一类Schrdinger算子的连续谱测度(英文)
量子力学索引英汉对照
21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫-玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, 衰变；Alpha particle, 粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥-沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色-爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克-高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, 势垒Delta-function well, 势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量-时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米-狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼-海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g-因子Gamma function, 函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, "好"量子数"Good" states, "好"的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆-施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克LLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g-因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维-西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦-玻尔兹曼分布Maxwell's equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, 子Muon-catalysed fusion, 子催化的聚变Muonic hydrogen, 原子Muonium, 子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置-动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋-轨道耦合Spin-orbit interaction, 自旋-轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋-自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番-玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩-盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。
Schrdingerequation薛定谔方程
Schrdingerequation薛定谔方程百科名片薛定谔方程推导薛定谔方程（Schrdingerequation）是由奥地利物理学家薛定谔提出的量子力学中的一个基本方程，也是量子力学的一个基本假定，其正确性只能靠实验来检验。
是将物质波的概念和波动方程相结合建立的二阶偏微分方程，可描述微观粒子的运动，每个微观系统都有一个相应的薛定谔方程式，通过解方程可得到波函数的具体形式以及对应的能量，从而了解微观系统的性质。
目录定义简介薛定谔方程的提出薛定谔简介薛定谔方程具体介绍薛定谔方程的数学表达形式展开编辑本段定义在量子力学中，体系的状态不能用力学量（例如某）的值来确定，而是要用力学量的函数Ψ（某,t），即波函数（又称概率幅，态函数）来确定，因此波函数成为量子力学研究的主要对象。
力学量取值的概率分布如何，这个分布随时间如何变化，这些问题都可以通过求解波函数的薛定谔方程得到解答。
这个方程是奥地利物理学家薛定谔于1926年提出的，它是量子力学最基本的方程之一，在量子力学中的地位与牛顿方程在经典力学中的地位相当。
薛定谔方程是量子力学最基本的方程，亦是量子力学的一个基本假定，它的正确性只能靠实验来检验。
编辑本段简介量子力学中求解粒子问题常归结为解薛定谔方程或定态薛定谔方程。
薛定谔方程广泛地用于原子物理、核物理和固体物理，对于原子、分子、核、固体等一系列问题中求解的结果都与实际符合得很好。
薛定谔方程仅适用于速度不太大的非相对论粒子，其中也没有包含关于粒子自旋的描述。
当计及相对论效应时，薛定谔方程由相对论量子力学方程所取代，其中自然包含了粒子的自旋。
.薛定谔提出的量子力学基本方程建立于1926年。
它是一个非相对论的波动方程。
它反映了描述微观粒子的状态随时间变化的规律，它在量子力学中的地位相当于牛顿定律对于经典力学一样，是量子力学的基本假设之一。
设描述微观粒子状态的波函数为Ψ（r，t），质量为m的微观粒子在势场V（r，t）中运动的薛定谔方程为。
Schrodinger operators generated by substitutions
a rXiv:c ond-mat/95161v112Oct1995SCHR ¨ODINGER OPERATORSGENERATED BY SUBSTITUTIONS ∗Anton BOVIERWeierstrass-Institut f¨u r Angewandte Analysis und StochastikMohrenstrasse 39,D-10117Berlin,GermanyE-mail:bovier@iaas-berlin.d400.deandJean-Michel GHEZ∗Work partially supported by EU contract CHRX-CT93-0411By means of the K-theory of C ∗-algebras,(ii)has been proven to be correct for all operators generated by primitive substitutions 9,with the limitation that in general one cannot exclude that some gaps are closed.A general perturbative approach to compute the opening of gaps has been proposed in 10.2.Cantor spectrumThe basic tool to establish (i)11is the transfer-matrix formalism.That is,the study of the properties of solutions of Eq.(1)leads to the analysis of products of two-by-two matrices of the form P n (E )= n k =1T E (w n −k ),where w k is the k -th letter in the substitution sequence w and T E :A →SL (2)is a map that assigns,for fixed energy E ,to each letter in the alphabet a unimodular two-by-two matrix.In the case of Eq.(1),T E (w n )= E −v (w n )−110 ,but the precise form of this map is not important,and therefore the same results hold for all types of second order equations that can be reduced to a problem of products of unimodular 2×2-matrices.This includes in particular the continuous Laplacian with piecewise constant potential of a finite number of different shapes (Kronig-Penney model,see 12,13).The main idea of the proof of the singularity of the spectrum σ(H )of H is its identification with the set O of zero Lyapunov exponents γ(E )=lim n ↑∞1of the spectrum of H.Note that the supplementary hypothesis for the second result is probably not necessary,since there is at least one example(Thue-Morse)for which it is not satisfied althoughσ(H)is singular continuous.3.ExamplesThe hypothesis of the theorem have been shown to be satisfied in a large number of particular cases in11:(i)The Fibonacci sequence:ξ(a)=ab,ξ(b)=a:σ(H)is singular continuous,as already proven in4,5;(ii)The Thue-Morse sequence:ξ(a)=ab,ξ(b)=ba:σ(H)is singular as already proven in 8,where we proved that it is in fact singular continuous;(iii)The period-doubling sequence:ξ(a)=ab,ξ(b)=aa:σ(H)is singular continuous,as already proven in8;(iv)The circle sequence:ξ(a)=cac,ξ(b)=accac,ξ(c)=abcac:σ(H)is singular continuous. More recently,in17a new class of examples was provided by sequences derived from circle maps with rotation numbers obtained from precious means;(v)The binary non-Pisot sequence:ξ(a)=ab,ξ(b)=aaa:σ(H)is singular.We can only conjecture that it is singular continuous;(vi)The ternary non-Pisot sequence:ξ(a)=c,ξ(b)=a,ξ(c)=bab:σ(H)is singular continuous;(vii)The Rudin-Shapiro sequence:ξ(a)=ac,ξ(b)=dc,ξ(c)=ab,ξ(d)=db:in this case,φis not semi-primitive and thus,presently,we cannot even prove the singularity of the spectrum,which is not so surprising if one recalls that it is the‘most random’of all these sequences9,10.Let us emphasize that our theorem provides explicit examples of operators with singular continuous plementary results have recently been obtained by Hof et al.18 in which for certain types of so-called“palindromic”substitutions it was shown that there exists an infinite number of unspecified translates of the original sequence for which the corresponding operator has singular continuous spectrum.4.Concluding remarksWe end this note with some remarks on the nature of the solutions of the Schr¨o dinger equation(1).The proof of the theorem shows in fact that for energies in the spectrum,no solution tends to zero at infinity,but these leaves room for a variety of behaviours.However, in most cases,there are countable subsets of energies in the spectrum,characterized bythe fact that the transfer matrices T(n)E (α),for given n,commute for allα∈A,at whichthe solutions are extended in a very regular way in that they consist of pieces of repeating patterns arranged according to the substitution13.Such solutions have actually been discovered already in7and8,but they have been recently re-discovered in numerical investigations several times and have given rise to erroneous claims of coexisting absolutely continuous spectrum.While this is nonsense,it is not unlikely that these states will be quite important for the transport properties of systems with singular continuous spectrum,a problem that still has not been satisfactorily investigated.5.References1.M.Queff´e lec,Substitution dynamical systems.Spectral analysis,Springer LectureNotes in Mathematics1294(1987)2.D.Schechtman,I.Blech,D.Gratias and J.V.Cahn,Phys.Rev.Lett.53(1984)1951-19533.M.Casdagli,Comm.Math.Phys.107(1986)295-3184.A.S¨u t˝o,J.Stat.Phys.56(1989)525-5315.J.Bellissard,B.Iochum,E.Scoppola and D.Testard,Comm.Math.Phys.,125(1989)527-5436.L.Raymond,A dynamical approach to the gap-labelling of a quasi-periodicSchr¨o dinger operator in Dynamique des syst`e mes complexes,aspects classique et quan-tique,Marseille,13-17/6/19947.J.Bellissard,in Number theory and physics,J.-M.Luck,P.Moussa,M.Waldschmidt,eds.(Springer Proceedings in Physics47(1990)140-150;see also F.Axel and J.Peyri`e re,J.Stat.Phys.57(1989)1013-10478.J.Bellissard,A.Bovier and J.-M.Ghez,Comm.Math.Phys.135(1991)379-3999.J.Bellissard,A.Bovier and J.-M.Ghez,Rev.Math.Phys.4(1992)1-3710.J.-M.Luck,Phys.Rev.B39(1989)5834-584911.A.Bovier and J.-M.Ghez,Comm.Math.Phys.158(1993)45-66and erratumComm.Math.Phys.166(1994)431-43212.J.Bellissard,A.Formoso,R.Lima and D.Testard,Phys.Rev.26(1982)3024-303013.A.Bovier and J.-M.Ghez:J.Phys.A:Math.Gen.28(1995)2313-23214.S.Kotani,Rev.Math.Phys.1(1990)129-13315.J.-P.Allouche and J.Peyri`e re,C.R.Acad.Sci.Paris302No18s´e rie II(1986)1135-113616.M.Kol´a r and F.Nori,Phys.Rev.B42(1990)1062-106517.M.H¨o rnquist and M.Johansson:J.Phys.A:Math.Gen.28(1995)479-49518.A.Hof,O.Knill and B.Simon:Singular continuous spectrum for palindromicSchr¨o dinger operators Preprint Caltech,to appear in Comm.Math.Phys.(1995)。
数学专业英语词汇（H）_数学物理英语词汇
h closed h闭的haar condition 哈尔条件haar measure 哈尔测度hadamard criterion 阿达玛判别准则hadamard gap condition 阿达玛间断条件hadamard matrix 阿达玛矩阵hadamard method of descent 阿达玛下降法hadamard multiplication theorem 阿达玛乘法定理hadamard three circles theorem 阿达玛三圆定理half 半half angle 半角half angle formulas 半角公式half axis 半轴half closed interval 半闭区间half exact 半正合的half line 半直线half neighborhood 半邻域half plane 半无限平面half plane of absolute convergence 绝对收敛半平面half plane of convergence 收敛半平面half round 半圆的half side formulas 半边公式half space 半空间halve 对分halving method 二等分法hamilton characteristic function 哈密顿特寨数hamilton formula 哈密顿公式hamilton function 哈密顿函数hamilton jacobi equation 哈密顿雅可比方程hamilton jacobi theory 哈密顿雅可比理论hamilton principle 哈密顿原理hamiltonian 哈密顿函数hamiltonian circuit 哈密顿回路hamiltonian group 哈密顿群hamiltonian operator 哈密顿算子hamiltonian path 合密顿道路hand 边handle 环柄handle of the second kind 交叉套handlebody 环柄体hankel transformation 汉克尔变换harmonic analysis 低分析harmonic analyzer 傅里叶分析仪harmonic conjugate 低共轭点harmonic constant 低常数harmonic curve 低曲线harmonic differential equation 低微分方程harmonic division 低分割harmonic function 低函数harmonic integral 低积分harmonic mapping 低映射harmonic mean 低平均harmonic measure 低测度harmonic motion 低运动harmonic oscillation 谐振动harmonic progression 低级数harmonic ratio 低比harmonic series 低级数harmonic synthesis 傅里叶综合法harmonicity 低性hasse diagram 哈塞图hausdorff group 豪斯道夫群hausdorff measure 豪斯道夫测度hausdorff metric 豪斯道夫度量hausdorff separation axiom 廉斯道夫分离公理hausdorff space 分离空间haversine 半正矢heat 热heat conduction 热传导hecke character 黑克特贞hecke operator 黑克算子hectoliter 百升hectometer 百米helicograph 螺旋规helicoid 螺旋面helicoidal surface 螺旋面helix 螺旋线hemi continuous 半连续的hemihedry 半对称hemipyramid 半棱锥体hemisphere 半球hemispherical 半球面的hemispherical shape 半球形hendecagon 十一边形henselization 享泽莱化heptagon 七边形heptahedron 七面体hereditarily enumerable set 遗传可数集hereditarily generating system 遗传的生成系hereditarily indecomposable continuum 遗传不可分解的连续统hereditarily normal space 遗传正规空间hereditary class 遗传类hereditary property 遗传性质hereditary set 遗传集hereditary system of sets 集的遗传系heredity 遗传性hermite function 埃尔米特函数hermite interpolation formula 埃尔米特插值公式hermite interpolation polynomial 埃尔米特插值多项式hermite normal form 埃尔米特正规形式hermite polynomial 埃尔米特多项式hermite reciprocity law 埃尔米待互反律hermitian bilinear functional 埃尔米特双线性泛函hermitian conjugate 埃尔米特共轭阵hermitian form 埃尔米特形式hermitian inner product module 埃尔米特内积模hermitian inner product space 埃尔米特空间hermitian kernel 埃尔米特核hermitian matrix 埃尔米特矩阵hermitian metric 埃尔米特度量hermitian operator 埃尔米特算子hermitian polynomiat 埃尔米特多项式hermitian transformation 埃尔米特变换hero formula 海伦公式hesse normal form 海赛正规形式hessian 海赛形式hessian group 海赛群hessian matrix 海赛矩阵hexagon 六边形hexagonal 六边形的hexagonal net 六边形网格hexagonal system 六角系hexahedral 六面体的hexahedron 六面体hexakistetrahedron 六四面体hierarchical classification 谱系分类hierarchy 分层high speed computer 高速计算机higher algebra 高等代数higher commutator 广义换位子higher derivative 高阶导数higher mathematics 高等数学higher order term 高阶项higher plane curve 高次平面曲线higher singularity 高次奇异性highest common divisor 最大公约highest common factor 最大公因子highest derivative 最高阶导数highest order 最高位highest weight 最高权hilbert basis theorem 希耳伯特基定理hilbert cube 希耳伯特超平行体hilbert inequality 希耳伯特不等式hilbert integral 希耳伯特积分hilbert matrix 希耳伯特矩阵hilbert modular form 希耳伯特模形式hilbert modular function 希耳伯特模函数hilbert modular group 希耳伯特模群hilbert nullstellensatz 希耳伯特零点定理hilbert parallelotope 希耳伯特超平行体hilbert problems 希耳伯特问题hilbert space 希耳伯特空间hill differential equation 希耳微分方程histogram 直方图history 履历hodograph 速端曲线hodograph transformation 速端曲线变换hodometer 路程表holding domain 解域holomorph convex manifold 全形凸廖holomorph separable manifold 全形可分廖holomorphic 正则的holomorphic completeness 全纯完全性holomorphic convexity 正则凸性holomorphic differential 全纯微分holomorphic differential form 全纯微分形式holomorphic divisor 全纯除子holomorphic function 全纯函数holomorphic manifold 复解析廖holomorphic mapping 全纯映射holomorphic part 全纯部分holomorphy 正则holonomic condition 完全性条件holonomic reference system 完整参考系holonomic system 完整系holonomy 完整holonomy group 完整群homeomorph 同胚象homeomorphic 同胚的homogeneity 齐性homogeneity formula 齐性公式homogeneity of variances 同方差性homogeneity relation 齐性关系homogeneous 均匀的homogeneous cartesian co ordinates 齐次笛卡儿坐标homogeneous coordinates 齐次笛卡儿坐标homogeneous distribution 均匀分布homogeneous element 齐次元素homogeneous equation 齐次方程homogeneous function 齐次函数homogeneous function of order k k阶齐次函数homogeneous ideal 齐次理想homogeneous integral equation 齐次积分方程homogeneous linear boundary value problem 齐次线性边值问题homogeneous linear differential equation 齐次线性微分方程homogeneous linear transformation 齐次线性变换homogeneous lineare transformation 齐次线性变换homogeneous markov chain 齐次马尔可夫链homogeneous markov process 齐次马尔可夫过程homogeneous operator 齐次算子homogeneous polynomial 齐次多项式homogeneous space 商空间homogeneous system of differential equations 齐次微分方程组homogeneous system of linear equations 齐次线性方程组homogeneous variational problem 齐次变分问题homographic function 单应函数homological algebra 同碟数homological dimension 同惮数homological invariant 同祷变量homologous mappings 同党射homologous simplicial map 同单形映射homologous to zero 同第零homology 同调homology algebra 同碟数homology class 同掂homology equivalence 同等价homology equivalent complex 同等价复形homology functor 同弹子homology group 同岛homology manifold 同滴homology module 同担homology operation 同邓算homology sequence 同凋列homology simplex 同单形homology spectral sequence 同底序列homology sphere 同凋homology theory 同帝homology type 同低homomorphic group 同态群homomorphic image 同态象homomorphism 同态homomorphism theorem 同态定理homoscedastic 同方差的homoscedasticity 同方差性homothetic transformation 相似扩大homothety 相似扩大homotopic 同伦的homotopic invariant 同伦不变量homotopic map 同伦映射homotopic path 同伦道路homotopically equivalent space 同伦等价空间homotopy associativity 同伦结合性homotopy category of topological spaces 拓扑空间同伦范畴homotopy chain 同伦链homotopy class 同伦类homotopy classification 同伦分类homotopy equivalence 同伦等价homotopy excision theorem 同伦分割定理homotopy extension 同伦扩张homotopy group 同伦群homotopy group functor 同伦群函子homotopy inverse 同伦逆的homotopy operator 同伦算子homotopy sequence 同伦序列homotopy set 同伦集homotopy sphere 同伦球面homotopy theorem 同伦定理homotopy theory 同伦论homotopy type 同伦型homotopyassociative 同伦结合的horizon 水平线horizontal axis 水平轴horizontal component 水平分量horizontal coordinates 水平坐标horizontal plane 水平面horizontal projection 水平射影horned sphere 角形球面horocycle 极限圆horosphere 极限球面horse power 马力hungarian method 匈牙利法hurewicz isomorphism theorem 胡列维茨同构定理hydrodynamics 铃动力学hydromechanics 铃力学hydrostatics 铃静力学hyper graeco latin square 超格勒科拉丁方格hyper octahedral group 超八面体群hyperabelian function 超阿贝耳函数hyperalgebraic manifold 超代数廖hyperarithmetical 超算术的hyperarithmetical relation 超算术关系hyperbola 双曲线hyperbolic 双曲线的hyperbolic automorphism 双曲代换hyperbolic catenary 双曲悬链线hyperbolic cosecant 双曲余割hyperbolic cosine 双曲余弦hyperbolic cotangent 双曲余切hyperbolic cylinder 双曲柱hyperbolic elliptic motion 双曲椭圆运动hyperbolic equation 双曲型方程hyperbolic function 双曲函数hyperbolic geometry 双曲几何学hyperbolic inverse point 双曲逆点hyperbolic involution 双曲对合hyperbolic line 双曲线hyperbolic motion 双曲运动hyperbolic orbit 双曲线轨道hyperbolic paraboloid 双曲抛物面hyperbolic plane 双曲平面hyperbolic point 双曲点hyperbolic riemann surface 双曲型黎曼曲面hyperbolic rotation 双曲旋转hyperbolic secant 双曲正割hyperbolic sine 双曲正弦hyperbolic space 双曲空间hyperbolic spiral 双曲螺线hyperbolic substitution 双曲代换hyperbolic system 双曲型组hyperbolic tangent 双曲正切hyperbolic tangent function 双曲正切hyperbolic type 双曲型hyperbolicity 双曲性hyperboloid 双曲面hyperboloid of one sheet 单叶双曲面hyperboloid of revolution 旋转双曲面hyperboloid of two sheets 双叶双曲面hypercohomology 超上同调hypercomplex 超复数hypercomplex number 超复数hypercone 超锥hyperconjugation 超共轭hypercyclic group 超循环群hypercyclide 超四次圆纹曲面hyperelliptic 超椭圆的hyperelliptic function 超椭圆函数hyperelliptic integral 超椭圆积分hyperelliptic theta function 超椭圆函数hyperfinite c* algebra 超有限c*代数hypergeometric differential equation 超几何微分方程hypergeometric distribution 超几何分布hypergeometric function 超几何函数hypergeometric function of the second kind 第二类超几何函数hypergeometric series 超几何级数hypergeometry 超几何学hypergroup 超群hypermatrix 超矩阵hypernormal dispersion 超正态方差hyperplane 超平面hyperplane coordinates 超平面坐标hyperplane of support 支撑超平面hyperplane section 超平面截面hyperquadric 超二次曲面hyperreal numbers 超实数hyperspace 超空间hypersphere 超球面hyperstonian space 超斯通空间hypersurface 超曲面hypocycloid 内摆线hypocycloidal 圆内旋轮线的hypoelliptic operator 次椭圆型算子hypoellipticity 次椭圆性hypotenuse 斜边hypothesis 假设hypothetical population 假言总体hypotrochoid 长短辐圆内旋轮线。
Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quant
with boundary conditions ψ (x) → 0 as x → ∞ and ψ (x) ∼ xl+1 as x → 0.∗ Over the years, many interesting features of this problem have been uncovered. For α < 0 and l(l+1) = 0, the theory corresponds to a double-well potential on the full real line, with l = −1 and l = 0 selecting the even and odd wavefunctions respectively. Such potentials have long been studied as toy models for instanton effects in quantum field theories [1]. Furthermore, at {α = −3, l = −1} the ground-state energy E0 is zero, with the remaining energy levels matching those at {α = 3, l = 0}. This reflects the fact that for these values of the parameters the model provides one of the simplest examples of supersymmetric quantum mechanics [2]. More peculiar are the properties of the spectrum at α = −(4J + 2l + 1): a finite number of energy levels can be exactly computed [3, 4] being roots of particular polynomials [5]. At these points, the model is said to be ‘quasi-exactly solvable’. Finally, at least a couple of physically interesting spectral problems are related to the above equation via simple variable and gauge transformations. The first is a linear combination of the harmonic and Coulomb potentials: σ γ (γ + 1) d2 ϕ(x) = Λ ϕ(x) , (1.2) − 2 + x2 − + dx x x2 with the two sets of parameters being related by γ = (l−1/2)/2, Λ = −α/2, σ = 2−3/2 E . The second theory is − β δ(δ + 1) 1 d2 + 3/2 + + χ(x) = ∆ χ(x) , 2 dx x x2 x (1.3)
微积分专有名词中英文对照
微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression (AP)算数级数asymptotes (vertical and horizontal)(垂直/水平)渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标(一般指直角坐标) Cartesian coordinates system 笛卡儿坐标系Cauch’s Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity (函数的)连续性concavity (函数的)凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant (二次函数)判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler's method (BC)欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression (GP)几何级数gradient 梯度Green formula 格林公式half-angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point (POI) 拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem ：中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left-hand derivative 左导数left-hand limit 左极限L'Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem (MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法，极值法origin 原点orthogonal 正交parametric equation (BC)参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece-wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point-slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域(related) rate of change with time (时间)变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right-hand limit 右极限Rolle's theorem 罗尔(中值)定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method (积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切(平)面tangent vector 切矢量taylor's series 泰勒级数three-dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形(积分)法则。
统计学专业词汇英语翻译
统计学专业词汇英语翻译H?lder不等式H?1der's inequalityHaldane差异测度Haldane's discrepancy measure半常态分布；半常态分配half normal distribution半常态描点图half normal plots半柯西分布；半柯西分配half-Cauchy distribution半不变式(量) half-invariant半常态机率纸half-normal probability paper半周期half-period半格子方阵half-plaid square半重复设计half-replicate design半宽度half-widthHammersley-Chapman-Robbins不等式Hammersley-Chapman-Robbins inequalityHardy总和法Hardy summation methodHardy公式Hardy's formulaHardy-Weinberg比例式Hardy-Weinberg proportions Harley逼近Harley approximation调和分析harmonic analysis调和分布；调和分配harmonic distribution调和平均数harmonic mean调和过程harmonic process国际商品统一分类Harmonized Commodity Description and Coding SystemHarris漫步Harris walkHarrison法Harrison's methodHartley检定Hartley's testHartley-Rao方案Hartley-Rao scheme故障分析hazard analysis故障函数hazard function故障图hazard plots故障率(分布) hazard rate (distribution)卫生统计health statistics重尾分布heavy tail distributionHellinger距离Hellinger distanceHelly第一定理Helly's first theoremHelly-Bray定理Helly-Bray theoremsHelmert准则Helmert criterionHelmert分布；Helmert分配Helmert distribution Helmert变换Helmert transformationHermite分布；Hermite分配Hermite distributions Hess矩阵Hessian matrix异偏态的heteroclitic异质性；非均齐性heterogeneity异质母体heterogeneous population异质层heterogeneous strata属量变数heterograde异峰度的heterokurtic不等变异的heteroscedastic不等变异数模型heteroscedastic model不等变异heteroscedastic variation不等变异性heteroscedasticity异质型的heterotypic潜伏周期hidden periodicity层次生灭过程hierarchical birth and death process 层次分类hierarchical classification层次可分组设计hierarchical group divisible design 层次hierarchy高度相关high correlation高低图high-low graph高阶自身回归方案higher order autoregressive scheme高阶混同higher order confounding高阶滞后结构higher order lag structures高阶递移矩阵higher transition matrix最高事后密度区间highest posterior density intervals登山法hill-climbing method直方图；矩形图histogram命中点hitting pointHodges二变量符号检定Hodges bivariate sign testHodges-Ajne检定Hodges-Ajne's testHodges-Lehmann估计值Hodges-Lehmann estimate Hodges-Lehmann估计量Hodges-Lehmann estimator Hodges-Lehmann单样本估计量Hodges-Lehmannone-sample estimatorHoeffding独立性检定Hoeffding's independence test Hoeffding不等式Hoeffding's inequalityHollander二变量对称性检定Hollander's bivariate symmetry testHollander平行性检定Hollander's parallelism test Hollander-Proschan「新比旧佳」检定Hollander-Proschan ' new better than used' testHolt法Holt's method同偏态的homoclitic均齐性；同质性homogeneity实验条件的均齐性homogeneity of experimental conditions变异数的均齐性homogeneity of variances均齐的；齐次的；同质的homogeneous均齐过程homogeneous process均齐层homogeneous stratum均齐递移机率homogeneous transition probability属性变数homograde等峰度的homokurtic等变异的homoscedastic等变异数模型homoscedastic model均齐变异homoscedastic variation等变异性homoscedasticity正当过程honest process横条图horizontal bar chart横尺度horizontal scaleHorvitz-Thompson估计量Horvitz-Thompson estimator Hotelling的T统计量Hotelling's T statisticHotelling的T2统计量Hotelling's T2 statisticHotelling检定(相依相关) Hotelling's test (depen dent correlations)Hotelling-Pabst检定Hotelling-Pabst test住户household住宅单位housing unitHuber估计值Huber estimateHuber损失函数Huber loss function人体工学；人因工程human engineering人为误差human error潮湿试验humidity test高峰hump百分比条图hundred-percent bar chart全数检验hundred-percent inspectionHunt-Stein定理Hunt-Stein theorem混合比率hybrid ratio混合制hybrid system超方格hyper square超希腊拉丁方格hyper-Graeco Latin square超卜瓦松分布；超卜瓦松分配hyper-Poisson distribution超球面常态分布；超球面常态分配hyper-spherical normal distribution双曲正割分布；双曲正割分配hyperbolic secant distribution 超立方体hypercube超几何分布；超几何分布hypergeometric distribution超几何机率hypergeometric probability超几何数列hypergeometric series超常态离势hypernormal dispersion超常态分布；超常态分配hypernormal distribution超常态性hypernormality假设；拟说hypothesis假设检定hypothesis testing统计假设hypothesis;statistical假设母体hypothetical population假设全域hypothetical universe。
Local scale-invariance in disordered systems
Recall that physical ageing as it is understood here comes from reversible microscopic processes, whereas chemical or biological ageing may come from the action of essentially irreversible (bio-)chemical processes.
1 Introduction
Understanding cooperative phenomena far from equilibrium poses one of the most challenging research problems of present-day many-body physics. At the same time, the practical handling of many of these materials has been pushed to great sophistication, and a lot of practical knowledge about them exists since prehistoric times. Paradigmatic examples of such system are glasses. In many cases, they are made by rapidly cooling (‘quenching’) a molten liquid to below some characteristic temperature-threshold. If this cooling happens rapidly enough, normal crystallization no longer takes place and the material remains in some non-equilibrium state. These non-equilibrium states may at first and even second sight look very stationary – everyone has probably seen in archaeological museums intact specimens of Roman glass or even older tools from the Paleolithic or old-stone-age – after all, obsidian or fire-stone is a quenched volcanic melt. But since the material is not at equilibrium, at least in principle it is possible (and it does happen very often in practice) that over time the properties of the material change - in other words, the material ages.1 The properties of such non-equilibrium systems depend on the time – their age – since they were brought out of equilibrium and this is colloquially referred to as ageing behaviour.
太原理工大学理工科研究生专业英语词汇篇
Unit 1outcome结果statistics统计statistical 统计的toss投die骰子dice 掷骰子intuitive直觉的analogously类似的conceptually概念的simultaneous同时的collection集合identical同一的，同样的individual个别的ensemble集，总和ensemble average集平均variable变量random variable随机变量stationary平稳的ergodic个态历经的ergodic process各态历经过程deterministic确定性的normalize使标准化的normalized规格化的，归一化的expectation期望值product乘积truncate截断，截短periodic周期的covariance协方差uncorrelated不相关的uniform均匀的overlap部分重叠separation间隔，距离spacing间隔，间距random process随机过程on the average 平均，一般来说make measurement 量度sample function 样本函数statistical average 统计平均值probability density function概率密度函数statistical characteristic统计特征autocorrelation function 自相关函数in connection with 与，，有关physical significance物理意义Fourier transform傅里叶变换power spectral density 功率谱密度be at liberty to被允许gaussian process高斯过程physical fact外界存在的事实upper-frequency频率上限fall off下降in the limit在极限情况下delta function狄拉克函数random pulse随机脉冲time invariant时不变的parseval's theorem巴塞瓦尔定理statistically independent统计独立successive pulse连续脉冲be representative of表示，代表词组sample function样本函数ensemble average集合平均physical significance物理意义a fourier transform pair傅里叶变换对deterministic waveform 确定波形in the limit 在极限情况下time invariant时不变的an upper-frequency limit频率上限parseval，s theorem帕斯瓦尔定理random pulses随机脉冲随机过程random process统计平均statistical average随机变量random variable自相关函数autocorrelation function傅里叶变换fourier transform功率谱密度power spectral density概率密度函数probability density function 高斯过程gaussian process平稳过程stationary process统计独立statistically时间平均值time average统计特征statistical character各态历经过程ergodic processUnit 2abstraction抽象encompass包括devise设计whereby凭那个，由此arbitarily任意地theorem定理infrequent 罕见的override压倒thereby由此gaussian高斯的unity单一complementary补充的derivation推导formidable困难的undertake进行，从事presumed假定的presumably推测起来likelihood可能likely可能interval间隔rounding舍入，四舍五入abrupt，急剧的，不连续reliably可靠的，安全的heuristic启发式的intuitive直觉的intuitively直觉的contemplate预期的vice versa反之亦然quantize量化reception接受extreme极端的，偏激的attenuate衰减distort失真distortion失真equalizer均衡器recoverable可恢复的enter into参加，涉及due to 由于be something of 有一点in principle 原则上provided that假如on principle按照原则probability of error 误码率be close to接近bandlimited gaussian channel限带高斯信道physical system物理系统turn out结果lower bound下界upper limit 上限root-mean-square均方根root-mean-square error 均方误差amount of information 信息量ideal lowpass filter理想低通滤波器rise time上升时间as a matter of convenience为方便起见white gaussian noise 高斯白噪声on the other hand 另一面trade off交替换位tradeoff折中signal-to-noise ratio信号噪声比be free to 随意make up for 补偿词组physical system物理系统rise time 上升时间amount of information信息量in principle原则上gaussian channel高斯信道probability density概率密度root-mean-square均方根trade off折中lower bound下界equalizer均衡器vice versa反之亦然upper limit上限通信理论communication theory香农定理Shannon，s theorem信道带宽channel bandwidth信号波形signal waveform理想低通滤波器ideal lowpass filter自相关函数autocorrelation function无噪音高斯信道noiseless gaussian channel 通信信道communication channel信息速率information rate信噪比signal-to-noise ratio信道容量channel capacity双边功率谱密度two-side power误码率probability of error那奎斯特采样速率nyquist sampling rate限带高斯信道bandlimited gaussian channel 高斯白噪声White gaussian noiseUnit 3analog模拟quantization量化volt伏特discrete离散的quantum量unipolar单级的polar极性的on-off开关的encoding编码encoder编码器decoder解码器likewise同样的amplitude振幅margin余量trinary三倍的immune免疫的hereafter今后unavailable不能避免的reconstructed重建的evaluation计算span横跨interval间隔compandor压扩器companding压扩uniformly均匀的compress压缩compressor压缩器compressor ratio压缩比quantizer均匀量化器uniform quantizer均匀量化器equivalent相等的serially连续的reshape改造distortion失真interconnect使相互连接filter滤波transition转换，转变PCM: pulse code modulation脉冲编吗调制PAM: pulse amplitude modulation脉冲振幅调制PPM: pulse phase modulation 脉宽调制pulse train脉冲序列amount to等于round off舍入step size步长quantum step size 量化步长positive pulse 正脉冲refer to as称为quantization error 量化步长peak magnitude峰值overall performance总性能crest factor振幅因数root-mean-square均方根Differential Pulse Code Modulation 差分脉冲编码调制adaptive DPCM 自适应DPCMpredictive coding预测编码correspond to相应，符合in practice 实际中in an attempt to 力图词组overall performance总性能crest factor振幅因数nonlinear operation 非线性炒作inverse operation 逆运算RMS均方根PAM脉福调制maximum magnitude最大幅值error intervals误差间隔entropy墒round off四舍五入quantum level量化水平DPCM差分脉码调制正脉冲positive pulse脉冲编码调制PCM量化步长quantum step size 峰值peak magnitude线性函数linear function脉冲序列pulse train均匀量化器uniform quantizer 预测编码predictive coding 压扩器compandor压缩比compression ratio Unit 4selectively仔细挑选formula公式formulas公式化sacrifice牺牲redundant冗余的demonstrate证明orthogonal正交的arbitrarily任意地raw未加工的raw data 原始数据denote 指示Retransmission转发sophisticated复杂的impractical不切实际的parity奇偶校验encoder编码器codeword码字modulo以。
schrodinger方程
Schrödinger方程简介Schrödinger方程是量子力学中最基本的方程之一，描述了量子系统的演化和波函数的行为。
由奥地利物理学家埃尔温·薛定谔于1925年提出，因此也被称为薛定谔方程。
Schrödinger方程是一个偏微分方程，用于描述粒子在势场中的运动。
它以波函数（或称为量子态）作为基本变量，并通过该波函数来计算粒子在不同位置和时间的概率分布。
通过解析或数值方法求解Schrödinger方程，我们可以得到粒子在不同状态下的能量、位置以及其他物理性质。
方程形式Schrödinger方程可以根据系统的性质和假设而有所不同。
下面是一般形式的时间依赖Schrödinger方程：其中，•ψ是波函数，表示粒子在空间中的状态；•i是虚数单位；•ℏ是约化普朗克常数；•∂ψ/∂t表示波函数随时间的变化；•H是哈密顿算符，描述了系统的总能量。
这个方程可以看作是对经典力学中的哈密顿-雅可比方程的量子化。
波函数解释波函数ψ是Schrödinger方程的解，它包含了关于粒子位置和动量的所有信息。
根据波函数的模值平方|ψ|^2，我们可以计算出粒子在不同位置上的概率分布。
这意味着波函数并不直接表示粒子的位置，而是给出了可能找到粒子在某个位置上的概率。
由于波函数是复数，我们无法直接观测到它。
但是通过测量物理量（如能量、动量等），我们可以得到与波函数相关的实际结果。
哈密顿算符哈密顿算符H在Schrödinger方程中起着关键作用。
它描述了系统的总能量，并且根据系统性质和假设有不同形式。
例如，在自由粒子情况下，哈密顿算符可以写为动能项和势能项之和：其中，•T表示动能算符；•V表示势能。
通过将哈密顿算符应用于波函数，我们可以得到Schrödinger方程的具体形式，并进一步求解波函数。
解Schrödinger方程求解Schrödinger方程是理解量子力学中物理系统行为的关键。
组合数学中的Stirling数性质
组合数学中的Stirling数性质Stirling数是组合数学中一系列重要的数学对象，它们在许多领域都有广泛的应用，包括代数学、离散数学、概率论等。
Stirling数分为第一类Stirling数和第二类Stirling数，它们都有一些重要的性质和应用。
一、第一类Stirling数第一类Stirling数，通常用符号${{S(n,k)}}$表示，表示将一个n个元素的集合划分成k个非空的环排列的方法数。
具体来说，${{S(n,k)}}$表示n个元素的集合被划分成k个环排列的方法数。
第一类Stirling数有一些基本的性质。
首先，当n等于k时，${{S(n,k)}}$等于1。
其次，当n大于k时，${{S(n,k)}}$即等于将n个元素划分成k个环排列的方法数。
第一类Stirling数还满足递推关系，即${{S(n,k) = S(n-1,k-1) + (n-1) \cdot S(n-1,k)}}$这个递推关系可以用来计算第一类Stirling数，并且可以用来证明第一类Stirling数的一些性质。
二、第二类Stirling数第二类Stirling数，通常用符号${{S(n,k)}}$表示，表示将一个n个元素的集合划分成k个非空的子集的方法数。
具体来说，${{S(n,k)}}$表示n个元素的集合被划分成k个非空的子集的方法数。
第二类Stirling数也有一些基本的性质。
首先，当n等于k时，${{S(n,k)}}$等于1。
其次，当n小于k时，${{S(n,k)}}$等于0，因为无法将n个元素划分成k个非空的子集。
第二类Stirling数也满足递推关系，即${{S(n,k) = k \cdot S(n-1,k) + S(n-1,k-1)}}$这个递推关系可以用来计算第二类Stirling数，并且可以用来证明第二类Stirling数的一些性质。
三、Stirling数的应用Stirling数在组合数学中具有广泛的应用。
数学专业英语词汇
数学专业英语词汇(S)s admissible s容许的s matrix s矩阵saddle point 鞍点saddle point method 鞍点法salient angle 凸角salient point 折点saltusfunction 跳跃函数sample 样本sample correlation coefficient 样本相关系数sample correlation matrix 样本相关阵sample covariance 样本协方差sample covariance matrix 样本协方差阵sample dispersion 样本方差sample function 样本函数sample mean 样本均值sample median 样本中位数sample moment 样本矩sample point 样本点sample quartile 样本四分位数sample size 样本的大小sample space 样本空间sample survey 样本甸sample unit 抽样单位sample variable 样本变量sample variance 样本方差sampling 抽样sampling distribution 样本分布sampling error 抽样误差sampling fraction 抽样比sampling inspection 抽样检查sampling method 抽样法sampling moment 样本矩sampling ratio 抽样比sampling survey 样本甸sampling unit 抽样单位sarrus rule 萨律法satisfiable 可满足的satisfy 满足saturated set 浸润集saturated subset 浸润子集saturated vertex 饱和顶点saturation curve 饱和曲线scalar 纯量scalar curvature 纯量曲率scalar density 纯量密度scalar field 纯量场scalar flow of vector field 向量场的纯量流scalar matrix 纯量阵scalar multiplication 纯量乘法scalar product 纯量积scalar quantity 纯量scalar triple product 纯量三重积scalar valued function 纯量值函数scalar valued map 纯量值映射scale 图度scale factor 标度因子scale mark 分核度scale parameter 尺度参数scale transformation 标度变换法scalene triangle 不规则三角形scalenohedron 偏三角面体scalenous triangle 不规则三角形scatter 散布scatter diagram 散布图scattered set 无核集scattergram 点状图scattering theory 散射理论schauder theorem 肖德不动点定理scheduling and production planning 生产计划理论scheduling problem 日程计划问题schematic diagram 简图scheme 模式schlicht 单叶的schlicht domain 单叶域schlicht function 单叶函数schlicht mapping 单叶映射schlicht surface 单叶曲面schmidt orthogonalization 施密特正交化法schwartz space 施瓦尔兹空间schwarz formula 施瓦尔兹公式schwarz inequality 施瓦尔兹不等式scope 辖域screw 螺旋体search process 搜她程secant 正割secant curve 正割曲线second axiom of countability 第二可数公理second boundary condition 诺伊曼边界条件second boundary value problem 诺伊曼问题second comparison test 第二比较检验second component 第二分量second countability axiom 第二可数公理second derivative 二次导数second factor 第二因子second fundamental form 第二基本形式second limit theorem 第二极限定理second variation 第二变分secondary diagonal 次对角线secondary extremal 配连极值secondary obstruction 第二障碍section 截口section functor 截面函子section graph 部分等距算子sectional area 截面积sectional curvature 截面曲率sector 扇形sectorial harmonic 扇低函数secular equation 长期方程sedenion 十六元数segment 线段segment of a circle 弓形segment of a curve 弧段segmentation 分割selection 选择self adjoint boundary value problem 自伴边值问题self adjoint differential equation 自伴微分方程self adjoint eigenvalue problem 自伴特盏问题self adjoint element 自伴元self adjoint linear mapping 自伴线性映射self adjointness 自伴性self checking code 自校验代吗self complementary graph 自补图self conjugate 自共轭的self conjugate partition 自共轭分拆self dual 自对偶的self dual category 自对偶范畴self dual group 自对偶群self intersection number 自交数self loop 自身环self orthogonal submodule 自成正交子模self polar curve 自配极曲线self polar tetrahedron 自配极四面形self polar triangle 自配极三角形self tangency 自切self weighting sample 自加权样本selfadjoint linear subspace 自伴线性子空间selfadjoint operator 自伴算子selfconjugate latin square 自共轭拉丁方selfosculating point of curve 曲线的自密切点semantic equivalence 语义等价semantical decision problem 语义判定问题semantical paradox 语义悖论semantics 语义学semi exact 半正合的semi interquartile 半内四分位数间距semi invariant 半不变式semi invariant exterior derivative 半不变外微分semi logarithmic representation 半对数表示semiadditive category 半加性范畴semianalytic set 半解析集合semiangle 半角semiaxis 半轴semibounded 半有界的semibounded operator 半有界算子semicircle 半圆semicircular 半圆的semicircular domain 半圆域semicircular protractor 量角器分度规semicontinuity 半连续性semicontinuous 半连续的semicontinuous function 半连续函数semicontinuum 半连续统semiconvergent series 半收敛级数semicubical parabola 半三次抛物线semidefinite eigenvalue problem 半定特盏问题semidefinite kernel 半定核semidefinite operator 半定算子semidefinite quadratic form 半定二次形式semidefinite variational problem 半定变分问题semidiameter 半径semidirect product 半直积semidiscretization 半离散化semifinite 半有限的semifinite trace 半有限迹semigroup 半群semigroup algebra 半群代数semigroup of operators 算子半群semihereditary ring 半遗传环semilinear 半线性的semilinear mapping 半线性映射semilinear substitution 半线性代换semilinear transformation 半线性变换semilocal ring 半局部环semilogarithmic diagram 半对数图semilogarithmic paper 半对数坐标纸semimagic square 半幻方semimajor axis 半长轴semimajorant 半强函数semimartingale 半semimean axis 半中轴semimetric 半度量semiminor axis 半短轴semiminorant 半弱函数semimodular lattice 半模格semimodularity 半模性seminorm 半范数semiorder 半有序semiordered banach space 半有序巴拿赫空间semiordered set 半有序集semipath 半通路semiperiod 半周期semipolar set 半极集semiprimary ring 半准素环semiprime ideal 半素理想semiprimitive ring 半本原环semiquaternion 半四元数semireductive 半可简约的semireflexive 半自反的semireflexive space 半自反空间semireflexivity 半自反性semiregular point 半正则点semiregular space 半正则空间semiregular topology 半正则拓扑semiscalar product 半纯量积semisimple algebra 半单代数semisimple group 半单群semisimple module 半单模semisimple representation 半单表示semisimple ring 半单环semisimplicial complex 半单纯复形semisimplicial map 半单纯映射semisphere 半球semispherical 半球的semitangent 半切线semitransverse axis 半贯轴semiuniform space 半一致空间sense of rotation 旋转指向senseclass 指向类sensepreserving mapping 保向映射sensitivity analysis 灵敏度分析sentence 命题sentential calculus 命题演算sentential connective 命题联结词sentential function 谓词separability 可分离性separable closure 可分闭包separable degree 分离度separable element 可分元separable extension 可分扩张separable field 可分域separable game 可分对策separable graph 可分图separable polynomial 可分多项式separable sets 可分集separable space 可分空间separable stochastic process 可分随机过程separable topological space 可分拓扑空间separable transcendental extension 可分超越扩张separate 分离separated equation 分离变数方程separated sets 隔离集separated space 分离空间separating edge 分离棱separating plane 分离平面separating transcendence basis 可分超越基separation 分离separation axiom 分离公理separation of the zeros 零点分离separation of variables 分离变量separation principle 分离原理separation relation 分离关系separation theorem 分离定理separator 分隔符sequence 序列sequence convergent almost everywhere 几乎处处收敛列sequence of arcs 弧序列sequence of complex numbers 复数序列sequence of differences 差分序列sequence of distinct points 一一序列sequence of functions 函数序列sequence of iterations 迭代序列sequence of numbers 数列sequence of partial sums 部分和列sequence of points 点序列sequence of sets 集序列sequence of signs 符号序列sequence space 序列空间sequencing 排序sequencing problem 日程计划问题sequent 串联的sequential analysis 序贯分析sequential compactness 列紧性sequential control 顺序控制sequential estimation 序列估计sequential likelihood ratio test 序贯似然比值检验sequential programming 顺序规划sequential sampling 序贯抽样sequential sampling plan 序列抽样法sequential switching circuit 依次转接电路sequential test 序贯检定sequential word function 序贯字函数sequentially compact set 列紧集sequentially complete space 序列完备空间serial correlation 自相关serial correlation coefficient 序列相关系数series 级数series development 级数展开series expansion 级数展开series of functions 函数级数series solution 级数解serpentine 蛇形线service model 服务模型sesquilinear form 半双线性形式set 集set algebra 集代数set function 集函数set of condensation points 凝聚点集set of continuum power 连续统势的集set of limit 极限集合set of lower bounds 下界集set of measure zero 零测度集set of numbers 数集set of points 点集set of sets 集的集set of strategies 策略集set of ternary numbers 三进制数集set of upper bounds 上界集set theoretic addition 集论的加法set theoretic image 集合论的象set theoretic intersection 集论的交set theoretic operation 集论的运算set theoretic proof 集论的证set theoretic union 集论的并集set theoretical 集论的set theory 集论set topology 集论拓扑set valued functor 集值函子set valued mapping 集值映射sexadecimal digit 十六进制数字sexadecimal notation 十六进记法sexadecimal number system 十六进制数系sexagesimal arithmetic 六十进算术sexagesimal system 六十进制sextant 六分仪sextic 六次曲线sextic equation 六次方程shadow 影shadow price 影子价格shape 形状shape theory 形状理论sheaf 层sheaf homomorphism 层同态sheaf of germs of continuous functions 连续函数的芽层sheaf of planes 平面束sheaf theoretic 层理论的shear 剪切shearing modulus 刚性模量shearing strain 切应变shearing stress 切应力sheet 叶sheets of rieman surface 黎曼曲面的叶shell 壳层shift 移动shift operator 位移算子shift register 移位寄存器shilov boundary 希洛夫边界short division 短除法shorten 缩短shortening 缩短shortest 最短线shortest confidence interval 最短置信区间shortest distance 最短距离shortest path 最短道路shortest path problem 最短道路问题shortest route 最短道路shortest route problem 最短道路问题side 边side condition 边条件side elevation 侧视图sieve 筛sieve method 筛法sieve of eratosthenes 厄拉多塞筛sigma additivity 可列可加性sigma algebra 代数sigma compact space 紧空间sigma compactness 紧性sigma complete filter 完备滤子sigma complete lattice 完全铬sigma complete lower semilattice 完备下半格sigma complete semilattice 完备半格sigma discrete family of subsets 子集的离散族sigma finite measure 有限测度sigma function 函数sigma locally finite family of subsets 子集的局部有限族sigma monogenic function 单演函数sigma space 空间sigmacompleteness 完备性sigmafield of sets 集的域sigmalattice 格sigmoid s形曲线sign 符号sign digit 符号数字sign of equality 等号sign of inclusion 包含记号sign of inequality 不等号sign of intersection 相交记号sign of multiplication 乘号sign of permutation 置换的符号sign of subtraction 减法sign of summation 连加号sign of the membership relation 从属关系记号sign of union 并号signal 信号signature of permutation 置换的符号signed numbers 带符号数signed rank test 符号秩检验signed tree 指定符号的树significance level 显著性水平significance of a deviation 偏差的显著性significance test 显著性检定significant 有效的significant digit 有效数字significant figure 有效数字signum 正负号函数similar figures 相似形similar function 相似函数similar matrix 相似矩阵similar ordered set 相似有序集similar region 相似域similar test 相似检验similar triangles 相似三角形similarity 相似similarity principle 相似性原理similarity theorem 相似性定理similarity transformation 相似变换similitude 相似similitude transformation 相似变换simple 单的simple abelian variety 单阿贝耳簇simple algebra 单代数simple arc 简单弧simple branch point 单分枝点simple broken line 单折曲线simple chain 简单链simple character 简单特贞simple circuit 简单围道simple closed curve 简单闭曲线simple component 单分量simple compression 单压缩simple connectedness 单连通性simple connectivity 单连通性simple continued fraction 正则连分数simple continued fraction expansion 简单连分数展开simple convergence 点态收敛simple correlation coefficient 单相关系数simple cycle 简单循环simple domain 单叶域simple eigenvalue 简单特盏simple elongation 单伸长simple event 简单事件simple extension 单扩张simple extension field 单扩张域simple fixed point 单纯不动点simple fraction 普通分数simple function 单叶函数simple graph 简单图simple group 单群simple harmonic motion 简谐运动simple hypothesis 简单假设simple integral 单积分simple intersection point 单纯交点simple iterative method 单迭代法simple lattice 单格simple lie algebra 单李代数simple module 单模simple object 简单对象simple path 简单道路simple pendulum 单摆simple point 单点simple polygon 简单多边形simple polyhedron 简单多面体simple product 简单积simple proper value 简单特盏simple quadrilateral 简单四边形simple random sampling 简单随机样本simple regression 简单回归simple regression coefficient 单回归系数simple ring 单环simple root 单根simple sample 简单样本simple sampling 简单抽样simple series 简单级数simple set 单集simple spectrum 单谱simple surface 简单曲面simple tangent 单切线simple theory of types 简单类型论simple transcendental extension 单超越扩张simplex 单形simplex method 单形法simplex multiplier 单形乘数simplex tableau 单形表simplex theorem 单形定理simplicial approximation 单纯逼近simplicial cell 单纯胞腔simplicial chainmapping 单纯链映射simplicial cochain complex 单纯上链复形simplicial cohomology group 单纯上同岛simplicial complex 单纯复形simplicial homology 单纯同调simplicial map 单形映射simplicial mapping cylinder 单形映射柱simplicial pair 单形对simplicialapproximation theorem 单纯逼近定理simplification 简化simplified fraction 简化分数simplified newton method 简化牛顿法simply connected group 单连通群simply connected region 单连通区域simply connected spatial domain 单连通空间域simply convergent filter 单收敛滤子simply ordered group 全有序群simply periodic function 单周期函数simplyconnected domain 单连通域simpson rule 辛卜生法则simulation 模拟simultaneity 同时性simultaneous confidence intervals 联合置信区间simultaneous diagonalization 同时对角化simultaneous differential equation 联立微分方程simultaneous differential equations 联立微分方程simultaneous equations 方程组simultaneous estimation 联立估计simultaneous invariant 联立不变式simultaneous substitution 同时代入sine 正弦函数sine curve 正弦曲线sine function 正弦函数sine integral 正弦积分sine integral function 正弦积分函数sine law 正弦定律sine spiral 正弦螺线sine theorem 正弦定理sine wave 正弦波single 单的single address 单地址的single address code 一地址代码single address instruction 单地址指令single address system 单地址系统single factor method 单因子法single step method 单步法single step process 单步法single valued 单值的single valued analytic function 单值解析函数single valued correspondence 单值对应single valued function 单值函数single valued operation 单值运算single valued relation 单值关系single valuedness 单值性singlevalued mapping 单值映射singly periodic function 单周期函数singular 奇异的singular automorphism 奇异自同构singular bivariate normal distribution 奇异二元正态分布singular boundary 奇异边界singular boundary point 奇异边界点singular chain 奇异链singular chain complex 奇异链复形singular cohomology group 奇异上同岛singular complex 奇异复形singular conic 奇二次曲线singular correspondence 奇对应singular cycle 连续循环singular distribution 退化分布singular element 奇元素singular elliptic function 奇异椭圆函数singular function 奇异函数singular function of bounded variation 有界变差奇异函数singular graph 奇异图singular homology 奇异下同调singular homology class 奇异同掂singular homology group 连续同岛singular integral 奇解singular integral element 奇异积分元素singular integral equation 奇异积分方程singular kernel 奇核singular line element 奇异线素singular linear operator 奇异线性算子singular linear transformation 奇异线性变换singular locus 奇轨迹singular mapping 奇异映射singular matrix 退化阵singular operator 奇异算子singular ordinal 特异序数singular part 奇异部分singular plane 奇异平面singular point 奇点singular proposition 特称命题singular quadric 奇异二次曲面singular series 奇异级数singular solution 奇解singular space 奇异空间singular submodule 奇子模singular subspace 奇异子空间singular surface 奇曲面singular transformation 奇异变换singular value 奇异值singular variational problem 奇异变分问题singular vector 奇异向量singularity 奇点sink 收点sinusoid 正弦摆线sinusoidal 正弦的sinusoidal function 正弦函数sinusoidal law 正弦定律sinusoidal spiral 正弦螺线situation 情况size 样本的大小skeleton 骨架skew curve 空间曲线skew derivation 斜微分skew determinant 斜对称行列式skew distribution 偏斜分布skew field 非交换域skew hermitian form 斜埃尔米德型skew hermitian matrix 斜埃尔米德矩阵skew lines 偏斜线skew position 歪扭位置skew quadrilateral 挠四边形skew surface 非可展直纹曲面skew symmetric 反对称的skew symmetric determinant 斜对称行列式skew symmetric matrix 斜对称矩阵skew symmetric tensor 斜对称张量slack variable 松弛变量slide rule 计算尺sliding vector 滑动向量slitregion 裂纹区域slope 斜率slope function 斜率函数slope intercept form 斜截式slope line 倾斜线slope of a curve 曲线的斜率slowly increasing sequence 缓增序列small circle 小圆small inductive dimension 小归纳维数small sample 小样本small set 小集smallest element 最小元smash 收缩smooth curve 平滑曲线smooth map 光滑映射smooth morphism 光滑射smooth projective plane curve 光滑射影平面曲线smoothing 光滑化smoothness 光滑度sobolev embedding theorem 水列夫嵌入定理sobolev space 水列夫空间sojourn time 逗留时间solenoidal group 螺线群solenoidal vector field 螺线向量场solid 立体;固体solid angle 立体角solid geometry 立体几何solid n sphere n维球体solid of revolution 旋转体soliton 孤立子soluble 可解的solution 解solution curve 积分曲线solution domain 解域solution formula 解公式solution set of equation 方程的解集solution space 解空间solution surface 积分曲面solution tree 解树解答树solution vector 解向量solvability 可解性solvable 可解的solvable equation 可解方程solvable group 可解群solvable ideal 可解理想solve 解sorter 分类器source 发点source free vector field 无源向量场source function 格林函数source of field 场源source program 源程序space 空间space coordinates 空间坐标space curve 空间曲线space integral 体积积分space like manifold 类空廖space of left cosets 左傍系空间space of matrices 矩阵空间space of quaternions 四元数空间space of right cosets 右傍系空间space quadratic transformation 空间二次变换space region 空间区域span 生成spanning tree 最大树生成树sparse matrix 稀巯阵spatial 空间的spatial co ordinate 空间坐标spatial isomorphism 空间同构spatial point 空间点special divisor class 特殊除子类special functional equations 特殊函数方程special functions 特殊函数special group 特殊群special homology manifold 特殊同滴special jordan algebra 特殊约当代数special linear group 特殊线性群special linear homogeneous group 特殊线性齐次群special orthogonal group 特殊正交群special purpose computer 专用计算机special representation 特殊表示special unitarian group 特殊酉群special valuation 特殊赋值specialization 特定化specific 特殊的specific address 绝对地址specific heat 比热specificity 特性spectral analysis of operators 算子的谱分析spectral decomposition 谱表示spectral density 谱线密度spectral distribution 谱分布spectral distribution curve 光谱分布曲线spectral function 谱函数spectral functor 谱函子spectral geometry 谱几何spectral integral 谱积分spectral invariant 谱不变量spectral line 谱线spectral mapping theorem 谱映射定理spectral measure 谱测度spectral multiplicity 谱重度spectral norm 谱模spectral point 谱点spectral property 谱性质spectral radius 谱半径spectral representation 谱表示spectral sequence 谱序列spectral set 谱集spectral space 谱空间spectral subspace 谱子空间spectral synthesis 谱综合spectral theorem 谱定理spectrum 谱spectrum of a matrix 阵的谱speed 速度sphere 球sphere of contact 相切球面sphere of the inversion 反演球spherical 球形的spherical angle 球面角spherical astronomy 球面天文学spherical asymptote 球面渐近线spherical bessel function 球贝塞耳函数spherical cap 球冠spherical cohomology class 球上同掂spherical coordinates 球极坐标spherical curvature 球面曲率spherical curve 球面曲线spherical cyclic curve 球面循环曲线spherical derivative 球面导数spherical domain 球面域spherical epicycloid 球面外摆线spherical excess 球面角盈spherical fiber 球面纤维spherical function 球函数spherical geometry 球面几何学spherical harmonic function 球面低函数spherical harmonics 球面低函数spherical helix 球面螺旋线spherical homology class 球面同掂spherical image 球面象spherical indicatrix 球面指标spherical indicatrix of binormal 副法线球面指标spherical indicatrix of principal normal 吱线球面指标spherical indicatrix of tangents 切线球面指标spherical mean 球中值spherical neighborhood 球形邻域spherical parallelogram 球面平行四边形spherical polar coordinates 球极坐标spherical polygon 球面多边形spherical pyramid 球面棱锥spherical sector 球心角体spherical segment 球截形spherical shell 球壳spherical space 球面空间spherical surface 球面spherical triangle 球面三角形spherical triangular coordinates 球面三角坐标spherical trigonometry 球面三角学spherics 球面几何学sphero quartic 球面四次曲线spheroid 回转椭圆面spheroidal coordinates 球体坐标spheroidal function 球体低函数spheroidal harmonic 球体低函数spheroidal wave function 球体波函数spinode 第一类尖点spinor 旋子spinor field 旋量场spinor genus 旋量狂spinor group 旋子群spinor representation 旋量表示spiral 螺线spiral point 螺线极点spiral surface 螺面spline 样条spline function 样条函数spline interpolation 样条内插splitting 分裂splittingfield 分裂域spur 迹squarable 可平方的square 正方形square bracket 方括弧square contingency 平方列联square deviation 平方偏差square matrix 方阵square measure 平方测度square mesh 正方网格square root 平方根square root transformation 平方根变换square summable function 平方可积函数squareintegrable function 平方可积函数stability 稳定性stability conditions 稳定条件stability criterion 稳定性判据stability group 稳定群stability number 稳定数stability region 稳定区stabilization 稳定stabilization method 稳定法stabilizer 迷向群stable convergence 稳定收敛stable equilibrium 稳定平衡stable homotopy group 稳定同伦群stable manifold 稳定廖stable orbit 稳定轨道stable point 稳定点stable process 稳定过程stable set 稳定集合stable solution 稳定解stable state 稳定状态stalk 茎standard complex 标准复形standard deviation 标准差standard equation 标准方程standard form 标准型standard isobaric surfaces 标准等压面standard n simplex 标准n单形standard regression coefficient 标准回归系数standard simplex 标准单形standard topology 标准拓扑standardization 标准化standardize 使标准化standardized normal distribution 标准化正态分布standardized normal variate 标准化正态变量standardized variable 标准化变量standardized variate 标准化变量star body 星形体star covering 星形覆盖star neighborhood 星形邻域star of a simplex 单形的星形star region 拟星形域star shaped domain 拟星形域starlike domain 拟星形域starlike mapping 拟星形映射starlike set 拟星集starrefinement 星型加细start time 开始时间state 状态state coordinates 状态坐标state function 状态函数state of equilibrium 平衡状态state region 状态区域state space 状态空间state variable 状态变数state vector 状态向量statement 命题static model 静态模型statics 静力学stationary 平稳的stationary curve 平稳曲线stationary distribution 平稳分布stationary flow 平稳流定常流stationary function 平稳函数stationary osculating plane 平稳密切面stationary point 平稳点stationary point process 平稳点过程stationary process 平稳随机过程stationary state 定态stationary time series 平稳时间序列stationary value 平稳值statistic 统计量statistical 统计的statistical accuracy 统计精确度statistical analysis 统计分析statistical data analysis 统计数据分析statistical decision function 统计判决函数statistical decision problem 统计判决问题statistical decision procedure 统计判决程序statistical decision process 统计判决过程statistical distribution 统计分布statistical ergodic theorem 平均遍历定理statistical error 统计误差statistical estimate 估计statistical estimation 统计估计statistical hypothesis 统计假设statistical hypothesis testing 统计假设检验statistical inference 统计推断statistical mechanics 统计力学statistical method 平均法statistical model 随机性模型statistical optimization 统计最佳化statistical quality control 统计质量管理statistical thermodynamics 统计热力学statistics 统计学statistics of extreme values 极值统计statistics of extremes 极值统计steady 平稳的steady state 定态steenrod algebra 斯丁洛特代数steenrod operation 斯丁洛特运算steepest descent method 最速下降法steering program 痔序steering routine 痔序steinberg group 斯坦因伯格群step 步长step function 阶梯函数step length 步长stepping stone method 起脚石法steradian 球面度stereoangle 立体角stereogram 立体频数stereographic projection 球极平面射影stereography 立体平画法stereometry 立体几何stiefel whitney class 斯蒂费尔惠特尼类stieltjes integral 斯蒂尔吉斯积分stirling formula 斯特林公式stochastic 随机的stochastic approximation 随机逼近stochastic automaton 随机自动机stochastic connection 随机联络stochastic control 随机控制stochastic dependence 随机相依stochastic differential equation 随机微分方程stochastic differentiation 随机微分法stochastic dynamic model 随机动态模型stochastic dynamic programming 随机动态规划stochastic filtering 随机滤波stochastic game 随机对策stochastic independence 随机独立stochastic integral 随机积分stochastic integral equation 随机积分方程stochastic integration 随机积分stochastic matrix 随机阵stochastic maximum principle 随机极大原理stochastic model 随机性模型stochastic optimization 随机规划法stochastic process 随机过程stochastic programming 随机规划法stochastic variable 随机变数stochastically dependent event 随机相依事件stochastically independent event 随机独立事件stokes integral theorem 斯托克斯定理stokes theorem 斯托克斯定理stop time 停止时间stopped process 停止过程stopping rule 停止规则storage 存储store 存储器store capacity 存储容量store cell 存储单元straight 直的straight angle 平角straighten 弄平straightline 直线strain 应变strain tensor 应变张量strategic equivalence 策略等价性strategic model 策略模型strategy 策略strategy of a game 对策的策略strategy polygon 策略多角形stratification 层化stratified sample 分层样本stratified sampling 分层抽样stratified selection 分层抽样stratum 层stream function 怜数streamline 吝strength 强度stress 应力stress ellipsoid 应力椭球stress function 应力函数stress of a body 体应力stress tensor 应力张量stretching transformation 伸缩变换strict convexity 严格凸性strict decreasing 严格递减strict epimorphism 严格满射strict extremum 严格极值strict implication 严格蕴涵strict increasing 严格递增strict inductive limit 严格归纳极限strict inequality 严格不等式strict isotonicity 严格保序性strict isotony 严格保序性strict minimum 严格极小strict morphism 严格射strict solution 严格解strict upper bound 严格上界strictly concave function 严格凹函数strictly convex function 严格凸函数strictly convex space 严格凸空间strictly decreasing 严格减少的strictly dominant strategy 严格优策略strictly finer topology 严格较细拓扑strictly increasing 严格递增的strictly increasing mapping 严格递增映射strictly lower triangular matrix 严格下三角矩阵strictly monotone decreasing 严格单递减的strictly monotone increasing 严格单递增的strictly monotonic function 严格单弹数strictly monotonic mapping 严格单党射strictly monotonic sequence 严格单凋列strictly normed linear space 严格赋范线性空间strictly positive measure 严格正测度strictly upper triangular matrix 严格上三角矩阵strip 带strip of conditional convergence 条件收敛带strip region 带形区域strong component 强分支strong convergence 强收敛strong convergent operator 强收敛算子strong deformation retract 强形变收缩核strong discontinuity 强间断strong dual 强对偶strong ellipticity 强椭圆型strong epimorphism 强满射strong extremum 强相对极值strong inequality 严格不等式strong law of large numbers 强大数定律strong markov process 强马尔可夫过程strong monomorphism 强单射strong operator topology 强算子拓扑strong solution 强解strong summability 强可和性strong topology 强拓扑strongly coercive 强强制的strongly connected compactum 强连通紧统strongly connected graph 强连通图strongly continuous map 强连续映射strongly elliptic operator 强椭圆算子strongly inaccessible cardinal 强不可达基数strongly inaccessible ordinal 强不可达序数strongly mixing transformation 强混合变换strongly paracompact space 强仿紧空间strongly plurisubharmonic function 强多重次低函数strongly pseudoconvex domain 强伪凸域strophoid 环诉structural morphism 结构射structural stability 构造稳定性structure 结构structure constant 构造常数structure equations 结构方程structure formula 结构公式structure function 结构函数structure group 结构群structure morphism 结构射structure sheaf 结构层structure theorem 结构定理structured complex 结构复形student t distribution 学生t分布sturm chain 斯图谟链sturm liouville eigenvalue problem 斯图谟刘维尔特盏问题subadditive function 次加性函数subadditive functional 次加性泛函subadditive interval function 次加性区间函数subadditive set function 次加性集函数subadditivity 次可加性subalgebra 子代数subautomaton 子自动机subbase 子基subbundle 子丛subcategory 子范畴subchain 子链subclass 子类subcoalgebra 子上代数subcomplex 子复形subcontinuum 子连续统subcovering 子覆盖subdeterminant 子行列式subdifferential 次微分subdirect sum 次直和subdivide 细分。
斯特林数（Stirlingnumber）
斯特林数（Stirlingnumber ）在组合数学，Stirling 数可指两类数，第⼀类Stirling 数和第⼆类 Stirling 数，都是由18世纪数学家 James Stirling 提出的。
Stirling 数有两种，第⼀类和第⼆类Stirling 数第⼀类斯特林数：形如n m，也写作 s (n ,k )组合意义：s (n ,k ) 表⽰吧n 个数分成k 组，每组是⼀个环，求分成的⽅案数。
也就是⼀个轮⼦，怎么转都是⼀样的，如：1,2,3,4 和 4,1,2,3 只算⼀种⽅案。
递推式：s (n +1,2)=s (n ,1)+s (n ,2)⋅n即要么⾃成⼀个环，要么加⼊其它k 个环，可以插⼊n −1个位置。
（每两个数之间）当然边界条件00=1性质：1. s (n ,1)=(n −1)!2. s (n ,2)=(n −1)!×∑n −1i =11i3. ∑n i =0s (n ,k )=n !证明：1. 显然，我们把n 个元素排列起来，有n !种可能，⾸尾相接即可得到⼀个环。
这⾥⾯每种情况重复了n 次，因为可以旋转n 次，所以除以n ，得到s (n ,1)=(n −1)!。
2. 通过数学归纳法可以证明。
s (n +1,2)=s (n ,1)+s (n ,2)⋅n=(n −1)!+n (n −1)!n −1∑i =11i=(n −1)!+n !n −1∑i =11i =n !n +n !n −1∑i =11i =n !n∑i =11i3. 这⾥有⼀个巧妙地“算两次”⽅法。
⾸先构造⼀个问题，求n 个数的所有排列。
⾸先⽤乘法原理直接得出结论，ans =n !。
我们知道，对于⼀个排列对应⼀个置换，即：12...n a 1a 2...a n把这个置换中的上下对应位置连边，可以得到许多的环。
由于排列和置换是⼀⼀对应的，所以我们要求排列的个数，就是求⽤n 个元素组成环的⽅案数，所以我们枚举环的个数:[][]()n!=n∑k=1s(n,k)由于我们有s(n,0)=0，所以也可以写成：n∑k=0s(n,k)=n !第⼆类斯特林数：形如nk，也写作S(n,k)组合意义：S(n,k) 表⽰吧n个数分成k组，组内⽆序，每组没有区别。
带非定域项schrodinger方程的jost解
带非定域项schrodinger方程的jost解在量子力学中，Schrödinger方程是研究量子系统的一个常用方程。
对于一维系统，Schrödinger方程可以写成如下形式：\[H\psi(x) = E\psi(x)\]其中H是系统的哈密顿算符，ψ(x)是波函数，E是能量。
通常情况下，Schrödinger方程是一个定域的方程，即和位置x有关系的项是局部的。
然而，在一些情况下，Schrödinger方程可以包含非定域项，即和位置x有关系的项是非局部的。
这种情况下的Schrödinger方程被称为带非定域项的Schrödinger方程。
\[H\psi(x) = E\psi(x) + V(x)F[\psi](x)\]其中V(x)是定域势能项，F[\psi](x)是非定域项，可能包含了波函数ψ在整个空间上的积分或导数等。
在研究带非定域项的Schrödinger方程时，一个重要的概念是Jost 解。
Jost解是一种特殊的解，它们具有一些特殊的性质。
Jost解可以通过一个无穷级数的形式来表示：\[\psi(x) = \sum_{n=0}^{\infty}a_ne^{ikx}\]其中k是一个实数，a_n是系数。
对于带非定域项的Schrödinger方程，Jost解可以写成如下形式：\[\psi(x) = \sum_{n=0}^{\infty}a_ne^{ikx} + \int dkF[k]\phi(k)e^{ikx}\]其中F[k]是一个与波矢k相关的函数，用来描述非定域项的影响。
φ(k)是一个与波矢k相关的函数，它是另外一个方程的特解，该方程不包含非定域项。
这个方程通常被称为约化方程。
Jost解的一个重要性质是它们满足正交归一条件。
即，Jost解的内积满足如下关系：\[\int dx \psi^*(x)\psi(x) = 1\]根据这个性质，可以推导出Jost解的系数a_n满足一组正交归一条件：\[\int dx \psi_n^*(x)\psi_m(x) = \delta_{nm}\]其中δ_nm是Kronecker delta符号，当n=m时为1，否则为0。
数学专有名词英文
数学专业英语词汇英汉对照1 概率论与数理统计词汇英汉对照表Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析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 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-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column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数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crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test 柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层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 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的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logistic 逻辑的lost function 损失函数Mmain 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model 模型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Nnegative relationship 负相关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Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数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predict 预测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probit analysis 概率分析proportion 比例Qqadratic 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，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数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 秩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reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本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separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激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 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验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total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数2. 最优化方法词汇英汉对照表Aactive 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Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Iindependent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange 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 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解Mmedium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method 多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性Sscalar 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solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点stationary point 驻点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination 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Uunconstrained 无约束的under-determined system 负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵3 样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline 分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点Ccoefficient/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Ddegree of freedom 自由度dimension 维数end conditions 约束条件Iinput argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squares approximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数Ppoint/points 数据点rational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条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Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量weight/weights 权重4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件eigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition 推广的Neumann边界条件geometry 几何形状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Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition 混合边界条件NNeuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 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Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的。
离散数学中英文名词对照表
离散数学中英⽂名词对照表离散数学中英⽂名词对照表外⽂中⽂AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作⽤addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平⾯algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent ⼏乎相等的alternating group 三次交代群annihilator 零化⼦antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合⼦asymmetric 不对称的(⾮对称的)atom 原⼦atomic formula 原⼦公式augmenting digeon hole principle 加强的鸽⼦笼原理augmenting path 可增路automorphism ⾃同构automorphism group of graph 图的⾃同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity ⽆穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation ⼆元关系binary symmetric channel ⼆进制对称信道binomial coefficient ⼆项式系数binomial theorem ⼆项式定理binomial transform ⼆项式变换bipartite graph ⼆分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对⾓线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡⼉积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中⼼characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number ⾊数chromatic polynomial ⾊多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着⾊coloring problem 着⾊问题combination number 组合数combination with repetation 可重组合common factor 公因⼦commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(⼦图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全⼆分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分⼦）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分⽀connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（⽆⽭盾性）continuum 连续统contraction of graph 图的收缩contradiction ⽭盾式（永假式）contravariant functor 反变函⼦coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通⽤映射countably infinite set 可列⽆限集（可列集）covariant functor (共变)函⼦covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对⾓线函⼦diameter 直径digraph 有向图dilemma ⼆难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选⾔三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring ⼦除环divison ring 除环divisor (factor) 因⼦domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离⼼率edge chromatic number 边⾊数edge coloring 边着⾊edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独⽴数eigenvalue of graph 图的特征值elementary divisor ideal 初等因⼦理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism ⾃同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧⼏⾥德算法Euclid domain 欧⽒整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多⾯体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推⼴规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number ⼴义 Fibonacci 数extended Lucas number ⼴义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface ⾯factor 因⼦factorable 可因⼦化的factorization 因⼦分解faithful (full) functor 忠实（完满）函⼦Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤⼦finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限⽣成模first order theory with equality 带符号的⼀阶系统five-color theorem 五⾊定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函⼦four-color theorem(conjecture) 四⾊定理（猜想）F-reduced product F-归纳积free element ⾃由元free monoid ⾃由单元半群free occurrence ⾃由出现free R-module ⾃由R-模free variable ⾃由变元free-?-algebra ⾃由?代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (G?del-Bernays-von Neumann system)GBN系统generalized petersen graph ⼴义 petersen 图generating function ⽣成函数generating procedure ⽣成过程generator ⽣成⼦（⽣成元）generator matrix ⽣成矩阵genus 亏格girth （腰）围长G?del completeness theorem G?del 完全性定理golden section number 黄⾦分割数（黄⾦分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第⼀类边⾊图graph of second class for edge coloring 第⼆类边⾊图graph rank 图秩graph sequence 图序列greatest common factor 最⼤公因⼦greatest element 最⼤元（素）Grelling paradox Grelling 悖论Gr?tzsch graph Gr?tzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函⼦homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超⼋⾯体图hypothelical syllogism 假⾔三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等⾃然变换imbedding 嵌⼊immediate predcessor 直接先⾏immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree ⼊次（⼊度）independent 独⽴的independent number 独⽴数independent set 独⽴集independent transcendental element 独⽴超越元素index 指数individual variable 个体变元induced subgraph 导出⼦图infinite extension ⽆限扩域infinite group ⽆限群infinite set ⽆限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函⼦injective (one to one mapping) 单射（内射）inner face 内⾯inner neighbour set 内（⼊）邻集integral domain 整环integral subdomain ⼦整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因⼦invariant factor ideal 不变因⼦理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆⾃然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换⼦）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-⾊图k-connected graph k-连通图k-critical graph k-⾊临界图k-edge chromatic graph k-边⾊图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman ⼥⽣问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin ⽅lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最⼩元least upper bound 上确界（最⼩上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因⼦left adjoint functor 左伴随函⼦left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻⽅many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表⽰maximal element 极⼤元maximal ideal 极⼤理想maximal outerplanar graph 极⼤外平⾯图maximal planar graph 极⼤平⾯图maximum matching 最⼤匹配maxterm 极⼤项（基本析取式）maxterm normal form(conjunctive normal form) 极⼤项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极⼩项minimal κ-connected graph 极⼩κ-连通图minimal polynomial 极⼩多项式Minimanoff paradox Minimanoff 悖论minimum distance 最⼩距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极⼩项（基本合取式）minterm normal form(disjunctive normal form)极⼩项范式（析取范式）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁⽅Nn-ary operation n-元运算n-ary product n-元积natural deduction system ⾃然推理系统natural isomorphism ⾃然同构natural transformation ⾃然变换neighbour set 邻集next state 下⼀个状态next state transition function 状态转移函数non-associative algebra ⾮结合代数non-standard logic ⾮标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规⼦群（不变⼦群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁⽅orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外⾯outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平⾯图Ppancycle graph 泛圈图parallelism 平⾏parallelism class 平⾏类parity-check code 奇偶校验码parity-check equation 奇偶校验⽅程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽⼦笼原理planar graph （可）平⾯图plane graph 平⾯图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表⽰法polynomial ring 多项式环possible world 可能世界power functor 幂函⼦power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着⾊proper factor 真正因⼦proper filter 真滤⼦proper subgroup 真⼦群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive ⾃反的regular graph 正则图regular representation 正则表⽰relation matrix 关系矩阵replacement theorem 替换定理representation 表⽰representation functor 可表⽰函⼦restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函⼦right cancellable 右可消的right factor 右因⼦right zero divison 右零因⼦ring 环ring of endomorphism ⾃同态环ring with unity element 有单元的环R-linear independence R-线性⽆关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满⾜的saturated 饱和的scope 辖域section 截⼝self-complement graph ⾃补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent ⽮列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原⼦）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category ⼩范畴smallest element 最⼩元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph ⽣成⼦图spanning tree ⽣成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系⼤集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra ⼦代数subcategory ⼦范畴subdirect product ⼦直积subdivison of graph 图的细分subfield ⼦域subformula ⼦公式subdivision of graph 图的细分subgraph ⼦图subgroup ⼦群sub-module ⼦模subrelation ⼦关系subring ⼦环sub-semigroup ⼦半群subset ⼦集substitution theorem 代⼊定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关⽹络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验⼦syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system ?3 , ?n , ??0 , ??系统?3 , ?n , ??0 , ??system L 公理系统 Lsystem ?公理系统?system L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system ?n 公理系统?nsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重⾔式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全⾊数total chromatic number conjecture 全⾊数猜想total coloring 全着⾊total graph 全图total matrix ring 全⽅阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅⾏商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡⼦群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤⼦ultrapower 超幂ultraproduct 超积unary operation ⼀元运算unary relation ⼀元关系underlying graph 基础图undesignated truth value ⾮特指值undirected graph ⽆向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯⼀分解unique factorization domain (Gauss domain) 唯⼀分解整域unique k-colorable graph 唯⼀k着⾊unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通⽤结构universal enveloping algebra 通⽤包络代数universal generalization 全称推⼴规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph ⽆标号图untorsion ⽆扭模upper (lower) bound 上（下）界useful equivalent 常⽤等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff) word 字Zzero divison 零因⼦zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。
A family of Schrodinger operators whose spectrum is an interval
a r Xiv:809.3434v1[mat h.SP ]19Se p28A F AMILY OF SCHR ¨ODINGER OPERATORS WHOSE SPECTRUM IS AN INTER V AL HELGE KR ¨UGER Abstract.By approximation,I show that the spectrum of the Schr¨o dinger operator with potential V (n )=f (n ρ(mod 1))for f continuous and ρ>0,ρ/∈N is an interval. 1.Introduction In this short note,I wish to describe a family of Schr¨o dinger operators on l 2(N )whose spectrum is an interval.To set the stage introduce for a bounded sequence V :N →R and u ∈l 2(N )the Schr¨o dinger operator H V defined by (H V u )(n )=u (n +1)+u (n −1)+V (n )u (n )n ≥2(1.1)(H V u )(1)=u (2)+V (1)u (1).We will denote by σ(V )the spectrum of the operator H V .It is well known that if V (n )is a sequence of independent identically distributed random variables with distribution µsatisfying supp(µ)=[a,b ],we have that the spectrum σ(V )is [a −2,b +2].For the Almost–Mathieu Operator with potential (1.2)V λ,α,θ(n )=2λcos(2π(nα+θ)),where λ>0,α/∈Q ,the set σ(V λ,α,θ)is a Cantor set [2].Bourgain conjectured in [4],that if one considers the potential (1.3)V λ,α,x,y (n )=λcos n (n −1)Date :September 19,2008.2000Mathematics Subject Classification.Primary 81Q10;Secondary 47B36.Key words and phrases.Schr¨o dinger Operators,Spectrum.H.K.was supported by NSF grant DMS–0800100.12H.KR¨UGERLemma1.2.Let r≥0be an integer and r<ρ<r+1.Given anyα,θ,K≥1,ǫ>0,a0,...,a r,there exists an integer n≥1such that(1.6)sup|k|≤K |α(n+k)ρ+θ−rj=0a j k j|≤ǫ.We will prove this lemma in the next section.Proof of Theorem1.1.By Lemma1.2,we canfind for any x∈[0,1)a sequence n l such thatsup|k|≤l|α(n l+k)ρ+θ−x|≤1/l.Hence,the sequence V l(n)=V(n−n l)converges pointwise to f(x).The claim now follows from strong convergence.It is remarkable that combined with the Last–Simon semicontinuity of the abso-lutely continuous spectrum[6],one also obtains the following resultTheorem1.3.For r≥0an integer,r<ρ<r+1,and f a continuous function on T,introduce the set B r(f)as(1.7)B r(f)= a0,...,a rσac(f(r j=0a j k j)).Then(1.8)σac(f(αnρ+θ))⊆B r(f).Hereσac(V)denotes the absolutely continuous spectrum of H V.We note that for r=0,we have that(1.9)B0(f)=[−2+max(f),2−min(f)].Under additional regularity assumptions on f,Stolz has shown in[7]that we have equality in(1.8).Furthermore note that(1.10)B r+1(f)⊆B r(f).In[6],Last and Simon have stated the following conjecture(1.11)B1(2λcos(2π.))=∅forλ>0.They stated this in poetic terms as Does Hofstadter’s Butterfly have wings?.The best positive result in this direction as far as I know,is by Bourgain [3]showing(1.12)|B1(2λcos(2π.))|→0,λ→0.It is an interesting question if Theorem1.1holds forρ≥2an integer.In the particular case of f(x)=2λcos(2πx),ρ=2,this would follow from Bourgain’s conjecture.However,there is also the following negative evidence.Consider the skew-shift T:T2→T2given byT(x,y)=(x+α,x+y),(1.13)T n(x,y)=(x+nα,n(n−1)A FAMILY OF SCHR¨ODINGER OPERATORS WHOSE SPECTRUM IS AN INTER V AL3 whereα/∈Q.Then Avila,Bochi,and Damanik have shown that for generic continuous f:T2→R,the spectrumσ(f(T n(x,y)))is a Cantor set.So,it is not clear what to expect in this case.As afinal remark,let me comment on a slight extension.If one replaces V by the following family of potentialsV(n)=f(αnρ+Kk=1αk nβk),whereβk<ρandαk are any numbers,then Theorem1.1and1.3remain still valid.2.Proof of Lemma1.2Let in the following r be an integer such that r<ρ<r+1.By Taylor expansion, we have that(2.1)α(n+k)ρ=rj=0x j(n)k j+αρ...(ρ−r)j!nρ−j.We nowfirst note the following lemmaLemma2.1.For any K≥1andǫ>0,there exists a N0(K,ǫ)such that(2.3)|α(n+k)ρ−rj=0x j(n)k j|≤ǫfor|k|≤K and n≥N0(K,ǫ).Proof.This follows from(2.1)and thatρ−r−1<0.We furthermore need the following consequence of Theorem1.8in[5].Recall for this,that a sequence x(n)is uniformly distributed in T r+1if for any0≤a j<b j≤1, j=0,...,r we have that(2.4)limn→∞14H.KR¨UGERProof of Lemma1.2.By Lemma2.1,there exists a N0such that|α(n+k)ρ−rj=0x j(n)k j|≤ǫ2rK l ,ǫ。
分数阶Schr(o¨)dinger算子的自伴性与本质谱
分数阶Schr（o|¨）dinger算子的自伴性与本质谱Schrodinger方程理论由于其直观的物理背景及其应用价值,一直以来都是分析数学的中心课题之一,而其自伴性与谱的研究作为Schrodinger方程理论的基本问题也是受到数学工作者们的普遍关注。
对二阶Schrodinger算子而言,关于它的研究已经产生了很多丰硕的成果,方法也比较成熟,当然自伴性与谱的研究也很完善。
比如针对自伴性的研究,有一些常用的方法,如扰动的方法、Kato不等式的方法、张量积分解的方法等。
作为二阶经典情形的自然延拓,分数阶Schrodinger 算子近来也引起了数学工作者们的兴趣,但由于缺乏必要的技术手段以及更直观的物理背景,关于它的研究要比二阶情形困难得多,但在数学工作者们的努力下,也取得了一定的成果,如方程的光滑估计等。
当然还有许多问题尚待研究,针对分数阶Schrodinger算子,我们主要研究其自伴性和本质谱。
针对分数阶Schrodinger算子(-△)α/2+V,我们主要是在总结前人关于二阶Schrodinger算子研究成果的基础上,根据不同的位势类对其自伴性与本质谱进行研究。
本文主要分为四个部分：首先,第一章为绪论,主要是对Schrodinger方程的背景及发展历程作了一个简单的介绍；其次,第二章对二阶经典情形Schrodinger算子的自伴性和本质谱的一些经典结论及方法进行引述；再次,第三章主要研究了在两类不同位势下分数阶Schrodinger算子的自伴性；最后一部分,第四章主要对分数阶Schrodinger算子的本质谱的一个性质进行了探讨。
其中在对本文的主要结论进行证明时,很大程度上是将经典情形的方法进行适当的推广,从而将其应用到分数阶上来。
在主要内容之后,还给出了本文结论的不足之处以及还可以继续改进和探讨的地方。
第一类Stirling数的递推公式的算子证明
第一类Stirling数的递推公式的算子证明徐春雷【摘要】第一类Stirling数与排列的一种组合化表示--圈结构密切相关。
无符号的第一类Stirling数是双射π：S→S中圈的个数。
本文通过引入一类算子来证明已知的第一类Stirling数的递推公式。
%There is a deeply relationship between stirling number of the first kind and the cycle structure of permutation which is the representive from combinatorics aspect. The unsigned stirling number of the first kind is the number of cycle for bijection π:S→S .We apply a functional to prove a recursion for the stirling number of the first kind in this paper.【期刊名称】《内蒙古民族大学学报（自然科学版）》【年(卷),期】2013(000)004【总页数】2页(P384-385)【关键词】第一类Stirling数;算子;递推公式【作者】徐春雷【作者单位】内蒙古民族大学计算机科学与技术学院，内蒙古通辽 028043; 内蒙古民族大学离散数学研究所，内蒙古通辽 028043【正文语种】中文【中图分类】O157.5计数组合学是计算集合中元素个数的学科.既然任何数学问题的本质上都可以归结为计数问题，那么上述定义本身并未包含很多该学科的信息.对于真正的计数问题，集合中的元素通常具有相当简单的组合学定义而且几乎没有什么附加结构.包括计数组合学在内，组合学近些年的发展速度十分惊人，其中一个重要原因是，组合数学作为计算机科学及其相关领域的一种工具发挥着基础性的作用〔1〕.本文讨论的第一类Stirling数与排列的一种组合化表示——圈结构密切相关.集合和重集的排列是计数组合学中内容最丰富的课题之一.一个根本原因在于排列具有各式各样的组合化表示，例如集合的排列可以表示为字或者是函数.如果将集合排列π看作一个双射π:S→S，那么很自然的对每个x∈S,我们会考虑序列x,π(x),π2(x),···.可以肯定，它一定回到x.这样，存在唯一的l≥1使πl(x)=x而元素互不相同.我们称序列为π的一个长为l的圈.我们认为圈是等价的.S的每个元素都出现在π的唯一圈中，因此我们可以把π视为一些不相交的圈的并或者互异的圈C1,C2,···,Ck的乘积，记为C1,C2,···,Ck.定义c(n,k)为恰好含有k个圈的π=σn的个数.数被称为第一类Stirling数，而c(n,k)被称为无符号的第一类Stirling数.第一类Stirling数还有如下的定义：定义1第一类Stirling数是满足如下公式的数s(n,k):多项式xn，[x]n，[x]n分别称为标准多项式，下降阶乘，上升阶乘.而序列{xn:n∈N0}，{[x]n:n∈N0}，{[x]n:n∈N0}是多项式序列的几个例子.还有其它类型的多项式序列.定义2一个多项式序列是一个实多项式族{pn(x):n∈N0}，使得对于所有的n，deg(pn(x))=n，以及deg(p(x))=0当且仅当p(x)≡a≠0，并且如果p(x)≡0时，deg(p(x))=-1.我们用R[x]表示以x为变量的全体实多项式的集合.具有一般加法和标量乘法的R[x]是R上的一个向量空间.从一个多项式序列的成员有严格递增的次数这一事实我们立即推出以下结论.命题3〔2〕任意一个多项式序列是向量空间R[x]的一个基.一个线性算子可以唯一地被在基上的赋值所确定.特别地，对于一个多项式序列pn(x)和一个实数序列an，有一个算子L:R[x]→R，满足对于全部的n，Lpn(x)=an.本文引入一类算子证明了[2]中的第一类Stirling数的递推公式.定理4对于所有的n,k≥0:（i）对于n＞0，s(0,0)=1，s(n,0)=1（ii）s(n+1,k)=s(n,k-1)-ns(n,k)，（iii）s(n+1,k)=证明考虑R[x]上的一个算子L:其中是Kronecker符号由第一类Stirling数的定义（定义1）可知从递归公式得综上有s(n+1,k)=s(n,k-1)-ns(n,k).为了证明（iii），由递归公式于是从而〔1〕Richard P.Stanley.Enumerative Combinatorics〔M〕.vol（I）.Camgbridge：Camgbridge press，1997.〔2〕Martin binatorial Theory.Reprint of the 1979-ed.Berlin 〔M〕.Berlin：Springer-Verlag Berlin Heidelberg，1997.〔3〕吉日木图.关于第一类Stirling数的一般表达式及其几个性质〔J〕.内蒙古民族师范学院学报，1995（1）：13-15.〔4〕Peter J.Cameron.组合数学—专题、技术与算法〔M〕.北京：人民邮电出版社，2009.〔5〕赵凌琪.关于一种二元集合计数问题的研究〔J〕.内蒙古民族大学学报（自然科学版），2009，24（6）：605-607.。
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Jou rnal of M athem atical R esearch&Expo siti onV o l.18,N o.1,11216,Feb ruary1998Con ti nuous Spectru m M easure of a Class of Schroβd i ngerOperatorsΞS hang P engj ian(D ep t.of A pp l.M ath.,N o rthern J iao tong U n iversity,Beijing100044)Abstract W e con sider the discrete Sch roβdinger operato r acting on l2(Z)w ith the po ten2 tial V n,the sequence con sisting of k+1sym bo ls:{0,1,2,…,k},and p rove that it ex2h ib its pu rely con tinuou s spectrum.Keywords discrete,Sch roβdinger operato r,po ten tial,con tinuou s spectrum.Cla ssif ica tion AM S(1991)47H12 CCL O177.61.I n troductionW e study one-di m en si onal discrete Sch roβdinger op erato r acting on l2(Z):(H U)n=U n+1+U n-1+ΚV n U n(1) w here V n is a sequence con sisting of k+1sym bo ls:{0,1,…,k}.V ari ou s such one di m en si onal m odels have been investigated[1,2,3],nam ely w ith random o r al m o st p eri odic po ten tial.A lso, m uch w o rk s have been devo ted to studying the Sch roβdinger op erato r w ith a po ten tial generated by the F ibonacci sequence.T he discovery of quasi-crystals gives p hysical grounds to study of such k ind of po ten tial w h ich can be though t as one-di m en si onal m odel[4,5,6].In[1],Kohm o to, Kadanoff and T ang p ropo sed a m odel w here V n is given byV n=ςA(nΞ+Η)(2) w hereςA is the characteristic functi on of an in terval A on the circle and the argum en t ofςA has to be understood m odu lo1.W hereΞw as the go lden m ean,A w as the in terval(-Ξ3,Ξ2]andΗw as 0.T hese au tho rs con jectu red that the sp ectral m easu re is singu lar con tinuou s.In th is p ap er,w e show that the op erato r(1)w ith the po ten tial V n,con sisting of k+1sym bo ls:{0,1,2,…,k}, exh ib its p u rely con tinuou s sp ectrum.2.Nota tion and Result2.1 O n con tinued fracti on exp an si on sΞR eceived July19,1994. Since w e w ill m ake an ex ten sive u se of the con tinued fracti on exp an si on (CFE )of the irra 2ti onal num ber ,w e m u st first recall som e basic no tati on .A ny irrati onal num ber Ξ>0can be w rit 2ten in a un ique w ay asΞ=a 0+1a 1+1a 2+…=[a 0;a 1,a 2,…],(3)w here the in tegers a i ′s are called the p artial quo tien ts of CFE .If w e truncate th is exp an si on ,w e ob tain a sequence of rati onal app rox i m an ts1a 1+1a 2+…+1a i=p iq i(4)called the p rinci p al convergen ts of the CFE .T heir num erato rs and denom inato rs bo th obey linear recu rsive relati on s asp i =a i p i -1+p i -2, q i =a i q i -1+q i -2.(5)In additi on ,w e have fo r i Ε0p i q i +1-p i +1q i +1=1.(6)2.2 Sequences generated by a circle m apL et u s first define characteristic functi on w ith ∃a given num ber betw een 0and 1:ς∃(x )=In t (x )-In t (x -∃)=1if 0ΦF rac (x )<∃,0if ∃ΦF rac (x )<1..(7)H ere In t (x )and F rac (x )=x -In t (x )deno te in teger and fracti onal p arts of x ,resp ectively .It is easily seen that ςA (x )is a 1-p eri odic functi on .Con sider sequence{ς∃(n Ξ+Η)},con sisting of tw o sym bo ls 0and 1.T h is sequence m ay also be generated by a circle m ap ,nam ely un it circum 2ference ,as show n by F ig .1F ig 1L et u s define an in term ediate labelling sequence on the circle (un it circum ference ).A se 2quence L n is associated w ith the sequence of num bers Αn =F rac (Η+n Ξ)by the ru leL n =k ,if0ΦΑn <x 1,k -1,if x 1ΦΑn <x 2,1,if x k -1ΦΑn <x k ,0,if x k ΦΑn <1,(8)w here 0<x 1<x 2<…<x k < 1.T h is sequence con sists of k +1sym bo ls :0,1,2,…,k .A nd it m ay also be generated by a circle m ap (nam ely unit circum ference ),as show n by F ig .2.T he sequence {ς∃(Η+n Ξ)}is recovered by choo sing k =1,x 1=∃F ig 22.3 R esu ltsW e study discrete Sch ro βdinger op erato rs (1).In the fo llow ing theo rem s V n is the sequence L nin (8),Ξan irrati onal num ber .W e u se the no tati on H (Η,Κ)to take in to accoun t the dep endence of H on Κand Η.Theore m F or L ebesque a l m ost every Ξand f or any x 1,x 2,…,x k (0<x 1<x 2<…<x k ),then f or L ebesque a l m ost every the sp ectra l m easu re of H (Η,Κ)is con tinuous f or any va lue of Κ.}3.L e mma s and Proof of Theore mL e mma 1 L et U be a solu tion of the eig enva lue equa tion H U =EU and supp ose tha t there ex ists anin teg er r such tha t Vn +ir=V n f or i =-1,1and 0<n Φr (H yp othesis H 1),then w e have f or a ll E :m ax (‖Y -r ‖,‖Y r ‖,‖Y 2r ‖)Ε‖Y 0‖ 2,(9)w here Y n is the vector (U n ,U n +1)T .Proof Fo r any so lu ti on U of H U =EU one can w riteU n +1+U n -1+ΚV n U n =EU n .B y the u se of tran sfer m atrices ,w e haveY n =T (n )Y n -1,(10)w hereY n=U nU n+1, T(n)=01-1E-ΚV n.(11)L etM(n)=T(n)T(n-1)…T(1),(12) thenY n=M(n)Y0.(13) H ypo thesis H1i m p lies thatY r=M(r)Y0Y0=M(r)Y-rY2r=M(r)Y r.(14)Since the determ inan t of M(r)is1,the m atrix M(r)satisfies:M2(r)-tr(M(r)) M(r)+I=0,(15) w here tr( )is the trace of m atrix( ),I is2×2un it m atrix.i) T he case tr(M(r)) Φ 1.A pp ly the characteristic equati on(15)to any vecto r x∈C2,and take the no rm,th is yields‖x‖=‖t r(M(r))M(r)x-M2(x)x‖Φ‖M(r)x‖+‖M2(r)x‖.L et x=Y0=U0U1,and con sider equalities(14),one gets‖Y0‖Φ2M ax{‖Y r‖,‖Y2r‖}.(16) ii) T he case t r(M(r)) > 1.T he equati on(15)i m p lies thatM(r)=1tr(M(r))M2(r)+1tr(M(r))I.Fo r any x∈C2,w e haveM(r)x=1tr(M(r))M2(r)x+1tr(M(r))x.T ake the no rm‖M(r)x‖=‖1tr(M(r))M2(r)x+1t r(M(r))x‖Φ‖M2(r)x‖+‖x‖.L et x=M-1(r)Y0and con sider equalities(14),one gets‖Y0‖Φ2M ax{‖Y r‖,‖Y-r‖}.(17)T hu s L emm a 1is a con sequence of (16)and (17).L e mma 2 F or a l m ost every Ξthere ex ists f or a l m ost every Ηa sequence r n (Η)f or w h ich the H y 2p othesis H1of L e m m a 1is f u lf illed .Proof L et p n q nbe the n thp rinci p al convergen t of the CFE of the irrati onal num ber Ξso that one getsq n +1=a n q n +q n -1, p n +1=a n p n +p n -1,w here a n is the n thp artial quo tien t of CFE of Ξ.T he rate of convergence of thep nq nto Ξis given by Ξ-p n q n Φ p n +1q n +1-p n q n =1q n q n +1Φ1a n q 2n.(18)N ow the set E (n )of value of Ηsuch that H ypo thesis H 1is fu lfilled w ith r =q n is given byE (n )=E 1(n )∩E 2(n )∩…∩E k (n )=∩k i =1E i (n ),E i (n )={Η:inf 0<m Φq n( F rac (m Ξ+Η)-x i )>q n Ξ-p nq n},(19)w here x i (i =1,2,3,…,k )are the endpo in ts in (8),see F ig .2.Indeed ,(19)exp resses the fact that F rac (m Ξ+Η)fo r m in [1,q n ]m u st be distan t from x i by at least the p hase sh ife co rrespond 2ing to a tran slati on of ±q n in o rder to en su re the exact rep etiti on of the po ten tial con sidered in L emm a 1.C learly w e haveΛ(E i (n ))Ε1-2q 2n Ξ-p nq n,(20)w here Λis the L ebesgue m easu re on [0,1],thu s u sing (18):Λ(E (n ))Ε1-2(k +1)q 2n Ξ-p nq nΕ1-2(k +1)a n.(21)N ow ,it is know n [7]that fo r a .e .Ξone gets :li m sup n →∞a n =+∞,(22)w h ich yields thatli m sup n →∞Λ(E (n ))=1,w henceΛ(li m sup n →∞E (n ))=1.Con sequen tly ,fo r a .e .Ξ,there ex ists fo r Λa .e .Ηan infin ite sequence r k (Η)=q n k (Η)such that H 1is fu lfilled .Proof of Theore m It suffices to show that any so lu ti on U of the eigenvalue equati on H U =EU canno t decay at infin ity . F rom L emm a2w e know that there ex ists,fo r a.e.Ξand fo r a.e.Η,an infin ite increasingsequence r n satisfying the H ypo thesis H1of L emm a1.In th is case m ax(‖Y-rn ‖,‖Y rn‖,‖Y2rn‖)does no t tend to o as n tend to+∞,therefo re H(Η,Κ)canno t have eigenvecto rs in l2 (Z)fo r a.e.Ηand allΚ.References[1] M.Kohmo to,L.Kadanof,C.T ang,L ocaliz ation p roble m in one d i m ension:M app ing and escap e,Phys.R ev.L ett.,50(1983),1870.[2] M.Kohmo to,Y.Oono,Cantor sp ectrum f or an alm ost p eriod ic sch roβd ing er equation and a dy nam2ical m ap,Phys.L ett.,145(1984),120A.[3] S.O stlund,R.Pandit,D.R ang,H.J.Schellnhuber,E.D.Siggia,O ne-d i m ensional sch roβd ing erequation w ith an alm ost p eriod ic p otential,Phys.R ev.L ett.,50(1983),1873.[4] A.Ya.Go rdon,U sp.M ath.N auk31(1976),257.[5] J.A vron,B.Si m on,Bull.Am.M ath.Soc.6(1982),81.[6] B.Si m on,A dv.A pp l.M ath.,3(1982),463.[7] A.Ya.Kh inςn,Continued fracti ons,T he U niversity of Ch icago P ress,1964,60.一类Sch roβd inger算子的连续谱测度商 朋 见(北方交通大学应用数学系,北京100044)摘 要 考虑l2(Z)上的离散Sch roβdinger算子,其势V n是一个由k+1个符号{0,1,2,…,k}构成的序列.证明了它具有纯连续谱.。