piecewise quadratic induced measure to approximatethe largest divergence rate of switched systems

摘要毒物兴奋效应是指在较低药物剂量水平时，随着剂量水平的增加，受试者对药物的不良反应降低．这种毒物兴奋效应现象暗示了门阀剂量的存在，如果存在毒物兴奋效应现象，必定发生在小于门阀剂量水平的剂量处．当剂量水平低于这个门阀剂量水平时，该剂量水平下的反应要比控制剂量下的反应即背景反应低，并且本文假定该门阀剂量水平下的反应与背景反应相同．因此我们这里用两个二次函数组成的分段函数构成了这种毒物兴奋效应现象的剂量一反应模型，其中较低剂量水平下的二次函数是Ｕ形曲线。

由于受试者对药物毒性的敏感程度的差异，本文用了混合正态模型来描述数据，在有限混合正态模型下用ＥＭ算法进行了参数估计，并用Ｂａｙｅｓ因子方法和Ｂｏｏｔｓｔｒａｐ置信区间方法进行了模型选择，其中Ｂｏｏｔｓｔｒａｐ置信区间方法选用的是简单实用的分位数置信区间方法．应用模型选择方法，对选用单分布还是混合分布问题，剂量一反应函数是否选用分段函数问题和如何选取分段函数的断点问题即门阀剂量值进行了研究，并对参数估计问题和三个模型选择问题进行了模拟，通过模拟结果可以看出混合模型要比单分布模型更适合对药物毒性的研究，先降后增的分段的剂量一反应函数要比传统的单一的单调增加的剂量一反应函数更能很好地描述这种毒物兴奋效应现象，选择恰当的门阀剂量值能使剂量一反应函数更适合数据．关键词：毒物兴奋效应；分段函数；模型选择；Ｂａｙｅｓ因子；Ｂｏｏｔｓｔｒａｐ置信区间ＥＭ算法；Ｓｃｈｗａｒｚ准则；ＢＩＣＡｂｓｔｒａｃｔＨｏｒｍｅｓｉｓｉｓｔｈａｔｄｅｃｒｅａｓｅｄｒｉｓｋｏｆａｄｖｅｒｓｅｅｆｆｅｃｔｓｗｏｕｌｄＯｃｃｕｒｗｉｔｈｉｎｅｒｅａｓｅｄｄｏｓｅｌｅｖｅｌｓｗｈｅｎｓｕｂｊｅｃｔｓａｒｅｅｘｐｏｓｅｄｔｏｔｏｘｉｎａｔｌｏｗｌｅｖｅｌｓ．Ｔｈｉｓｉｍｐｌｉｅｓｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｔｈｒｅｓｈｏｌｄｌｅｖｅｌ，ａｓｈｏｒｍｅｓｉｓ，ｉｆｅｘｉｓｔｓ，ｗｏｕｌｄｅｘｉｓｔｂｅｌｏｗｔｈｉｓｌｅｖｅｌ．Ｂｅｌｏｗｔｈｅｔｈｒｅｓｈｏｌｄｄｏｓｅｌｅｖｅｌ，ｎｏａｄｖｅｒｓｅｅｖｅｎｔｓａｂｏｖｅｔｈｅｒｅｓｐｏｎｓｅａｔｔｈｅｃｏｎｔｒｏｌｌｅｖｅｌｗｈｉｃｈｉｓｃａｌｌｅｄｔｈｅｂａｃｋｇｒｏｕｎｄｒｅｓｐｏｎｓｅｏｃｃｕｒ，ｔｈｅｎｂａｃｋｇｒｏｕｎｄｒｅｓｐｏｎｓｅａｌｓｏｏｃｃｕｒｓａｔｔｈｅｔｈｒｅｓｈｏｌｄｌｅｖｅｌ．Ｉｎｏｒｄｅｒｔｏｄｅｓｃｒｉｂｅｔｈｉｓｐｈｅｎｏｍｅｎｏｎ．ｗｅｃｈａｒａｃｔｅｒｉｚｅｔｈｅｏｖｅｒａｔｌｄｏｓｅ—ｒｅｓｐｏｎｓｅｒｅｌａｔｉｏｎｓｈｉｐｗｉｔｈａｐｉｅｃｅｗｉｓｅｆｕｎｃｔｉｏｎｔｈａｔｃｏｎｓｉｓｔｓｏｆａｑｕａｄｒａｔｉｃｃｕｒｖｅｗｈｉｃｈｉｓａｓｓｕｍｅｄｔｏｂｅＵ—ｓｈａｐｅｃｕｒｖｅａｎｄａｎｏｔｈｅｒｑｕａｄｒａｔｉｃｃｕｒｖｅａｔｈｉｇｈｄｏｓｅｌｅｖｅｌｓ．Ｈｅｒｅｗｅｕｓｅｆｉｎｉｔｅｍｉｘｔｕｒｅｎｏｒｍａｌｍｏｄｅｌｓｂｅｃａｕｓｅｔｈｅｅｘｐｅｒｉｍｅｎｔａｌｓｕｂｊｅｃｔｓｍａｙｈａｖｅｄｉｆｆｅｒｅｎｔｒｅｓｐｏｎｓｅｓｌｉｋｅ“ｍｏｒｅｓｕｓｃｅｐｔｉｂｌｅ．１ｅｓｓｓｕｓｃｅｐｔｉｂｌｅ”ａｆｔｅｒａｔｏｘｉｃａｎｔｅｘｐｅｒｉｍｅｎｔ．ｗｅｆｉｒｓｔｕｓｅｔｈｅＥＭａｌｇｏｒｉｔｈｍｔｏｏｂｔａｉｎｔｈｅｅｓｔｉｍａｔｅｓｏｆｔｈｅｐａｒａｍｅｔｅｒｓｉｎｔｈｅｍｉｘｔｕｒｅｎｏｒｍａｌｍｏｄｅｌｓ，ａｎｄｔｈｅｎｄｉｓｃｕｓｓｔｈｅｍｏｄｅｌｓｅｌｅｃｔｉｏｎｐｒｏｂｌｅｍｓｕｓｉｎｇＢａｙｅｓｆａｃｔｍ‘ｍｅｔｈｏｄａｎｄＢｏｏｔｓｔｒａｐｃｏｎｆｉｄｅｎｃｅｉｎｔｅｒｖａｌｍｅｔｈｏｄ，ｗｅａｄｏｐｔｔｈｅｒｏｕｇｈｂｕｔｓｉｍｐｌｅｐｅｒｃｅｎｔｉｌｅｍｅｔｈｏｄｆｏｒＢｏｏｔｓｔｒａｐｃｏｎｆｉｄｅｎｃｅｉｎｔｅｒｖａｌ．ｗｅｓｔｕｄｙｔｈｅｆｏｌｌｏｗｉｎｇｔｈｒｅｅｍｏｄｅｌｓｅｌｅｃｔｉｏｎｐｒｏｂｌｅｍｓ：ｆｉｒｓｔ，ｔｈｅｍｏｄｅｌｗｅｃｈｏｏｓｅｉｓｔｈｅｍｉｘｔｕｒｅｍｏｄｅｌｓｏｒｔｈｅｓｉｎｇｌｅ—ｃｏｍｐｏｎｅｎｔｍｏｄｅｌ；ｓｅｃｏｎｄ．ｔｈｅｏｖｅｒａｌｌｄｏｓｅ—ｒｅｓｐｏｎｓｅｆｕｎｃｔｉｏｎｗｅｃｈｏｏｓｅｉｓａｐｉｅｃｅｗｉｓｅｆｕｎｃｔｉｏｎｏｒｎｏｔ；１ａｓｔ．ｉｆｗｅｃｈｏｏｓｅｔｈｅｐｉｅｃｅｗｉｓｅｄｏｓｅ．ｒｅｓｐｏｎｓｅｆｕｎｃｔｉｏｎ；ｈｏｗｔｏｄｅｔｅｒｍｉｎｅｔｈｅｃｕｔｐｏｉｎｔｗｈｉｃｈｉｓａｌｓｏｃａｌｌｅｄｔｈｅｔｈｒｅｓｈｏｌｄｌｅｖｅｌ．Ｔｈｅｒｅｓｕｌｔｓｏｆｔｈｅｓｉｍｕｌａｔｉｏｎｓｗｉｌｌｉｎｄｉｃａｔｅｔｈａｔｔｈｅｍｉｘｔｕｒｅｍｏｄｅｌｓａｒｅｓｉｇｎｉｆｉｃａｎｔｌｙｂｅｔｔｅｒｔｈａｎｔｈｅｓｉｎｇｌｅ—ｃｏｍｐｏｎｅｎｔｐｏｐｕｌａｔｉｏｎｆｏｒｔｈｅｄｅｖｅｌｏｐｍｅｎｔａｌｔｏｘｉｃｉｔｙｓｔｕｄｉｅｓ，ｔｈｅｐｉｅｃｅｗｉｓｅｄｏｓｅ—ｒｅｓｐｏｎｓｅｆｕｎｃｔｉｏｎｉｓｓｉｇｎｉｆｉｃａｎｔｌｙｂｅｔｔｅｒｔｈａｎｔｈｅｃｕｒｒｅｎｔｄｏｓｅ—ｒｅｓｐｏｎｓｅｆｕｎｃｔｉｏｎｗｈｉｃｈｉｓｎａｉｖｅｌｙａｓｓｕｍｅｄｓｔｒｉｃｔｌｙｉｎｃｒｅａｓｉｎｇｏｖｅｒｔｈｅｒａｎｇｅｏｆｔｏｘｉｃｄｏｓｅｓｆｏｒｔｈｅｈｏｒｍｅｓｉｓ，ｔｈｅｓｕｉｔａｂｌｅｔｈｒｅｓｈｏｌｄｌｅｖｅｌｃａｎｉｍｐｒｏｖｅｔｈｅｄｏｓｅ—ｒｅｓｐｏｎｓｅｍｏｄｅｌ’ｓａｂｉｌｉｔｙｔｏｆｉｔｔｈｅｈｏｒｍｅｔｉｃｄａｔａ．Ｋｅｙｗｏｒｄｓ：ｈｏｒｍｅｓｉｓ；ｐｉｅｃｅｗｉｓｅｆｕｎｃｔｉｏｎ；ｍｏｄｅｌｓｅｌｅｃｔｉｏｎ；Ｂａｙｅｓｆａｃｔｏｒ；Ｂｏｏｔｓｔｒａｐｃｏｎｆｉｄｅｎｃｅｉｎｔｅｒｖａｌ；ＥＭａｌｇｏｒｉｔｈｍ；Ｓｃｈｗａｒｚｃｒｉｔｅｒｉｏｎ；ＢＩＣＩＩ独创性声明本人声明所呈交的学位论文是本人在导师指导下进行的研究工作及取得的研究成果。

隶属度grade of membership 隶属函数membership function 均方差mean square deviation 先验分布 a priori distribution先验估计 a priori estimate先验概率 a priori probability稳定性stability矩阵A的根radical of matrix A积分方程integral equation射影projection横坐标abscissa收敛convergence横轴abscissa axis连续的continuous凸闭集convex closed set单调的monotonic无穷infinity加性additive多项式polynomial算子operator函子functor微分方程differential equation微分式differential expression 转置伴随adjugacy肯定的affirmative代数范数algebraic norm任意平均数arbitrary average任意常数arbitrary constant任意递归arbitrary recurrence树形图arborescence算术距离arithmetic distance不等式inequality算术平均arithmetic mean几何平均geometric mean人工神经网络artificial neural network相伴图associated diagram度量空间metric space格lattice自协方差auto-covariance自相关autocorrelation自动机理论automata theory公理axiom公理的axiomatic坐标轴axis of coordinates纵轴axis of ordinates整数部分integer part不动点定理fixed point theorem Bayes序贯决策函数Bayessequenetial decision function基向量base vector基本权base weight重心barycenter最优调和优函数best harmonicmajorant双轴坐标biaxial coordinates有偏估计biased estimate双调和biharmonic二项分布binomial distribution边界集boundary set有界闭包bounded closure弱有界的bounded-weak大括号brace括号，方括号bracket分支树branch tree差分法calculus of differences微积分calculus of fluxion典型的，标准的canonical标准型canonical form标准同构canonical isomorphism基数cardinal number基数积cardinal product支集carried set不变量invariant引理lemma舍去casting out范畴，类型category圆心center of a circle点划线chain line重心center of gravity必然事件certain event等价的equivalent变换change基变换change of base信道channel特征超平面characteristichyperplane特征超曲面characteristichypersurface特征根characteristic root特征列characteristic series特征解characteristic solution固有值，特征值characteristic value最佳逼近特征characteristic of bestapproximation分层，谱系hierarchy2χ检验chi square criterion2χ分布chi square distribution中国邮递问题Chinese postmanproblem主列chief series中国剩余定理Chinese remaindertheorem论题thesis误差error组平均class mean经典概率论classical theory ofprobability类，组，级，班class闭开的clopen闭开集clopen set闭集closed collection闭包closure聚类分析cluster analysis粗糙的coarse子集subset偏相关系数coefficient of partialcorrelation回归系数coefficient of regression上包络coenvelope偏序集partially ordered set协调系统coherent system重合coincidence列矩阵column matrix列置换column permutation公共基common base紧凸集compact convex set均衡equilibrium相容的compatible余，补complementary完全，完备completeN维空间n-dimension space复合数composite number合成测度compound measure计算复杂性computation complexity不稳定性instability凹的concave条件熵conditional entropy条件概率密度conditionalprobability density条件完备的conditionally complete置信区间confidence interval置信下限confidence lower limit转置矩阵，伴随矩阵transposematrix相容条件consistency condition约束问题constrained problem上连续continuity from above 相反命题contrary proposition变量variable收敛证明convergence proof规划program卷积convolution坐标基向量coordinate basis vector 相关指数correlation indices陪集coset余弦cosine耦合couple耦合变量coupling variable耦合关系coupled relation协方差covariance临界点critical point临界值critical value临界集critical set三次的cubic割集cutset割平面cutting plane最小二乘法least square method疏散化decentralization疏散求和decentral summary解藕decoupling递减函数decreasing function定义公设defining postulate逆fuzzy defuzzify退化degenerateδ分布delta distribution线性linear行列式determinant确定性系统deterministic system展开development可展函数developable function差集difference set直接推导direct derivation正半锥direct half conoid直接方法direct manner直积direct product拓扑topology有向的directedDirichlet 边界条件Dirichlet boundary conditionDirichlet 级数Dirichlet series间断性，不连续性discontinuity 离散discrete直和direct sum不相交集disjoint subset不对称dissymmetry特异序列distinguished sequence扰动理论disturbed theory发散的divergent有效性域domain of validity对偶约束dual constraint对偶最优解dual optimal solution动态规划流程图dynamicprogramming flow chart有效性分析efficiency analysis有效估计efficiency estimate不变量invariant初等的elementary经验假定empirical assumption空集empty set端end终端条件end condition端结点end node端值end value可枚举的enumerableε循环epsilon-cycle相等约束equated constraint同等强度equally strong等价类equivalence class等价测度equivalent measures等值投影equivalent projection等价集equivalent set等变化的equivariant遍历ergode遍历的ergodic遍历类ergodic class估计值estimate value公式formula偶数even integer奇偶效验even-odd check精确解exact solution存在性证明existence proof互斥事件exclusive event显式公式explicit formula指数增长exponential growth广义基extended basis极值点extreme point失效率failure rate子集簇family of subsets故障树fault tree可行基feasible basis反馈feedback集合域field of sets有穷自动机finite automaton数学归纳法finite induction增量increment前向差分forward difference函数族function family多元函数function of severalvariables函数级数function series基本类fundamental classFuzzy聚类fuzzy clustering贴近度nearness松弛relaxation增益gain恒等式identity一般集合论general set theory广义解generalized solution全局约束global constraint全局可靠性global reliability图论graph theory群论group theory分组抽样grouping sampling调和分析harmonic analysis半连续的hemi-continuous插值interpolation双线性泛函bilinear functional启发式搜索heuristic search谱系分类hierarchical subdivision高阶差分higher difference立方cube齐次的homogeneous超平面hyperplane第i分量i-th component恒等的identical恒等identically equal病态方程ill-conditioned equation不可比的incomparable不相容的，非一致的inconsistent不确定性indeterminacy不确定的indeterminate归纳induction归纳推理inductioninference/reasoning归纳学习inductive learning无穷大infinity不稳定性instability整数部分integral part逆函数inverse function可逆函数invertible function可逆性invertibility不可约的irreducible等熵的isentropic孤立边界点isolated boundary point等距的isometric迭代iteration迭代的iterative迭代奇异值分解iterative singular value decomposition判断judgement滤波filtering扩张extension必要条件Lagrange necessary conditionlambda特征根latent root等价律law of equivalence最不利分布least favorable distribution最小二乘曲线拟合least square curve fitting左陪集left coset左极限left limit左递归规则left recursive rule小于或等于less than or equal to 似然函数likelihood function线性假设line hypothesis线性演绎linear deduction局部分析local analysis半群semigroup对数logarithm对数的logarithmic主码majority code多值函数many-valued function 多值性many-valuedness匹配matching矩阵求逆matrix inversion矩阵变换matrix transformation 矩阵转置matrix transposition 极大极小原则maximin criterion 极大极大maximax极小极大Minimax均方误差mean square error可测的measurable 元知识metaknowledge等效法method of equal effects归纳法method of induction极小极大定理min-max theorem最小生成树minimal spanning tree冗余度极小化minimization ofredundancy混合问题mixed problem模函数modular function模，模数modulo多维分布multidimensional allocation超几何的hypergeometric正态分布normal distribution多值，重值multiple value重数multiplicity多阶段决策multistage decision多元判别分析multivariatediscriminant analysis互斥问题mutual exclusion problem互斥(不相容)集mutually exclusivesets互不相交子集mutually disjointsubsets互素的mutually prime互相分离集mutually separated sets互相奇异的mutually singularN元n-aryN元关系n-ary relationN连通图n-connected graphN维随机变量n-dimensionalrandom variableN倍，N重n-foldN元组n-tupleN次幂n-th power负二项分布negative binomial否命题negative proposition嵌套递归式nested recursion非等距的non-isometric非交叉集non-overlapping set非自反关系non-reflexive relation非对称的non-symmetrical非稳定的non-stable非对称关系non-symmetric relation非平凡的non-trivial非空的non-vacuous非零向量non-vanishing vector非经典的nonclassical非相容nonconforming非紧的noncompact非合作noncooperative非退化nondegenerateNondegenerative不确定的nondeterministic非空集nonempty set不可行的nonfeasible不完整约束nonholonomicconstraint非增nonincreasing非劣解noninferior solution非迭代的noniterative非极限的nonlimit非线性的nonlinear正规化方程normalized equation数字的numerical一对多的one-to-many单值函数one-valued function单向函数one-way function开集opener逆范畴，对偶范畴oppositecategory异号opposite sign最优的optimal停止stopping有序和ordered sum序数，有序的ordinal定向orientation正交基orthogonal basic两两的pairwise两两独立pairwise independence平行parallel部分展开partial expansion偏序partial order峰值peak value罚函数penalty function完全相关perfect correlation完全对偶性perfect duality性能评价performance evaluation允许值permissible value扰动问题perturbation problem分段连续piecewise continuity逐点遍历定理pointwise ergodictheorem势，基数potency优先图precedence diagram偏爱序preference ordering素数prime先验分布prior distribution范围range线性linearity伪随机的pseudo-random纯策略pure strategy二次对偶quadratic dual四重的，四倍的quadruple，quadrupling，quadruplex径向基函数radial basis function径向偏差radial deviation径向开集radially open set可归约的reducible正则基数regular cardinal多面体polyhedron可靠性计算reliability calculation 可去集removable set累极限repeated limit剩余类residue class可解的resoluble回溯集retraceable set可逆的reversible直角right angle严格上界rigorous upper bound 鲁棒性，稳健性robustness均方根root-mean-square粗糙集rough set粗逻辑rough logic舍入round-off舍入的rounded程序routine行坐标row-coordinates行向量row vector消元规则rule of elimination鞍点saddle point同类same kind样本中心矩sample central moment 样本方差sample variance散布的scattered单叶函数schlicht function二次多项式second degree of polynomial二次导数算法second derivative algorithm二阶必要条件second order necessary condition自相容性self-consistency自相关性self-correlation自对偶self-dual 半同余semi-congruence半等价的semi-equivalent偏序集semi-ordered set半完全的semi-perfect可分的separable七倍的，七维的septuple七的septimal时序sequence in time子集序列sequence of subsets实数集set of real numbers集值函数set-valued function收缩算法shrinking algorithmσ域sigma-field单纯形法simplex method单值函数single-valued function稀疏索引sparse index谱估计spectral estimate样条拟合spline fitting稳态误差steady state error步长，阶段step步长step size逐步法stepwise随机分布stochastic distribution严格不等式strict inequality严格单调递增strictly monotoneincreasing严格正定算子strictly positivedefinite operator强对偶strong duality较强一致性stronger uniformity连通有向图connected directedgraph次加性的subadditive次调和解subharmonic solution下标subscript下标变量subscript variable辅助条件subsidiary condition充分集sufficient set上确界sup上标superscript支撑集support set表面散度surface divergence前向搜索sweep forward后向搜索sweep backward常微分方程组system of ordinarydifferential equation剩余系system of residues逐项term by term三元的，三进制的ternary假设检验test of hypothesis正态性检验test of normality显著性检验test of significance最后决定terminal decision择一定理theorem of alternative互补松弛定理theorem ofcomplementary slackness赋值论theory of valuations消除论theory of elimination数论theory of numbers置换理论theory of permutation排队论theory of queue集合论theory of sets象限quadrant自顶向下top-down全部单调的totally monotonic变换范围transformation range瞬态响应transient response超限评估transfinite evaluation传递闭包transitive closure转移概率矩阵transition probabilitymatrix转置，对换transposition梯形的trapezoidal梯形公式trapezoidal formula遍历traverse类树的tree-like树图tree graph试错搜索trial-and-error search三角数triangular number平凡序trivial ordered双重的two-foldTwo-person nash game超因子ultra-divisor超指数ultra-exponential超无穷点ultra-infinite point超弱的ultra-weak无偏置置信区间unbiasedconfidence interval无偏估计unbiased estimate无偏估计量unbiased estimator无偏α水平检验unbiased levelalpha test不确定推理uncertain reasoning无条件的unconditional不相关的uncorrelated不等方差unequal variance非本质的unessential一致性uniformity酋矩阵unitary matrix普通范畴universal category泛算子universal operator无序的unordered全域，总体universe无下标变量unsubscripted variable 非零解，非平凡解untrivial solution 上界upper limit变分类variational class向量不变式vector invariant通用的，多用途的versatile/versal 锥顶vertex of a cone顶点图vertex graph抛物线的顶点vertex of a parabola 垂直射影vertical projection小波wavlet弱闭包weak closure弱完备性weak completion弱耦合weak coupling弱序化weak ordering弱耦合系统weak coupled system 价格加权综合weighted aggregate of price良序子集well-ordered subset整数whole numberzeta零和对策zero sum game零序列zero-sequence零熵zero entropy笛卡尔积Cartesian product折扣tradeoff交集intersection推论corollary引理lemma证明proof公式formula定理theorem。

随机性优化算法有效性定量对比评价方法李鹏飞;石正军【摘要】To compare effectiveness of different stochastic optimization algorithms quantitatively, a method is proposed, which is based on statistical analysis of multiple independent results of those algorithms solving a set of typical test samples. Probability quantifying the relative superiority of effectiveness of two algorithms is conducted. Using this method, effec-tiveness comparison of different stochastic optimization algorithms is easy to make. Further, this method is implemented on two patterns of standard Particle Swarm Optimization(PSO)algorithm, in which the global best particle’s information is updated synchronously as well as asynchronously. As a result, a relatively effective pattern of standard PSO algorithm solving unconstrained single-objective optimization problems is advised.%针对随机性优化算法寻优结果不可重复的特点，为该类优化算法提供了一种定量对比评价算法有效性的方法。

数学专业词汇英汉对照汇编数学英汉词汇AAbelian group 阿贝尔群abscissa axis 横轴absolute continuity 绝对连续absolute convergence 绝对收敛absolute value 绝对值abstract algebra 抽象代数addition 加法affine 仿射Aleph-zero 阿列夫零algebra topology 代数拓扑algorithm 算法almost everywhere ⼏乎处处almost surely ⼏乎必然alternative 互斥性analogy 类似analytic expression 解析式anomalous 反常的apex 顶点approximate calculation 近似计算associative law 结合律asymmetric line 渐近线axiom of choice 选择公理axis of abscissas 横坐标轴axis of imaginary 虚轴axis of real 实轴Bbase number 底数base of logarithmic function 对数函数的底base vectors 基向量basic element 基元素bijection 双射bilinear 双线性binary ⼆元的binomial ⼆项式biunivocal ⼀对⼀的bondage 约束boundary compact space 有界紧空间boundary condition 边界条件bounded continuous function有界连续函数bounded interval 有界区间brace ⼤括号Ccalculus of proposition 命题演算canonical form 标准型cardinal number of set 集的基数Cartesian product 笛卡尔积catalog ⽬录category of a space 空间的筹数causality 因果律center of compression 压缩中⼼central limit theorem 中⼼极限定理certain event 必然事件characteristic equation 特征⽅程characteristic value 特征值chart 图check procedure 检验步骤circumscribed 外切的class field 类域closure axioms 闭包公理cluster point 聚点丛点coefficient of autocorrelation ⾃相关系数coefficient of regression 回归系数cofactor of a determinant ⾏列式的余⼦式cohomology 上同调collinear 共线column matrix 列矩阵column rank 列秩common factor 公因⼦commutative law 交换律commutative law of addition 加法交换律commutative law of multiplication乘法交换律compact convex set 紧凸集complement of a set 集的余集complement law 补余律complete matrix space 完备度量空间complete orthogonal system 完全正交系complex analysis 复分析complex conjugate 复共轭complex field 复数域compound function 复合函数concave 凹的conclusion 结论condition 条件conditional of inequality 条件不等式conditional of equivalence 等价条件conditional of integrability 可积条件conditional convergence 条件收敛confidence interval 置信区间conjugate 共轭connectivity 连通性consistency principle ⼀致原则constant factor 常数因⼦construction 作图构造continuous function 连续函数contradiction ⽭盾contrary propositions 相反命题convergence almost everywhere⼏乎处处收敛convergence in measure 依测度收敛convergence rate 收敛速度convergence region 收敛区域converse proposition 逆命题convex closure 凸包coplanar 共⾯cosine law 余弦定律countable additivity 可列可加性critical point 临界点Ddecision theory 决策论degenerate quadratic form 退化⼆次型dense 稠密derivate 导数differential 微分dimensionality 维数discriminant 判别式disjoint 不相交的distributive law 分配率divergent infinite series 发散⽆穷级数double integral ⼆重积分dual operations 对偶运算Eefficiency estimation 有效估计eigenelement 本征元素elementary event 基本事件endomorphism ⾃同态envelope 包络equivalence class 等价类equivalent relation 等价关系even number 偶数existence and uniqueness 存在且唯⼀性expansion in series 级数展开exponent 指数extreme point 极值点Ffeasible solution 可⾏解finite additivity 有限可加性fraction 分数frequency 频率fundamental assumption 基本假定fuzzy 模糊game theory 对策论general remark ⼀般说明generalized derivatives ⼴义导数geometric significance ⼏何意义global convergence 全局收敛Hharmonic analysis 调和分析harmonics 调和函数homology 同调homotopy 同伦homomorphism 同态hyperbolic plane 双曲平⾯hypothesis 假设Iideal 理想identical element 单位元identity law 同⼀律illustrate 说明阐释implicit function 隐函数in like manner 同理in the large 全局的in the small 局部的inclusion of sets 集的包含关系incompatible 互斥的不相容的independence test 独⽴性检验independent variable ⾃变量induction 归纳法归纳infinite ⽆穷⼤integral calculus 积分学integral divisor 整因⼦integrate 积分interior mapping 开映射inverse of matrix 矩阵的逆irrational root ⽆理根irreducible fraction 不可约分数irreducible polynomial 不可约多项式isolated point 孤⽴点isometric 等距的iteration method 迭代法Jjoint 连接jump discontinuity 跳跃不连续性Kkernel 核knee 拐点known quantity 已知量large by comparison 远⼤于latent vector 特征向量law of association 结合律law of causality因果律law of commutation 交换律law of contradiction ⽭盾律law of distribution 分配律law of mean 中值定理leader ⾸项limited function 有界函数linear dependence 线性相关logarithm 对数lower bound 下界Mmapping space 映射空间marginal value 临界值measure 测度metric space 度量空间monotone 单调multinomial 多项式multiplication 乘法mutual correlation 互相关mutually conjugate 相互共轭的mutually disjoint 互不相交的mutually inverse 互逆的mutually prime 互素的Nnatural logarithm ⾃然对数negate 取否定negative index 负指数negate proposition 否定命题nest of intervals 区间套neutral element 零元nonhomogeneous differential equation⾮齐次微分⽅程nonlinear boundary value problem⾮线性边值问题non-vanishing vector ⾮零向量normal space 正规空间normalized form标准型n-th power n次幂numerical analysis 数值分析Oobjective function ⽬标函数oblique line 斜线odd number 奇数odd symmetry 奇对称odevity 奇偶性one degree of freedom ⼀个⾃由度onto mapping ⾃⾝映射open covering 开覆盖opposite sign 异号optimal solution 最优解optimize 最优化order of infinity ⽆穷⼤的解ordered pair 有序偶ordinary differential equation常微分⽅程ordinary solution 通常解ordinate 纵坐标oriented circle 有向圆oriented segment 有向线段orthogonal 正交orthonormal basis 标准正交基outer measure 外测度overfield 扩张域overview 概述Ppairwise orthogonal 两两正交parabolic asymptotes 渐进抛物线parabolic curve 抛物曲线parallel 平⾏线parametric equation 参数⽅程parity 奇偶性partial sum 部分和passive 被动的path curve 轨线periodicity 周期性permutation 排列perpendicular line 垂直线piecewise 分段的plus 加point of intersection 交点population 总体的positive definite quadratic form 正定⼆次型positive number 正数potential 位势power formula 乘⽅公式prime ideal 素理想prime ring 素环primitive equation 本原⽅程primitive term 原始项principal factor method 主因⼦法principal minor 主⼦式principal of the point of accumulation聚点定理prior estimate 先验估计probability curve 概率曲线probability distribution 概率分布process of iteration 迭代法proper polynomial 特征多项式proper subset 真⼦集pure imaginary 纯虚数QQED(quod erat demonstrandum) 证毕quadrant 象限quadratic ⼆次quadratic equation with one unknown⼀元⼆次⽅程quadratic root 平⽅根qualitative analysis 定性分析quantitative analysis 定量分析quarter 四分之⼀queue discipline 排队规则quotation 引⽤引证quotient group 商群Rradial deviation 径向偏差radian 弧度radical sign 根号radius of a circle 圆的半径radius of convergence 收敛半径radius of curvature 曲率半径radix point ⼩数点random distribution 随机分布random sampling distribution 随机抽样分布randomness test 随机性检验rang of distribution 分布域rang of points 点列rank of linear mapping 线性映射的秩rank of quadratic form ⼆次型的秩rarefaction 稀疏rational proper fraction 有理真分式rationalizing denominators 有理化分母raw data 原始数据real analytic function 实解析函数real axis 实轴real variable function 实变函数reasoning by analogy 类⽐推理相似推理reciprocal 倒数reciprocal automorphism 反⾃同构reciprocal ratio 反⽐rectangular coordinates 直⾓坐标recurrence formula 递推公式recurrence relations 关系recursive function 递归函数reduction of a fraction 约分reduction to a common denominator 通分reduction to absurdity 反证法reference system 参考系reflection 反射region convergence 收敛区域regression analysis 回归分析regular function 正则函数rejection region 拒绝域relation of equivalence 等价关系relative error 相对误差relative minimum 相对极⼩值repeated integral 累积分residue class 剩余类resolution 分解reverse theorem 逆定理reversible transformation 可逆变换rigorous upper bound 严格的上界rotation axis 旋转轴roundoff error 舍⼊误差Ssample average 样本均值satisfy 满⾜scalar multiplication 数乘secondary ⼆次的辅助的次级的section 截⾯截线截点sectionally smooth 分段光滑self-conjugate subgroup 正规⼦群self-evident 显然不证⾃明semi-closure 半闭sensitivity 灵敏度separability 可分性sequence 序列series 级数series expansion 级数展开series of positive terms 正项级数shaded region 阴影区域significance level 显著性⽔平significant digits 有效数字similarity isomorphic 相似同构的simply connected region 单连通区域simulated data 模拟数据simultaneous inequalities 联⽴不等式sine curve 正弦曲线singular element 奇元素退化元素skew matrices 斜对称矩阵solid figure ⽴体形spanning set ⽣成集spherical neighborhood 球形领域stability condition 稳定性条件standard deviation 标准差stationary curve 平稳曲线statistical dependence 统计相关stochastic allocation 随机分配subadditivity 次可加性subbasis ⼦基subsequence ⼦列subtotalling 求部分和successive approximation 逐次逼近法sufficient and necessary condition 充要条件supplementary set 补集surface 曲⾯surplus variable 剩余变量symbolic function 符号函数symmetric center 对称中⼼symmetry transformation 对称变换synchronism 同步synthetic proof 综合证明Ttable of random numbers 随机数表tangent line 切线tends to infinity 趋于⽆穷term by term differentiation 逐项微分terminal check 最后校验termwise integration 逐项积分totally bounded 完全有界的transcendental equation 超越⽅程transposition 转置transverse surface 横截⾯triangle computations 三⾓形解法trisection 三等分Uultimate 最后的最终的极限unbiased estimation ⽆偏估计unconditional stability ⽆条件稳定uniform boundness ⼀致有界unilateral limits 单侧极限unique solution 唯⼀解universal proposition 全称命题unordered ⽆序的unreduced 不可约的untrivial solution ⾮零解upper integral 上积分Vvalid 有效真确valuation 赋值variance ⽅差偏差vector of unit length 单位向量velocity-time graph 速度-时间图verify 检验校验versal 通⽤的vertex 极点顶vertical 垂直的vibration 震动visual proof 图像证明直观证明volume 体积Wwave form 波形weak boundary condition 弱边界条件weighted arithmetic mean 加权算术平均whole number 整数Xx-axis x轴x-component x分量x-coordinate x坐标x-direction x⽅向Yyield estimation 合格率估计Zzero correlation 零相关zero divisor 零因⼦zone of preference for acceptance 合格域zoom up 放⼤。

AxiomaPortfolio Optimizer The most flexible portfolio-construction tool on the marketAxioma Portfolio Optimizer supports a wide range of investment management approaches, from quantitative to fundamental. With virtually limitless objectives and an equally unlimited range of constraints, Axioma Portfolio Optimizer delivers maximum flexibility to model even the most complex strategies.Plus, Axioma Portfolio Optimizer incorporates a key unique feature: our patented Axioma Alpha Alignment methodology prevents the risk underestimation common to optimizers,while effectively managing uncertainty with Axioma’s Robust Optimization technology. And with its open platform, clientshave the freedom to choose any combination of Axioma Robust Risk Models and/or third-party and internal risk models for their portfolio construction process.Flexible is better…For index-tracking portfolios…• Take advantage of flexible options for risk control.Minimize the tracking error of your portfolio in theobjective or place a hard limit on tracking error using a riskconstraint.Use more than one risk model to incorporate severalperspectives on risk.Include risk elements that use more than one benchmarkor model portfolio.• Incorporate all the transaction-related costs that impactperformance. The transaction cost types available in theoptimizer can be used in any combination to accuratelyreflect overall costs.Capture market impact using non-linear market-impactmodels (quadratic, 3/2, and 5/3 powers) or a piecewise-linear approximation.Use the fully integrated Goldman Sachs Shortfall Model.Include commissions and brokerage fees using linearcosts.• Place explicit limits on trading activity.Limit overall portfolio turnover or limit turnover of a set ofassets.Place asset-specific limits on trading; for example, limittrade size to a fraction of average daily volume.• Control portfolio beta using Axioma-provided historicalbetas, predicted betas computed relative to any benchmark you choose, or betas you supply.• Limit the number of names held or traded.For actively managed portfolios…In addition to the risk and transaction cost controls available for passive portfolio management, the modeling library contains many options designed to get the most from your alpha signal.• Augment your risk control using the Alpha Factor. The Alpha Factor compensates for components of your alpha signalthat are not included in the risk model, providing moreaccurate predicted risk estimates.• Limit risk contributions at the factor or asset level.• Use Robust Optimization to explicitly incorporate uncertainty in your alpha estimates.• Incorporate additional tilts in your objective. Tilts can beimplemented on any factor, including risk factors, factorsfrom one of Axioma’s factor libraries, or user-suppliedfactors.• Prevent small positions and small trades with thresholdconstraints.For long/short portfolios…Manage your long/short portfolio, create a hedging basket, or create an overlay using Axioma’s extensive set of options for long/short optimization.• Control the long, short, and net exposure to any factor,including user-defined factors.• Fix the portfolio leverage or let the optimizer select the best leverage value within a range you specify.• Limit the ratio of long to short holdings in the total portfolio or over any subset of assets.• Capture trading costs unique to long/short portfolios.Include asset-specific borrowing costs to capturedifferences in the short rebate resulting from holding ashort position.Include asset-specific short sell costs to capture costsassociated with taking a short position.• Limit short selling at the asset, group, or portfolio level.• Prevent short positions in “hard to borrow” assets.• Limit the number of assets held on the long and short sideseparately.For global portfolios…Global portfolios present many challenges for optimizers. Asset universes are larger, there are often multiple assets in the universe from a given issuer, global risk models contain many more factors, and risk and exposure need to be managed over more dimensions.• Incorporate risk from the currency perspective of yourchoice.• Control exposures to countries and currencies, in addition to sectors, industries, and styles.• Accurately incorporate active specific risk for portfolios that hold multiple lines from the same issuer using Axioma’sIssuer Specific Covariance.• Track a benchmark holding foreign ordinaries by holdingdepository receipts.• Control holdings by issuer.Include “issuer bet” constraints that aggregate assetholdings by issuer.• Control trading by issuer:Limit the total buys and sells by issuer.Prevent simultaneous buying and selling of assets from thesame issuer.• Capture ticket charges for buys, sells or all transactions.Ticket charges can vary by country, exchange or on anasset-by-asset basis.• Set round lot values by country, exchange or individual asset.The Superior Features of Axioma Portfolio Deliver Superior Performance Modeling flexibility provided by powerful optimization algorithms…Axioma Portfolio uses Second Order Cone Programming (SOCP), a state-of-the-art approach capable of solving complex optimization problems exactly and efficiently. With Axioma Portfolio, you can move beyond simple mean-variance optimization scenarios to construct models that accurately capture all the complexities of your investment process.An open platform…The Axioma Portfolio platform is completely open. Use content from Axioma, a third party or your own research team to power your construction process.Incorporate ETFs, futures and other composite instruments…• Utilize integrated content from Axioma or create your own custom asset compositions.• Take advantage of full look-through to accurately control portfolio risks.• Control total exposure at the asset or factor level resulting from holding a combination of individual stocks and ETFs or futures.Diagnose infeasibilities quickly and easily with the Constraint Hierarchy…Axioma Portfolio’s Constraint Hierarchy provides a comprehensive approach to dealing with infeasibilities. Simply provide a priority for each constraint and when a conflict between constraints occurs, the Optimizer will provide a solution that is as close as possible to satisfying all the constraints. If a constraint is violated, the constraint with the lowest possible priority is always chosen.Use Constraint Attribution to understand the costs of your constraints…Axioma’s Constraint Attribution provides a dashboard view of the impact of individual constraints, objectives or groups of constraints on the objective function value (usually portfolio return) in your strategy. This tells you which elements of your strategy have an overall impact and which do not. It also provides a relative measure of the potential for improving the objective by relaxing different constraints.Explore strategy trade-offs with the Frontier…Frontier optimization explores the impact of varying either a constraint limit or an objective term weight. Used in conjunction with Constraint Attribution, it enables the user to validate the impact of relaxing constraints in the strategy. Objective frontiers can be used to create a classical mean variance frontier or to explore trade-offs, such as return vs. transaction costs or risk vs. tax liability. The Frontier automates the execution of a series of rebalancings covering a range of strategy settings. It provides a consolidated view of the results for each rebalancing enabling the user to select the optimum trade-off.Evaluate small changes to the optimized portfolio with the Buy/Sell Editor…The Buy/Sell Editor enables the user to evaluate the impact of changes to the portfolio holdings or optimized trade list. These changes can be created manually or by an automated process external to the Optimizer. The Buy/Sell Editor provides an easy-to-use interface for specifying these changes and evaluating their impact on the characteristics of the resulting portfolio. All portfolio analytics are accessible for portfolios created using the Buy/Sell Editor.Flexible backtesting…Add the Axioma Backtester module and utilize all the model building flexibility in the Optimizer for time series simulations. Like the Optimizer, the Backtester is an open platform enabling you to use content from any source you choose in your backtest analysis. Automate your portfolio construction processes…The execution of large batches of portfolio rebalancings can be automated and scheduled to take place at your convenience. Portfolios can share a common strategy, or each portfolio can have a unique set of optimization settings.Further customization and automation can be accomplished using one of Axioma Portfolio’s API interfaces. In addition to the GUI, there are APIs in C++, Java, MATLAB, and R. There is also an XML-based interface that enables easy integration with existing systems via a command-line interface.Minimum system requirements…Axioma Portfolio runs on Windows 200x, XP, or Windows 7. A Pentium 4 or better processor is recommended. For best performance, 4GB or more of RAM is recommended.For more information about Axioma*****************************or call us.United States and Canada: +1 212-991-4500Europe: +44 (0)20 7856 2424Asia: +852-8203-2790Axioma, Inc. is the leading provider of innovative risk-managementand portfolio-construction solutions to financial institutions worldwide.With an emphasis on innovation, outstanding service and open-platform technology, Axioma gives its clients more choice, greaterflexibility and increased performance. Founded in 1998, Axioma isheadquartered in New York, with offices in Atlanta, Geneva, HongKong, London, San Francisco and Singapore.© Copyright, Axioma, Inc. 2014 - All rights reserved。

美国大学生数学建模MCM 数学专用名词augmented matrix增广矩阵asymptotic渐进的asymptote渐进线asymmetrical非对称的associative law结合律ascending上升的arrangement排列arithmetic算术argument幅角，幅度，自变量，论证area面积arc length弧长apothem边心距apex顶点aperiodic非周期的antisymmetric反对称的antiderivative原函数anticlockwise逆时针的annihilator零化子angular velocity角速度angle of rotation旋转角angle of incidence入射角angle of elevation仰角angle of depression俯角angle of circumference圆周角analytic space复空间analytic geometry解析几何analytic function解析函数analytic extension解析开拓amplitude幅角，振幅alternative互斥的alternate series交错级数almost everywhere几乎处处algebraic topology代数拓扑algebraic expression代数式algebraic代数的affine仿射(几何学)的admissible error容许误差admissible容许的adjugate伴随转置的adjoint operator伴随算子adjoint伴随的adjacency邻接additive加法，加性acute angle锐角accumulation point聚点accidential error偶然误差accessible point可达点abstract space抽象空间abstract algebra抽象代数absolute value绝对值absolute integrable绝对可积absolute convergent绝对收敛Abelian阿贝尔的，交换的balance equation平衡方程bandwidth带宽barycenter重心base基base vectors基向量biased error有偏误差biased statistic有偏统计量bilinear双线性的bijective双射的bilateral shift双侧位移的binomial二项式bisector二等分线，平分线boundary边界的，边界bounded有界的broken line折线bundle丛，把，卷calculus微积分calculus of variations变分法cancellation消去canonical典型的，标准的canonical form标准型cap交，求交运算capacity容量cardinal number基数Cartesian coordinates笛卡尔坐标category范畴，类型cell单元，方格，胞腔cell complex胞腔复形character特征标characterization特征circuit环路，线路，回路circular ring圆环circulating decimal循环小数clockwise顺时针方向的closed ball闭球closure闭包cluster point聚点coefficient系数cofinal共尾的cohomology上同调coincidence重合，叠和collinear共线的collective集体的columnar rank列秩combinatorial theory组合理论common tangent公切线commutative交换的compact紧的compact operator紧算子compatibility相容性compatible events相容事件complementary余的，补的complete完全的，完备的complex analysis复变函数论complex potential复位势composite复合的concave function凹函数concentric circles同心圆concurrent共点conditional number条件数confidence interval置信区间conformal共形的conic圆锥的conjugate共轭的connected连通的connected domain连通域consistence相容，一致constrained约束的continuable可延拓的continuity连续性contour周线，回路，轮廓线convergence收敛性convexity凸形convolution对和，卷积coordinate坐标coprime互质的，互素的correspondence对应coset陪集countable可数的counterexample反例covariance协方差covariant共变的covering覆盖critical临界的cubic root立方根cup并，求并运算curl旋度curvature曲率curve曲线cyclic循环的decade十进制的decagon十边形decimal小数的，十进制的decision theory决策论decomposable可分解的decreasing递减的decrement减量deduction推论，归纳法defect亏量，缺陷deficiency亏格definition定义definite integral定积分deflation压缩deflection挠度，挠率，变位degenerate退化的deleted neighborhood去心邻域denominator分母density稠密性，密度density function密度函数denumerable可数的departure偏差，偏离dependent相关的dependent variable因变量derangement重排derivation求导derivative导数descent下降determinant行列式diagram图，图表diameter直径diamond菱形dichotomy二分法diffeomorphism微分同胚differentiable可微的differential微分differential geometry微分几何difference差，差分digit数字dimension维数directed graph有向图directed set有向集direct prodect直积direct sum直和direction angle方向角directional derivative方向导数disc圆盘disconnected不连通的discontinuous不连续的discrete离散的discriminant判别式disjoint不相交的disorder混乱，无序dissection剖分dissipation损耗distribution分布，广义函数divergent发散的divisor因子，除数division除法domain区域，定义域dot product点积double integral二重积分dual对偶dynamic model动态模型dynamic programming动态规划dynamic system动力系统eccentricity离心率econometrics计量经济学edge棱，边eigenvalue特征值eigenvector特征向量eigenspace特征空间element元素ellipse椭圆embed嵌入empirical equation经验公式empirical assumption经验假设endomorphism自同态end point端点entropy熵entire function整函数envelope包络epimorphism满同态equiangular等角equilateral等边的equicontinuous等度连续的equilibrium平衡equivalence等价error estimate误差估计estimator估计量evaluation赋值，值的计算even number偶数exact sequence正合序列exact solution精确解excenter外心excision切割，分割exclusive events互斥事件exhaustive穷举的expansion展开，展开式expectation期望experimental error实验误差explicit function显函数exponent指数extension扩张，外延face面factor因子factorial阶乘fallacy谬误fiducial置信field域，场field theory域论figure图形，数字finite有限的finite group有限群finite iteration有限迭代finite rank有限秩finitely covered有限覆盖fitting拟合fixed point不动点flag标志flat space平旦空间formula公式fraction分数，分式frame架，标架free boundary自由边界frequency频数，频率front side正面function函数functional泛函functor函子，算符fundamental group基本群fuzzy模糊的gain增益，放大率game对策gap间断，间隙general topology一般拓扑学general term通项generalized普遍的，推广的generalized inverse广义逆generalization归纳，普遍化generating line母线genus亏格geodesic测地线geometrical几何的geometric series几何级数golden section黄金分割graph图形，网格half plane半平面harmonic调和的hexagon六边形hereditary可传的holomorphic全纯的homeomorphism同胚homogeneous齐次的homology同调homotopy同伦hyperbola双曲线hyperplane超平面hypothesis假设ideal理想idempotent幂等的identical恒等，恒同identity恒等式，单位元ill-condition病态image像点，像imaginary axis虚轴imbedding嵌入imitation模仿，模拟immersion浸入impulse function脉冲函数inclination斜角，倾角inclined plane斜面inclusion包含incomparable不可比的incompatible不相容的，互斥的inconsistent不成立的indefinite integral不定积分independence无关（性），独立（性）index指数，指标indivisible除不尽的inductive归纳的inductive definition归纳定义induced诱导的inequality不等式inertia law惯性律inference推理，推论infimum下确界infinite无穷大的infinite decimal无穷小数infinite series无穷级数infinitesimal无穷小的inflection point拐点information theory信息论inhomogeneous非齐次的injection内射inner point内点instability不稳定integer整数integrable可积的integrand被积函数integral积分intermediate value介值intersection交，相交interval区间intrinsic内在的，内蕴的invariant不变的inverse circular funct反三角函数inverse image逆像，原像inversion反演invertible可逆的involution对合irrational无理的，无理数irreducible不可约的isolated point孤立点isometric等距的isomorphic同构的iteration迭代joint distribution联合分布kernel核keyword关键词knot纽结known已知的large sample大样本last term末项lateral area侧面积lattice格子lattice point格点law of identity同一律leading coefficient首项系数leaf蔓叶线least squares solution最小二乘解lemma引理Lie algebra李代数lifting提升likelihood似然的limit极限linear combination线性组合linear filter线性滤波linear fraction transf线性分linear filter线性滤波式变换式变换linear functional线性泛函linear operator线性算子linearly dependent线性相关linearly independent线性无关local coordinates局部坐标locus(pl.loci)轨迹logarithm对数lower bound下界logic逻辑lozenge菱形lunar新月型main diagonal主对角线manifold流形mantissa尾数many-valued function多值函数map into映入map onto映到mapping映射marginal边缘master equation主方程mathermatical analysis数学分析mathematical expectati数学期望matrix(pl. matrices)矩阵maximal极大的，最大的maximum norm最大模mean平均，中数measurable可测的measure测度mesh网络metric space距离空间midpoint中点minus减minimal极小的，最小的model模型modulus模，模数moment矩monomorphism单一同态multi-analysis多元分析multiplication乘法multipole多极mutual相互的mutually disjoint互不相交natural boundary自然边界natural equivalence自然等价natural number自然数natural period固有周期negative负的，否定的neighborhood邻域nil-factor零因子nilpotent幂零的nodal节点的noncommutative非交换的nondense疏的，无处稠密的nonempty非空的noncountable不可数的nonlinear非线性的nonsingular非奇异的norm范数normal正规的，法线normal derivative法向导数normal direction法方向normal distribution正态分布normal family正规族normal operator正规算子normal set良序集normed赋范的n-tuple integral重积分number theory数论numerical analysis数值分析null空，零obtuse angle钝角octagon八边形octant卦限odd number奇数odevity奇偶性off-centre偏心的one-side单侧的open ball开球operations reserach运筹学optimality最优性optimization最优化optimum最佳条件orbit轨道order阶，级，次序order-preserving保序的order-type序型ordinal次序的ordinary寻常的，正常的ordinate纵坐标orient定方向orientable可定向的origin原点original state初始状态orthogonal正交的orthonormal规范化正交的outer product外积oval卵形线overdetermined超定的overlaping重叠，交迭pairity奇偶性pairwise两两的parabola抛物线parallel平行parallel lines平行线parallelogram平行四边形parameter参数parent population母体partial偏的，部分的partial ordering偏序partial sum部分和particle质点partition划分，分类path space道路空间perfect differential全微分period周期periodic decimal循环小数peripheral周界的，外表的periphery边界permissible容许的permutable可交换的perpendicular垂直perturbation扰动，摄动phase相，位相piecewise分段的planar平面的plane curve平面曲线plane domain平面区域plane pencil平面束plus加point of intersection交点pointwise逐点的polar coordinates极坐标pole极，极点polygon多边形polygonal line折线polynomial多项式positive正的，肯定的potency势，基数potential位势prime素的primitive本原的principal minor主子式prism棱柱proof theory证明论probability概率projective射影的，投影proportion比例pure纯的pyramid棱锥，棱锥体quadrant像限quadratic二次的quadric surface二次曲面quantity量，数量quasi-group拟群quasi-norm拟范数quasi-normal拟正规queuing theory排队论quotient商radial径向radical sign根号radication开方radian弧度radius半径ramified分歧的random随机randomize随机化range值域，区域，范围rank秩rational有理的raw data原始数据real function实函数reciprocal倒数的，互反的reciprocal basis对偶基reciprocity互反性rectangle长方形，矩形rectifiable可求长的recurring decimal循环小数reduce简化，化简reflection反射reflexive自反的region区域regular正则regular ring正则环related function相关函数remanent剩余的repeated root重根residue留数，残数resolution分解resolvent预解式right angle直角rotation旋转roundoff舍入row rank行秩ruled surface直纹曲面runs游程，取遍saddle point鞍点sample样本sampling取样scalar field标量场scalar product数量积，内积scale标尺，尺度scattering散射，扩散sectorial扇形self-adjoint自伴的semicircle半圆semi-definite半定的semigroup半群semisimple半单纯的separable可分的sequence序列sequential相继的，序列的serial序列的sheaf层side face侧面similar相似的simple curve简单曲线simplex单纯形singular values奇异值skeleton骨架skewness偏斜度slackness松弛性slant斜的slope斜率small sample小样本smooth manifold光滑流形solid figure立体形solid geometry立体几何solid of rotation旋转体solution解solvable可解的sparse稀疏的spectral theory谱论spectrum谱sphere球面，球形spiral螺线spline function样条函数splitting分裂的statistics统计，统计学statistic统计量stochastic随机的straight angle平角straight line直线stream-line流线subadditive次可加的subinterval子区间submanifold子流形subset子集subtraction减法sum和summable可加的summand被加数supremum上确界surjective满射的symmetric对称的tabular表格式的tabulation列表，造表tangent正切，切线tangent space切空间tangent vector切向量tensor张量term项terminal row末行termwise逐项的tetrahedroid四面体topological拓扑的torsion挠率totally ordered set全序集trace迹trajectory轨道transcendental超越的transfer改变，传transfinite超限的transformation变换式transitive可传递的translation平移transpose转置transverse横截、trapezoid梯形treble三倍，三重trend趋势triad三元组triaxial三轴的，三维的trigon三角形trigonometric三角学的tripod三面角tubular管状的twist挠曲，扭转type类型，型，序型unbiased无偏的unbiased estimate无偏估计unbounded无界的uncertainty不定性unconditional无条件的unequal不等的uniform一致的uniform boundness一致有界uniformly bounded一致有界的uniformly continuous一致连续uniformly convergent一致收敛unilateral单侧的union并，并集unit单位unit circle单位圆unitary matrix酉矩阵universal泛的，通用的upper bound上界unrounded不舍入的unstable不稳定的valuation赋值value值variation变分，变差variety簇vector向量vector bundle向量丛vertex顶点vertical angle对顶角volume体积，容积wave波wave form波形wave function波函数wave equation波动方程weak convergence弱收敛weak derivatives弱导数weight权重，重量well-ordered良序的well-posed适定的zero零zero divisor零因子zeros零点zone域，带</Words>。

IEEE JOURNAL OF SOLID-STATE CIRCUITS,VOL.38,NO.3,MARCH 2003561A 2-V 23- A5.3-ppm/m CMOS technology with V thn 0C,the proposed voltage reference can operate down to a 2-V supply and consumes a maximum supply current of231.43mV/V at 27The Hong Kong University of Scien Kong (e-mail:eemok@t.hk).Digital Object Identifier 10.1109/Fig.1.Proposed bandgap reference with curvature compensation.(a)Structure.(b)Schematic.III.P ROPOSED C URV ATURE -C OMPENSATEDCMOS B ANDGAP R EFERENCEThe idea of the proposed curvature-compensation technique is illustrated in Fig.1(a).Theresistorsin this proposed structure,a PTATvoltage is formedacross.These two voltagesare added togetherwithis,thus,generatedis the emitter-area ratio of Q1and Q2,andandistemperature dependent due to different types of materials.From the data from AMS0.6-and,re-spectively.The circuit design mainly involves the design of precision current and frequency compensation.In addition,temperaturetrimming is also an important post-fabrication step for obtaining a low-drift reference voltage.A.Current MirrorPrecision voltage references are generally degraded by cur-rent mirror and device mismatches.To eliminate the use of cas-code current mirrors,aresistor27while the high-order tempera-ture trimming is performedbyFig.2.Chip micrograph of the proposed bandgap voltage reference.TABLE IS UMMARYOF THEP ERFORMANCE OF THE P ROPOSED B ANDGAP V OLTAGE REFERENCEsecond step involves the minimization of the linear tempera-ture dependence by finding anoptimumm CMOS technology.The thresholdvoltage of this technology isaboutC.The chip micrograph is shown in Fig.2and the active chip area is 0.057mmC.The maximum supply current is23C,respectively.Fig.3shows the mea-sured temperature dependence with supply voltages of 1.8,2.0,3.0,and 4.0V .Results show that the proposed reference pro-vides low temperature coefficients starting from a 2-V supply and,thus,the minimum supply voltage is 2V .It should be noted that there is only little change on the temperature coefficients for the change of supply voltage from 2to 4V .This is due to the well-matched current branches provided by the proposed two-branches structure.The measured supply dependence at 0,27,and 1001.43mV/V inthe range 0C.The variation on the reference voltage is due to the systematic offset voltage caused by the finite gain of the error amplifier.A cascode error amplifier can further im-prove the line regulation.The measured power-supply rejection ratio (PSRR)is shown in Fig.5.The measurement conditions are with a 2-V supply voltage and at room temperature without a filtering capacitor.The PSRR at 10Hz isFig.5.Measured PSRR of the proposed bandgap voltage reference at room temperature,with a 2-V supply,and without a filteringcapacitor.Fig.6.Measured noise spectrum of the proposed bandgap voltage reference at room temperature with a 2-V supply.cascode error amplifier is used.However,the required supply voltage increases,and this is not suitable for low-voltage voltage-reference design.At high frequencies,the PSRR isA supply current to provide a temperature coefficient of 25ppm/A.It provides atemperature coefficient of 5.3ppm/20ppmA 5.3-ppm/C4th-order curvature-compensated CMOS bandgap reference,”in Proc.IEEE Custom Integrated Circuits Conf.,May 2002,pp.457–460.[8] A.Hastings,The Art of Analog Layout .Englewood Cliffs,NJ:Pren-tice-Hall,2001.[9]K.N.Leung and P.K.T.Mok,“A sub-1-V 15-ppm/。