1. 标准正态分布:标准正态分布是一个连续的对称随机变量,其概率密度函数关于均值为0的对称轴对称。
2. 均匀分布:均匀分布是一个连续的对称随机变量,其概率密度函数在中心对称轴上的值相等。
3. 二项分布:二项分布是一个离散的对称随机变量,其概率质量函数关于其期望值对称。
对称随机变量的性质:对称随机变量有一些重要的性质:1. 对称随机变量的期望值(均值)是其中心轴的位置。
2. 对称随机变量关于中心轴具有对称性。
也就是说,如果x是对称随机变量的一个取值,那么与x关于对称轴对称的另一个取值的概率与x取值的概率相等。
随机厄米矩阵本征值英文回答:Random Hermitian matrices are a special class of matrices that have proven to be useful in a variety of applications, including quantum mechanics, statistical physics, and machine learning. One of the most important properties of random Hermitian matrices is the distribution of their eigenvalues.The eigenvalues of a random Hermitian matrix are knownto be real and distributed according to the Wignersemicircle law. The Wigner semicircle law states that the probability density function of the eigenvalues is given by:P(\lambda) = \frac{1}{2\pi \sigma^2} \sqrt{4\sigma^2-\lambda^2}。
where $\sigma^2$ is the variance of the matrix entries.The Wigner semicircle law has been proven in a number of different ways, including using the theory of orthogonal polynomials and the method of moments. It is a fundamental result in random matrix theory and has a number of important applications.中文回答:随机厄米矩阵是厄米矩阵的一种特殊类型,在量子力学、统计物理和机器学习等领域中有着广泛的应用。
一、PolynomialFeatures简介PolynomialFeatures是scikit-learn中的一个函数,用于生成一个新的特征矩阵,其中包含所有的组合特征,用于多项式回归或其他高阶模型。
在机器学习中,有时候通过添加高次项自变量的幂函数来拟合一些非线性模型。
这时候可以使用PolynomialFeatures函数来自动生成这些高次项的特征变量。
二、PolynomialFeatures的基本用法在scikit-learn中使用PolynomialFeatures函数非常简单,只需按照以下步骤即可:1.导入PolynomialFeatures函数在代码中首先需要导入PolynomialFeatures函数,一般位于`sklearn.preprocessing`模块中。
2.创建PolynomialFeatures对象接下来需要创建一个PolynomialFeatures对象,可以通过指定`degree`参数来设置生成的多项式阶数,还可以设置`include_bias`参数来决定是否包含截距项。
3.拟合与转换将原始的特征矩阵传入PolynomialFeatures对象的`fit_transform`方法,即可得到包含多项式特征的新特征矩阵。
4.应用到模型中可以将生成的多项式特征矩阵应用到机器学习模型中进行训练和预测。
三、PolynomialFeatures的参数说明PolynomialFeatures函数主要包含以下几个常用参数:1. degree:整数型参数,表示多项式的阶数。
2. interaction_only:布尔型参数,默认为False,表示是否只生成特征的交互项,若为True则只生成特征之间的乘积项。
3. include_bias:布尔型参数,默认为True,表示是否在生成的多项式特征中包含截距项。
四、示例代码以下是一个简单的示例代码,演示了如何使用PolynomialFeatures生成多项式特征。
随机高斯过程结构方程模型一、高斯过程模型高斯过程模型是一种概率模型,用于描述连续随机变量或随机向量之间的依赖关系。
在时间序列分析、统计学和机器学习等领域中,高斯过程模型被广泛用于建模和预测。
二、随机过程随机过程是一系列随机变量的集合,这些随机变量随着时间或其他参数的变化而变化。
随机过程在许多领域都有应用,如统计学、经济学、通信和自然语言处理等。
三、结构方程模型结构方程模型是一种用于描述变量间因果关系的统计模型。
它通过估计一组潜在变量的因果关系来描述这些变量之间的关系。
结构方程模型通常用于探索性因子分析、路径分析和因果推理等领域。
四、隐含变量模型隐含变量模型是一种统计模型,用于描述不可观测的潜在变量与可观测的变量之间的关系。
这种模型通常用于揭示隐藏在数据背后的结构和模式,例如在语音识别、图像处理和自然语言处理等领域中。
五、线性高斯模型线性高斯模型是一种特殊的线性回归模型,其中因变量是连续的且符合高斯分布。
这种模型在许多领域中都有应用,例如时间序列分析和统计质量管理。
线性高斯模型具有简单和易于解释的特点,同时还可以使用高效的算法进行估计和预测。
六、连续参数模型连续参数模型是一种数学模型,用于描述连续变量的变化规律。
这种模型通常用于模拟和预测物理现象、化学反应和生物系统等领域的动态过程。
连续参数模型的参数通常是未知的,需要通过实验数据或观测数据来估计。
七、非参数模型非参数模型是一种统计模型,它不假设任何特定的数据分布或结构。
这种模型的优点是灵活性高,可以适应各种不同的数据类型和分布情况。
然而,非参数模型的估计和推断通常比较复杂,需要使用复杂的算法和技术。
Open Problems ListArising from MathsCSP Workshop,Oxford,March2006Version0.3,April25,20061Complexity and Tractability of CSPQuestion1.0(The Dichotomy Conjecture)Let B be a relational structure.The problem of deciding whether a given relational structure has a homomorphism to B is denoted CSP(B).For which(finite)structures is CSP(B)decidable in polynomial time?Is it true that for anyfinite structure B the problem CSP(B)is either decidable in polynomial time or NP-complete?Communicated by:Tomas Feder&Moshe Vardi(1993) Question1.1A relational structure B is called hereditarily tractable if CSP(B )is tractable for all substructures B of B.Which structures B are hereditarily tractable?Communicated by:Pavol Hell Question1.2A weak near-unanimity term is defined to be one that satisfies the following identities:f(x,...,x)=x and f(x,y,....y)=f(y,x,y,....y)=...=f(y,...,y,x).Is CSP(B)tractable for any(finite)structure B which is preserved by a weak near-unanimity term?Communicated by:Benoit Larose,Matt Valeriote Question1.3A constraint language1S is called globally tractable for a problem P,if P(S)is tractable,and it is called(locally)tractable if for everyfinite L⊆S,P(L)is tractable.These two notions of tractability do not coincide in the Abduction problem(see talk by Nadia Creignou).•For which computational problems related to the CSP do these two notions of tractability coincide?•In particular,do they coincide for the standard CSP decision problem?Communicated by:Nadia Creignou 1That is,a(possibly infinite)set of relations over somefixed set.1Question1.4(see also Question3.5)It has been shown that when a structure B has bounded pathwidth duality the corresponding problem CSP(B)is in the complexity class NL (see talk by Victor Dalmau).Is the converse also true(modulo some natural complexity-theoretic assumptions)?Communicated by:Victor Dalmau Question1.5Is there a good(numerical)parameterization for constraint satisfaction problems that makes themfixed-parameter tractable?Question1.6Further develop techniques based on delta-matroids to complete the com-plexity classification of the Boolean CSP(with constants)with at most two occurrences per variable(see talk by Tomas Feder).Communicated by:Tomas Feder Question1.7Classify the complexity of uniform Boolean CSPs(where both structure and constraint relations are specified in the input).Communicated by:Heribert Vollmer Question1.8The microstructure graph of a binary CSP has vertices for each variable/value pair,and edges that join all pairs of vertices that are compatible with the constraints.What properties of this graph are sufficient to ensure tractability?Are there properties that do not rely on the constraint language or the constraint graph individually?2Approximability and Soft ConstraintsQuestion2.1Is it true that Max CSP(L)is APX-complete whenever Max CSP(L)is NP-hard?Communicated by:Peter Jonsson Question2.2Prove or disprove that Max CSP(L)is in PO if the core of L is super-modular on some lattice,and otherwise this problem is APX-complete.The above has been proved for languages with domain size3,and for languages contain-ing all constants by a computer-assisted case analysis(see talk by Peter Jonsson).Develop techniques that allow one to prove such results without computer-assisted analysis.Communicated by:Peter Jonsson Question2.3For some constraint languages L,the problem Max CSP(L)is hard to approximate better than the random mindless algorithm on satisfiable or almost satisfiable instances.Such problems are called approximation resistant(see talk by Johan Hastad).Is a single random predicate over Boolean variables with large arity approximation resistant?What properties of predicates make a CSP approximation resistant?What transformations of predicates preserve approximation resistance?Communicated by:Johan Hastad2Question2.4Many optimisation problems involving constraints(such as Max-Sat,Max CSP,Min-Ones SAT)can be represented using soft constraints where each constraint is specified by a cost function assigning some measure of cost to each tuple of values in its scope.Are all tractable classes of soft constraints characterized by their multimorphisms?(see talk by Peter Jeavons)Communicated by:Peter Jeavons 3AlgebraQuestion3.1The Galois connection between sets of relations and sets of operations that preserve them has been used to analyse several different computational problems such as the satisfiability of the CSP,and counting the number of solutions.How can we characterise the computational goals for which we can use this Galois connection?Communicated by:Nadia Creignou Question3.2For any relational structure B=(B,R1,...,R k),let co-CSP(B)denote the class of structures which do not have a homomorphism to B.It has been shown that the question of whether co-CSP(B)is definable in Datalog is determined by P ol(B),the polymorphisms of the relations of B(see talk by Andrei Bulatov).Let B be a core,F the set of all idempotent polymorphisms of B and V the variety generated by the algebra(B,F).Is it true that co-CSP(B)is definable in Datalog if and only if V omits types1and2(that is,the local structure of anyfinite algebra in V does not contain a G-set or an affine algebra)?Communicated by:Andrei Bulatov Question3.3Does every tractable clone of polynomials over a group contain a Mal’tsev operation?Communicated by:Pascal Tesson Question3.4Classify(w.r.t.tractability of corresponding CSPs)clones of polynomials of semigroups.Communicated by:Pascal Tesson Question3.5Is it true that for any structure B which is invariant under a near-unanimity operation the problem CSP(B)is in the complexity class NL?Does every such structure have bounded pathwidth duality?(see also Question1.4)Both results are known to hold for a2-element domain(Dalmau)and for majority operations(Dalmau,Krokhin).Communicated by:Victor Dalmau,Benoit Larose3Question3.6Is it decidable whether a given structure is invariant under a near-unanimity function(of some arity)?Communicated by:Benoit Larose Question3.7Let L be afixedfinite lattice.Given an integer-valued supermodular func-tion f on L n,is there an algorithm that maximizes f in polynomial time in n if the function f is given by an oracle?The answer is yes if L is a distributive lattice(see“Supermodular Functions and the Complexity of Max-CSP”,Cohen,Cooper,Jeavons,Krokhin,Discrete Applied Mathemat-ics,2005).More generally,the answer is yes if L is obtained fromfinite distributive lattices via Mal’tsev products(Krokhin,Larose–see talk by Peter Jonsson).The smallest lattice for which the answer is not known is the3-diamond.Communicated by:Andrei Krokhin Question3.8Find the exact relationship between width and relational width.(It is known that one is bounded if and and only if the other is bounded.)Also,what types of width are preserved under natural algebraic constructions?Communicated by:Victor Dalmau 4LogicQuestion4.1The(basic)Propositional Circumscription problem is defined as fol-lows:Input:a propositional formulaφwith atomic relations from a set S,and a clause c.Question:is c satisfied in every minimal model ofφ?It is conjectured(Kirousis,Kolaitis)that there is a trichotomy for this problem,that it iseither in P,coNP-complete or inΠP2,depending on the choice of S.Does this conjecturehold?Communicated by:Nadia Creignou Question4.2The Inverse Satisfiability problem is defined as follows: Input:afinite set of relations S and a relation R.Question:is R expressible by a CNF(S)-formula without existential variables?A dichotomy theorem was obtained by Kavvadias and Sideri for the complexity of this problem with constants.Does a dichotomy hold without the constants?Are the Schaefer cases still tractable?Communicated by:Nadia Creignou4Question4.3Let LFP denote classes of structures definable infirst-order logic with a least-fixed-point operator,let HOM denote classes of structures which are closed under homomorphisms,and let co-CSP denote classes of structures defined by not having a homomorphism to somefixed target structure.•Is LFP∩HOM⊆Datalog?•Is LFP∩co-CSP⊆Datalog?(forfinite target structures)•Is LFP∩co-CSP⊆Datalog?(forω-categorical target structures)Communicated by:Albert Atserias,Manuel BodirskyQuestion4.4(see also Question3.2)Definability of co-CSP(B)in k-Datalog is a sufficient condition for tractability of CSP(B),which is sometimes referred to as having width k. There is a game-theoretic characterisation of definability in k-Datalog in terms of(∃,k)-pebble games(see talk by Phokion Kolaitis).•Is there an algorithm to decide for a given structure B whether co-CSP(B)is definable in k-Datalog(for afixed k)?•Is the width hierarchy strict?The same question when B isω-categorical,but not necessarilyfinite?Communicated by:Phokion Kolaitis,Manuel BodirskyQuestion4.5Find a good logic to capture CSP with“nice”(e.g.,ω-categorical)infinite templates.Communicated by:Iain Stewart 5Graph TheoryQuestion5.1The list homomorphism problem for a(directed)graph H is equivalent to the problem CSP(H∗)where H∗equals H together with all unary relations.•It is conjectured that the list homomorphism problem for a reflexive digraph is tractable if H has the X-underbar property(which is the same as having the bi-nary polymorphism min w.r.t.some total ordering on the set of vertices),and NP-complete otherwise.•It is conjectured that the list homomorphism problem for an irreflexive digraph is tractable if H is preserved by a majority operation,and NP-complete otherwise. Do these conjectures hold?Communicated by:Tomas Feder&Pavol Hell5Question5.2“An island of tractability?”Let A m be the class of all relational structures of the form(A,E1,...,E m)where each E i is an irreflexive symmetric binary relation and the relations E i together satisfy the following‘fullness’condition:any two distinct elements x,y are related in exactly one of the relations E i.Let B m be the single relational structure({1,...,m},E1,...,E m)where each E i is the symmetric binary relation containing all pairs xy except the pair ii.(Note that the relations E i are not irreflexive.)The problem CSP(A m,B m)is defined as:Given A∈A m,is there a homomorphism from A to B m?When m=2,this problem is solvable in polynomial time-it is the recognition problem for split graphs(see“Algorithmic Graph Theory and Perfect Graphs”,M.C.Golumbic, Academic Press,New York,1980)When m>3,this problem is NP-complete(see“Full constraint satisfaction problems”,T.Feder and P.Hell,to appear in SIAM Journal on Computing).What happens when m=3?Is this an“island of tractability”?Quasi-polynomial algorithms are known for this problem(see“Full constraint satisfaction problems”,T. Feder and P.Hell,,to appear in SIAM Journal on Computing,and“Two algorithms for list matrix partitions”,T.Feder,P.Hell,D.Kral,and J.Sgall,SODA2005).Note that a similar problem for m=3was investigated in“The list partition problem for graphs”, K.Cameron,E.E.Eschen,C.T.Hoang and R.Sritharan,SODA2004.Communicated by:Tomas Feder&Pavol Hell Question5.3Finding the generalized hypertree-width,w(H)of a hypergraph H is known to be NP-complete.However it is possible to compute a hypertree-decomposition of H in polynomial time,and the hypertree-width of H is at most3w(H)+1(see talk by Georg Gottlob).Are there other decompositions giving better approximations of the generalized hypertree-width that can be found in polynomial time?Communicated by:Georg Gottlob Question5.4It is known that a CSP whose constraint hypergraph has bounded fractional hypertree width is tractable(see talk by Daniel Marx).Is there a hypergraph property more general than bounded fractional hypertree width that makes the associated CSP polynomial-time solvable?Are there classes of CSP that are tractable due to structural restrictions and have unbounded fractional hypertree width?Communicated by:Georg Gottlob,Daniel Marx Question5.5Prove that there exist two functions f1(w),f2(w)such that,for every w, there is an algorithm that constructs in time n f1(w)a fractional hypertree decomposition of width at most f2(w)for any hypergraph of fractional hypertree width at most w(See talk by Daniel Marx).Communicated by:Daniel Marx6Question5.6Turn the connection between the Robber and Army game and fractional hypertree width into an algorithm for approximating fractional hypertree width.Communicated by:Daniel Marx Question5.7Close the complexity gap between(H,C,K)-colouring and (H,C,K)-colouring (see talk by Dimitrios Thilikos)Find a tight characterization for thefixed-parameter tractable(H,C,K)-colouring problems.•For the(H,C,K)-colouring problems,find nice properties for the non-parameterisedpart(H−C)that guaranteefixed-parameter tractability.•Clarify the role of loops in the parameterised part C forfixed-parameter hardnessresults.Communicated by:Dimitrios Thilikos6Constraint Programming and ModellingQuestion6.1In a constraint programming system there is usually a search procedure that assigns values to particular variables in some order,interspersed with a constraint propagation process which modifies the constraints in the light of these assignments.Is it possible to choose an ordering for the variables and values assigned which changes each problem instance as soon as possible into a new instance which is in a tractable class? Can this be done efficiently?Are there useful heuristics?Question6.2The time taken by a constraint programming system tofind a solution toa given instance can be dramatically altered by modelling the problem differently.Can the efficiency of different constraint models be objectively compared,or does it depend entirely on the solution algorithm?Question6.3For practical constraint solving it is important to eliminate symmetry,in order to avoid wasted search effort.Under what conditions is it tractable to detect the symmetry in a given problem in-stance?7Notes•Representations of constraints-implicit representation-effect on complexity•Unique games conjecture-structural restrictions that make it false-connectionsbetween definability and approximation•MMSNP-characterise tractable problems apart from CSP7•Migrate theoretical results to tools•What restrictions do practical problems actually satisfy?•Practical parallel algorithms-does this align with tractable classes?•Practically relevant constraint languages(”global constraints”)•For what kinds of problems do constraint algorithms/heuristics give good results?8。
matlab中random函数用法1. 什么是random函数在matlab中,random函数是一种用于生成随机数的函数。
随机数在计算机科学和数学领域有着广泛的应用,可以帮助我们模拟现实生活中的随机事件。
2. random函数的基本用法使用random函数可以生成一个介于0和1之间的随机数。
语法如下:r = random其中,r是生成的随机数。
下面是一个示例代码:r = randomdisp(r)运行以上代码,matlab会输出一个介于0和1之间的随机数。
3. 生成指定范围的随机数除了生成介于0和1之间的随机数,random函数还可以生成指定范围内的随机数。
语法如下:r = random('distribution', a, b)其中,distribution是指定的分布类型,a和b是范围的上下界。
常见的分布类型包括’normal’(正态分布)、‘uniform’(均匀分布)等。
下面是一个示例代码:r = random('normal', 0, 1)disp(r)运行以上代码,matlab会输出一个符合标准正态分布的随机数。
4. 生成多个随机数除了生成单个随机数,random函数还可以生成一个包含多个随机数的矩阵或向量。
语法如下:r = random('distribution', a, b, m, n)其中,m和n是生成矩阵或向量的行数和列数。
下面是一个示例代码:r = random('uniform', 1, 10, 3, 4)disp(r)运行以上代码,matlab会生成一个3行4列的矩阵,其中的元素都是介于1和10之间的随机数。
5. 控制随机数生成种子在matlab中,可以通过设置随机数生成种子来控制随机数的生成过程。
这样可以使得随机数的生成结果可复现。
使用random函数生成随机数时,可以添加一个额外的参数来指定随机数生成种子。
act数学与高考知识点ACT(American College Testing)考试是美国大学招生中广泛使用的一种标准化考试,其中包括数学科目。
本文将详细介绍ACT数学考试的知识点,以帮助考生有效备考。
1. 代数 (Algebra)1.1 线性方程与不等式 (Linear Equations and Inequalities)1.1.1 一元一次方程 (One-variable linear equations)1.1.2 一元一次不等式 (One-variable linear inequalities)1.1.3 线性方程组 (Systems of linear equations)1.2 函数 (Functions)1.2.1 函数定义与图像 (Function definition and graphs)1.2.2 函数的运算 (Operations with functions)1.2.3 函数的反函数 (Inverse functions)1.3 多项式与因式分解 (Polynomials and Factoring)1.3.1 一元多项式 (One-variable polynomials)1.3.2 因式分解 (Factoring)1.3.3 二次方程与二次多项式 (Quadratic equations and polynomials)2. 几何 (Geometry)2.1 平面几何 (Plane Geometry)2.1.1 直线与角度 (Lines and angles)2.1.2 三角形与四边形 (Triangles and quadrilaterals)2.1.3 圆与圆环 (Circles and annuli)2.2 空间几何 (Spatial Geometry)2.2.1 空间中的点、直线、面 (Points, lines, and planes in space)2.2.2 空间几何体的体积与表面积 (Volumes and surface areas of spatial figures)2.2.3 空间几何体的旋转与投影 (Rotations and projections of spatial figures)3. 数据分析与概率 (Data Analysis and Probability)3.1 图表解读与数据分析 (Interpreting graphs and data analysis)3.1.1 条形图、折线图与饼状图 (Bar graphs, line graphs, and pie charts)3.1.2 平均数、中位数与众数 (Mean, median, and mode)3.2 概率 (Probability)3.2.1 随机事件与概率计算 (Random events and probability calculations)3.2.2 排列与组合 (Permutations and combinations)4. 比例、百分数与利率 (Ratios, Percentages, and Rates)4.1 比例与比率 (Ratios and rates)4.2 百分数 (Percentages)4.3 利率与利息 (Interest rates and interest)5. 数字、指数与对数 (Number, Exponents, and Logarithms)5.1 整数与有理数 (Integers and rational numbers)5.2 指数 (Exponents)5.3 对数 (Logarithms)6. 函数与三角 (Functions and Trigonometry)6.1 线性函数与二次函数 (Linear functions and quadratic functions)6.2 三角函数 (Trigonometric functions)6.3 三角方程与三角恒等式 (Trigonometric equations and identities)通过掌握以上知识点,考生能够在ACT数学考试中取得优异的成绩。
在化学中,原子排列的规则性对于物质的性质和功能至关重要。
然而,在某些情况下,原子可能以无序的方式排列,这种现象被称为化学无序。
为了模拟化学无序,可以使用统计量子化学(SQS)模型。
SQS模型是一种计算方法,它通过考虑原子排列的统计分布来预测分子的性质。
在SQS模型中,原子排列被视为一种随机过程,其中每个原子都可以在给定的空间位置上随机出现。
通过计算不同原子排列的概率分布,可以确定分子的平均性质。
使用SQS模型模拟化学无序的步骤如下:
确定分子结构和原子类型。
确定每个原子可能的坐标位置。
使用蒙特卡洛方法生成一组随机原子排列。
对每个原子排列进行量子化学计算,以确定分子的能量、电子结构和其他相关性质。
计算所有原子排列的平均性质,以获得分子的平均性质。
需要注意的是,SQS模型是一种近似方法,它只能提供分子的平均性质。
对于某些特定的分子或系统,可能需要使用更精确的方法来模拟化学无序。
一维伊辛模型严格解
一维伊辛模型是一种描述自旋系统的模型,其中自旋在一维链上
排布。
每个自旋只能处于两种状态中的一种,记为自旋向上和自旋向下。
伊辛模型的基本假设是自旋之间存在相互作用,并且系统的能量
由相邻自旋的相互作用决定。
我们考虑一维含有N个自旋的链,自旋可以在格点上取值为+1或-1。
系统的总能量可以用以下哈密顿量来描述:
H = -J * ∑(i=1到N) Si * Si+1 - h * ∑(i=1到N) Si
其中,Si表示第i个自旋的值,Si*Si+1表示自旋之间的相互作用,J是自旋间相互作用的耦合常数,h是外场的强度。
对于一维伊辛模型,我们可以使用解析的方式求解该系统的严格解。
首先,我们可以使用巴塞尔函数和傅里叶变换来方便地处理问题。
通过使用傅里叶变换,我们可以将自旋的链上的问题转化为动量空间
中的积分问题。
在解出哈密顿量的本征值和本征态后,我们可以计算系统的各种
性质,如自旋的关联函数、磁化强度和比热等。
这些性质可以用于研
究相变的行为,例如系统的临界温度和相变点。
需要注意的是,在一维伊辛模型中,由于不存在严格相变,因此
没有明确的临界温度。
但是我们可以通过计算性质的导数来观察到相
变的迹象。
总之,一维伊辛模型的严格解提供了对自旋链系统的深入理解,
可以帮助我们研究自旋系统的性质和行为。
A guide to use GeoGebra when teaching AS and A level Further Mathematics Below you can find links to GeoGebra files designed to help you teach the content of AS and Alevel Further Mathematics qualifications with the aid of GeoGebra.These can be used for teaching, or as students’ aided or independent learning materials with Pearson Textbooks.Core Pure MathematicsCore Pure AS/Year 1 Core Pure Year 2OptionsFurther Pure 1 Further Pure 2 Further Statistics 1 Further Statistics 2 Further Mechanics 1 Further Mechanics 2 Decision 1 Decision 2AS/Year 1 Further Mathematics – Core PureArgand diagrams (Chapter 2)Explore adding and subtracting complex numbers on an Argand diagram. (Page 19)•GeoGebra interactiveExplore multiplying and dividing complex numbers on an Argand diagram. (Page 25)•GeoGebra interactiveExplore the locus of z, when |z − z1| = r (Page 28)•GeoGebra interactiveExplore the locus of z, when |z − z2| = |z − z2| (Page 29)•GeoGebra interactiveExplore the locus of z, when arg (z − z1) = θ(Page 32)•GeoGebra interactiveExplore shaded region on an Argand diagram. (page 37)•GeoGebra interactiveVolumes of revolution (Chapter 5)•Page 76: Explore volumes of revolution around the x- and y-axes using GeoGebra.•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2AS/Year 1 Further Mathematics – Core Pure (Cont’d)Linear transformations (Chapter 7)Explore rotations of the unit square. (Page 134)•GeoGebra interactiveExplore enlargements and stretches of triangle T. (Page 137)•GeoGebra interactiveExplore rotations about the coordinate axes. (Page 145)•GeoGebra interactiveVectors (Chapter 9)Explore the vector equation of a line. (Page 168)•GeoGebra interactiveExplore the vector and Cartesian equations of a plane. (Page 176)•GeoGebra interactiveUse GeoGebra to consider the scalar product as the component of one vector in the directionof another. (Page 179)•GeoGebra interactiveExplore the angle between a line and a plane. (Page 186)•GeoGebra interactiveVisualise the angle between two planes. (Page 187)•GeoGebra interactiveExplore the perpendicular distance between two lines. (Page 194)•GeoGebra interactiveExplore reflections in a plane using GeoGebra. (Page 198)•GeoGebra interactiveYear 2/ A level Further Mathematics – Core PureComplex numbers (Chapter 1)Explore nth roots of complex numbers in an Argand diagram. (Page 25)•GeoGebra interactiveSeries (Chapter 2)Explore graphs of the successive Maclaurin polynomials. (Page 42)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2Year 2/ A level Further Mathematics – Core Pure (Cont’d)Methods in calculus (Chapter 3)Explore the integral. (Page 54)•GeoGebra interactiveVolumes of revolution (Chapter 4)Explore volumes of revolution around the x-axis. (Page 78)•GeoGebra interactiveExplore volumes of revolution around the y-axis. (Page 81)•GeoGebra interactivePolar coordinates (Chapter 5)Explore curves given in polar form using GeoGebra. (Page 106)•GeoGebra interactiveExplore the area enclosed by a loop of the polar curve with the form r = a sin θ. (Page 110)•GeoGebra interactiveHyperbolic functions (Chapter 6)Explore graphs of hyperbolic functions. (Page 122)•GeoGebra interactiveMethods in differential equations (Chapter 7)Explore families of solution curves. (Page 148)•GeoGebra interactiveModelling with differential equations (Chapter 8)Explore simple harmonic motion. (Page 175)•GeoGebra interactiveExplore damped harmonic motion. (Page 181)•GeoGebra interactiveExplore forced harmonic motion. (Page 183)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Pure 1Vectors (Chapter 1)Explore the cross product of two vectors. (Page 2)•GeoGebra interactiveExplore the area of a parallelogram using vector notation. (Page 8)•GeoGebra interactiveExplore the scalar triple product. (Page 11)•GeoGebra interactiveExplore the vector equation of a line, written using a cross product. (Page 16)•GeoGebra interactiveConic sections 1 (Chapter 2)Explore the focus-directrix properties of a parabola. (Page 35)•GeoGebra interactiveExplore the locus of M. (Page 56 Question 7)•GeoGebra interactiveConic sections 2 (Chapter 3)Explore conic sections. (Page 63)•GeoGebra interactiveExplore the foci and directrices of an ellipse. (Page 69)•GeoGebra interactiveExplore the foci and directrices of a hyperbola. (Page 72)•GeoGebra interactiveExplore the locus of the midpoint of AB. (Page 84)•GeoGebra interactiveInequalities (Chapter 4)Explore the solution to the inequality. (Page 97)•GeoGebra interactiveExplore the solution to the inequality. (Page 101)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Pure 1 (Cont’d)Taylor series (Chapter 6)Explore the Taylor series expansion of f(x) = tan x (Page 133)•GeoGebra interactiveMethods in calculus (Chapter 7)Explore the graph of the function t = e ln t (Page 154)•GeoGebra interactiveNumerical methods (Chapter 8)Explore tangent fields. (Page 162)•GeoGebra interactivePage 174: Explore the use of Simpson's rule to estimate the integral. (Page 174)•GeoGebra interactiveA level Further Mathematics – Further Pure 2Number theory (Chapter 1)Implement the Euclidean algorithm. (Page 6)• GeoGebra interactiveCarry out divisibility tests. (Page 17)•GeoGebra interactiveExplore permutations. (Page 31)•GeoGebra interactiveExplore combinations. (Page 33)•GeoGebra interactiveGroups (Chapter 2)Explore groups of symmetries. (Page 57)•GeoGebra interactiveExplore generators. (Page 69)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Pure 2 (Cont’d)Complex numbers (Chapter 3)Explore the locus of z when |z - a| = k|z - b| (Page 88)•GeoGebra interactiveExplore the locus of z when arg((z - a)/(z - b)) = θ (Page 91)•GeoGebra interactiveExplore this region. (Page 98)•GeoGebra interactiveExplore these transformations. (Page 101)•GeoGebra interactiveRecurrence relations Chapter 4)Play the Tower of Hanoi. (Page 130)•GeoGebra interactiveMatrix algebra (Chapter 5)Explore eigenvalues and eigenvectors. (Page 155)•GeoGebra interactiveIntegration techniques (Chapter 6)Explore the use of integration to find the arc length between two points. (Page 199) •GeoGebra interactiveExplore the use of integration to find the length of an arc on a curve with a polar equation.(Page 201)•GeoGebra interactiveExplore the use of integration to find the area of a surface of revolution. (Page 208) •GeoGebra interactiveA level Further Mathematics – Further Statistics 1Discrete random variables (Chapter 1)Explore probability distributions of a discrete random variable and compare the theoreticaldistribution with observed results generated from that discrete random variable. (Page 5)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Statistics 1 (Cont’d)Poisson distributions (Chapter 2)Explore the Poisson distribution. (Page 21)•GeoGebra interactiveGeometric and negative binomial distributions (Chapter 3)Explore the geometric distribution. (Page 44)•GeoGebra interactiveExplore the cumulative geometric distribution. (Page 45)•GeoGebra interactiveExplore the negative binominal distribution. (Page 50)•GeoGebra interactiveHypothesis testing (Chapter 4)Explore critical regions for a Poisson distribution. (Page 63)•GeoGebra interactiveExplore critical regions for a geometric distribution. (Page 70)•GeoGebra interactiveChi-squared tests (Chapter 6)Explore the chi-squared distribution to determine critical values for goodness of fit. (Page 98)•GeoGebra interactiveQuality of tests (Chapter 8)Explore probabilities of Type I and Type II errors in a normal distribution. (Page 154)•GeoGebra interactiveA level Further Mathematics – Further Statistics 2Linear regression (Chapter 1)Explore the calculation of a least squares regression line. (Page 3)•GeoGebra interactiveExplore residuals of data points and reasonableness of fit. (Page 11)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Statistics 2 (Cont’d)Correlation (Chapter 2)Explore linear correlation between two variables, measured by the PMCC. (Page 22)•GeoGebra interactiveExplore Spearman’s rank correlation coefficient. (Page 27)•GeoGebra interactiveContinuous distributions (Chapter 3)Explore probability density functions. (Page 46)•GeoGebra interactiveExplore cumulative distribution functions. (Page 51)•GeoGebra interactiveEstimation, confidence intervals and tests using a normal distribution (Chapter 5) Explore biased and unbiased estimators. (Page 114)•GeoGebra interactiveFurther hypothesis tests (Chapter 6)Explore the F-distribution using GeoGebra and use it to determine critical values of the samplevariances. (Page 151)•GeoGebra interactiveConfidence intervals and tests using the t-distribution (Chapter 7)Explore the t-distribution using GeoGebra and use it to determine critical values of the samplevariances. (Page 165)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Mechanics 1Momentum and impulse (Chapter 1)Explore particle collisions. (Page 4)•GeoGebra interactiveExplore collisions with two moving particles. (Page 5)•GeoGebra interactiveExplore particle collisions with known impulse. (Page 7)•GeoGebra interactiveExplore particle collisions in two dimensions. (Page 10)•GeoGebra interactiveElastic strings and springs (Chapter 3)Explore Hooke's law in equilibrium problems involving two elastic springs. (Page 41) •GeoGebra interactiveExplore Hooke's law in equilibrium problems involving one elastic spring. (Page 42) •GeoGebra interactiveExplore Hooke's law in dynamics problems. (Page 47)•GeoGebra interactiveElastic collisions in one dimension (Chapter 4)Explore direct impact. (Page 72)•GeoGebra interactiveExplore direct impact with a known impulse. (Page 73)•GeoGebra interactiveExplore the direct collision of a falling particle with a smooth plane. (Page 77)•GeoGebra interactiveExplore the loss of kinetic energy in a collision. (Page 80)•GeoGebra interactiveExplore successive collisions. (Page 85)•GeoGebra interactiveExplore successive impacts of a falling particle. (Page 88)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Mechanics 1 (Cont’d)Elastic collisions in two dimensions (Chapter 5)Explore oblique impact with a fixed surface. (Page 97)•GeoGebra interactiveExplore successive oblique impacts with a fixed surface. (Page 105)•GeoGebra interactiveExplore oblique impacts of smooth spheres. (Page 111)• GeoGebra interactiveA level Further Mathematics – Further Mechanics 2Circular motion (Chapter 1)Explore circular motion of a particle attached to a light inextensible string. (Page 6) •GeoGebra interactiveExplore circular motion in three dimensions. (Page 11)•GeoGebra interactiveExplore vertical circular motion. (Page 20)•GeoGebra interactiveExplore motion of a particle not constrained on a circular path. (Page 26)•GeoGebra interactiveCentres of mass of plane figures (Chapter 2)Explore the centre of mass of systems of particles. (Page 37)•GeoGebra interactiveExplore the centre of mass of particles arranged in a plane. (Page 39)•GeoGebra interactiveExplore centres of mass of standard uniform plane laminas. (Page 45)•GeoGebra interactiveExplore centres of mass of a framework. (Page 55)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Further Mechanics 2 (Cont’d)Further centres of mass (Chapter 3)Explore the centre of mass of a solid of revolution. (Page 90)• GeoGebra interactiveExplore toppling and sliding. (Page 110)•GeoGebra interactiveKinematics (Chapter 4)Explore terminal or limiting velocity. (Page 162)•GeoGebra interactiveDynamics (Chapter 5)Explore simple harmonic motion. (Page 184)•GeoGebra interactiveExplore calculations for simple harmonic motion with a reference circle. (Page 191) •GeoGebra interactiveExplore the simple harmonic motion of a vertical spring. (Page 201)•GeoGebra interactiveA level Further Mathematics – Decision 1Algorithms (Chapter 1)See the operation of the first-fit algorithm using GeoGebra. (Page 17)•GeoGebra interactiveSee the operation of the first-fit decreasing algorithm using GeoGebra. (Page 18)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Decision 1 (Cont’d)Algorithms on graphs (Chapter 3)Example 1: Explore Kruskal's algorithm using GeoGebra. (Page 53)•GeoGebra interactiveExample 2: Explore Kruskal's algorithm using GeoGebra. (Page 54)• GeoGebra interactiveExplore Prim's algorithm using GeoGebra. (Page 57)•GeoGebra interactiveExplore Dijkstra's algorithm using GeoGebra. (Page 66)•GeoGebra interactiveExplore Floyd's algorithm using GeoGebra. (Page 76)• GeoGebra interactiveLinear programming (Chapter 7)Explore graphical solutions to linear programming problems. (Page 147)•GeoGebra interactiveExplore how the optimal solution can be found using the objective line method. (Page 150) •GeoGebra interactiveExplore how the optimal solution can be found using the objective line method. (Page 153) • GeoGebra interactiveExplore how the optimal solution can be found using the objective line method. (Page 154) •GeoGebra interactiveExplore how the optimal solution can be found using the objective line method. (Page 155) •GeoGebra interactiveExplore how the optimal solution can be found using vertex testing. (Page 157)•GeoGebra interactiveExplore how the optimal solution can be found using vertex testing. (Page 158)•GeoGebra interactiveExplore how the optimal solution can be found using the objective line method. (Page 163)•GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Decision 1 (Cont’d)Critical path analysis (Chapter 8)Explore event times in activity networks. (Page 230)•GeoGebra interactiveExplore critical paths. (Page 233)•GeoGebra interactiveA level Further Mathematics – Decision 2Transportation problems (Chapter 1)Explore how the north-west corner method can be used to find an initial solution. (Page 4) •GeoGebra interactiveExplore how to calculate shadow costs. (Page 13)•GeoGebra interactiveExplore how to calculate improvement indices. (Page 16)•GeoGebra interactiveExplore how to obtain an improved solution with the stepping stone method. (Page 16) •GeoGebra interactiveFlows in networks 1 (Chapter 3)Explore cuts and their capacities in this network. (Page 81)•GeoGebra interactiveFind feasible flows on this network. (Page 89)•GeoGebra interactiveFlows in networks 2 (Chapter 4)Explore feasible flows through a directed network with lower and upper capacities. (Page 113) •GeoGebra interactiveExplore cuts in a directed network with upper and lower capacities. (Page 116)•GeoGebra interactiveGame theory (Chapter 6)Find stable solutions for zero-sum games. (Page 186)•GeoGebra interactiveExplore the optimal solution to two-player zero-sum game with no stable solution. (Page 199) •GeoGebra interactiveCP1 CP2 FP1 FP2 FS1 FS2 FM1 FM2 D1 D2A level Further Mathematics – Decision 2 (Cont’d)Recurrence relations (Chapter 7)Play the Tower of Hanoi. (Page 224)•GeoGebra interactiveDecision analysis (Chapter 8)Explore EMV. (Page 246)•GeoGebra interactive。
球状变异函数模型计算例子球状变异函数模型(Spherical VariogramModel)是地统计学中常用的空间插值方法之一,用于估计未知地点的属性值。
它的数学表达式如下:γ(h) = C + (A - C) * [1 - exp(-3h2/2r2)]其中,γ(h)表示变异函数值,h表示两个地点之间的距离,C表示基线值(或称为“nugget“),A表示变异的最大值,r表示变异函数的尺度参数。
为了进行具体的计算例子,我们以简化的情况为例,假设我们有一个地理区域,其中有几个采样点,并且我们想要估计一个未知位置处的属性值。
假设我们有以下采样点的属性值和坐标信息:采样点1:属性值 = 10,坐标 = (0, 0)采样点2:属性值 = 20,坐标 = (1, 0)采样点3:属性值 = 15,坐标 = (0, 1)现在我们想要估计一个未知位置 (1, 1) 处的属性值。
首先,我们需要计算各个采样点之间的距离,即h。
我们可以使用欧氏距离公式进行计算。
h1 = sqrt((1-0)2 + (1-0)2) = sqrt(2)h2 = sqrt((1-1)2 + (1-0)2) = 1h3 = sqrt((1-0)2 + (1-1)2) = 1接下来,我们需要确定模型的参数。
参数的选择可以通过经验或者拟合数据得到。
在这个例子中,我们假设参数如下:C = 5A = 25r = 1根据球状变异函数模型的公式,我们可以计算出未知位置处的属性值。
γ(h1) = 5 + (25 - 5) * [1 - exp(-3 * 2 / (2 * 12))] ≈ 23.13γ(h2) = 5 + (25 - 5) * [1 - exp(-3 * 1 / (2 * 12))] ≈ 20.25γ(h3) = 5 + (25 - 5) * [1 - exp(-3 * 1 / (2 * 12))] ≈ 20.25因此,在位置 (1, 1) 处的属性值的估计结果为约 20.25。
matelabe知识点总结Matlab基本概念Matlab是Matrix Laboratory的缩写,是一种用于数值计算和技术计算的软件工具。
Matlab的主要特点包括:1. 跨平台性:Matlab可以在Windows、Mac OS和Linux等操作系统上运行。
2. 高性能计算:Matlab通过多线程、并行计算和GPU计算等方式实现高性能计算,适用于大规模数据处理和复杂计算任务。
3. 丰富的函数库:Matlab拥有丰富的函数库,包括数学、信号处理、图像处理、统计分析等方面的函数,方便用户进行数值计算和数据处理。
4. 可视化功能:Matlab提供了丰富的数据可视化工具,包括绘图、图像处理、动画等功能,可以方便用户进行数据可视化和结果展示。
5. 仿真建模:Matlab可以用于建立仿真模型,包括控制系统、通信系统、电力系统等方面的仿真模型,用于系统设计和性能分析。
Matlab常用语法和函数Matlab语言是一种高级脚本语言,具有类似C语言的语法结构,并且具有丰富的内置函数库。
下面介绍Matlab中的一些常用语法和函数:1. 变量和数据类型:Matlab的变量可以是数字、字符串、矩阵等类型,支持整数、浮点数、复数等不同的数据类型。
2. 控制结构:Matlab支持if-else、while、for等常见的控制结构,用于实现条件判断和循环操作。
3. 函数定义:Matlab中可以定义自定义函数,使用function关键字定义函数,并且支持多个输入参数和输出参数。
4. 矩阵操作:Matlab是Matrix Laboratory的缩写,矩阵运算是Matlab的核心功能之一,支持矩阵的加减乘除、转置、逆矩阵、特征值等操作。
5. 统计分析:Matlab提供了丰富的统计分析函数,包括均值、方差、相关系数、回归分析等功能,用于数据分析和统计建模。
6. 信号处理:Matlab拥有丰富的信号处理函数库,包括傅里叶变换、滤波、时频分析等功能,适用于信号处理和通信系统建模。
一元四次方程与su(3)群一元四次方程与SU(3)群一元四次方程,在数学上是一个非常常见的方程形式,通常表示为ax^4 + bx^3 + cx^2 + dx + e = 0。
这个方程有四个未知数,a, b, c, d, e,并且至少有一个系数非零。
它的解通常通过一些数学技巧,如降次,因式分解,或者使用一些特定的公式来找到。
然而,当我们把这种一元四次方程与物理学中的SU(3)群联系起来时,它就具有了更为丰富的内涵。
SU(3)群,特殊酉群的一种,在量子力学和粒子物理学中有着广泛的应用。
它描述的是一种由三个生成元(或称为矩阵)构成的操作,这三个生成元通常被称为Gell-Mann矩阵。
在某种程度上,一元四次方程和SU(3)群之间可能看起来并没有明显的联系。
但事实上,它们之间存在一种深度的数学和物理联系。
例如,SU(3)群在某种特定的量子力学系统中可以被用来描述粒子的状态变化。
同样地,一元四次方程的解可以被用来描述在特定条件下系统的状态变化。
具体来说,SU(3)群可以被用来描述强相互作用下的粒子行为,如夸克和胶子。
这些粒子在强相互作用下会经历一些复杂的变换,而这些变换可以通过SU(3)群的数学操作来描述。
另一方面,一元四次方程的解可以描述一个系统在受到某些外部干扰后如何变化。
例如,一个简单的弹簧振荡器在一受到外力后就开始振动,这个振动的过程可以通过一元四次方程来描述。
因此,我们可以看到,一元四次方程和SU(3)群在本质上都在描述一种变化的过程。
它们之间的联系并不是表面上的,而是深层次的,反映了自然世界中某种深度的规律和现象。
通过将这两种理论结合在一起,我们可以更深入地理解自然世界中的一些复杂现象。