Sobolev orthogonal polynomials in two variables and second order partial differential equations

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有界对称域上Ω代数中的Jackson定理

有界对称域上Ω代数中的Jackson定理王志军;陈英伟【摘要】对多复变Cn中有界对称域Ω上球代数的中心逼近性质进行了研究.通过建立多项式偏差估计,最终获得了Jackson型定理.【期刊名称】《河北大学学报（自然科学版）》【年(卷),期】2014(034)004【总页数】5页(P342-346)【关键词】Ω代数;Jackson定理;光滑模;有界对称域【作者】王志军;陈英伟【作者单位】河北经贸大学数学与统计学学院,河北石家庄050061;河北经贸大学数学与统计学学院,河北石家庄050061【正文语种】中文【中图分类】O174.41在逼近论中，中心逼近定理即为Jackson定理和Bernstein定理，其揭示了插值空间理论和整函数理论的紧密联系.Jackson定理［1］是逼近论中处理函数关于多项式偏差的重要结果.经典的结果主要是在连续函数及周期连续函数，然而，通过全纯扩展或考虑周期连续函数在单位圆盘中的调和延拓，自然会考虑单位圆盘上函数的逼近定理.基于此，Lipschitz函数类中的Jackson定理已拓展到复平面上的Jordan域［2］及单位圆盘上Qp空间［3］中.更多的结果可见文献［1，4－8］.对于多复函数空间，最近，多复变专家利用向量形式把新的Jackson定理延拓到多圆柱上的一些全纯函数空间［9］，例如Bergman型空间，Hardy空间［10］.笔者也曾推广至另一些空间［11－12］.令U表示复平面C上的单位圆盘.Sewell考虑了圆盘代数上的多项式逼近A（U）：＝H（U）∩C定理1［13］对任f∈A（U），k∈N，有其中Mk为次数不超过k的多项式集.问题是如何利用高阶光滑模在更广的定义域上建立更广义Jackson定理.本文中，Ω表示Cn中的有界对称域.本文目的是引入球代数A（Ω）函数空间，拓展了Jackson定理，并得到Lipschitz函数类的正逼近结果.定义1 代数A（Ω）为Ω中的全纯函数集合且连续开拓至Ω边界，模为在Ω代数中，函数逼近采用了如下高阶光滑模.定义2 令χ为Ω上具有半模‖·‖x的函数空间.对任f∈χ，δ＞0，r∈N，f的r阶光滑模定义3 定义Ek（f，χ）：＝inf‖f－Mk‖χ为最佳多项式逼近，其中下确界取为遍历次数不超过k的多项式集Mk.本文中，C表示与k和z无关的正常数，不同的地方取值可能不同.1 多项式逼近为构造最佳逼近多项式，引入上的一种复测度，其和Jackson核联系紧密.固定a＞0，ρ∈（0，1］和k∈N.该复测度定义为测度（φ）为上单位化测度，可从稍后引理1（f≡1和r＝0）知.当ρ＝1，记dvk：＝（φ）.其全变差可表示成广义Jackson核形式令r∈N∪｛0｝，引入重要积分算子将作为最佳逼近多项式.这里（φ）为广义Jackson核引理1 令为f∈H（Ω）的齐次展开式，则其中证明：对任固定ρ∈（0，1），取变量变换λ＝ρeiφ，可得对任｜ω｜＜1，由二项展开，易得故，上述积分可分为2部分注意到第2项在单位圆盘｜λ｜＜1上全纯，由留数定理可知其在｜λ｜＝ρ上积分为0.从第1项可得易知gm（λ）在单位圆盘上除原点外全纯，由留数定理得为计算留数，利用f的Taylor展开和二项展开可得式（2）右侧的Laurent展开从而结合式（3）和式（1），知故此即证明了ρ∈（0，1）的情况.当ρ＝1时可通过极限得到.引理得证.引理2［1］令k，β∈N，1）存在常数Cl，β（k）满足2）存在常数Cβ满足引理3［1］设0＜δ，λ＜＋∞，f∈A（U），则引理4 设0＜δ，λ＜＋∞，f∈A（Ω），则证明：对任ζ∈∂Ω，考虑f的slice函数fζ，其中fζ（w）＝f（wζ），w∈U，则引理3可得由定义2，对任ζ∈∂Ω，ωr（δ，fζ，A（U））≤ωr（δ，f，A（Ω）），故再由定义2得2 偏差估计回忆上的测度，对任0＜ρ＜1，η＞0和a＞0，由定义，知定理2 对任k－1∈N，f∈H（Ω），有证明：对任ρ∈（0，1），由引理1得其中从而其中易知［14－15］，对任0＜η≤1都存在非负常数C（η）满足：对任h∈H和0＜r＜1，有在式（5）中取h（λ）＝g（λ，z）并结合式（4），得其中需要指出定理2对情况0＜η≤1也是成立的，其一般情况也具有其应用价值.3 Jackson定理现在给出本文的主要结果（即A（Ω）中的Jackson定理）如下.定理3 设f∈A（Ω），则对任k－1，r∈N，有证明：对任ζ∈∂Ω，由的定义和定理1知故即注意到Ik［f］（z）为次数不超过k－1的多项式.再由引理4（取λ＝k｜φ｜和δ＝1／k）可得故，结合引理2可得故得证.最后给出球代数空间A（Ω）中的一类子空间具体的Jackson不等式.定义4Lipschitz型空间Lipγ（A（Ω）），0＜γ≤1，包含所有全纯函数f∈A（Ω）且满足这里L＞0被称为Lipschitz常数.易知对任f（z）∈A（Ω）和0＜γ≤1，推论1 设f∈Lipγ（A（Ω）），则对任k－1∈N，r＝1，有参考文献：［1] DEVORE R A，LORENTZ G G.Constructive approximation［M］.Berlin：Springer－Verlag，1993.［2] ANDERSSON J M，HINKKANEN A，LESLEY F D.On theorems ofJackson and Bernstein type in the complex plane［J］.Constr Approx，1988，4：307－319.［3] CHEN Yingwei，REN Guangbin.Jackson's rheorem in Qpspaces ［J］.Science in China Series A：Mathematics，2010，53：367－372. ［4] ANDRIEVSII V.Harmonic version of Jackson's theorem in the complex plane［J］.J Approx Theory，1997，90：224－234.［5] KRYAKIN Y，TREBELS W.q－moduli of continuity in Hp（D），p＞0，and an inequality of Hardy and Littlewood［J］.J Approx Theory，2002，115：238－259.［6] KOSTOVA V A.On a generalization of Jackson's theorem in Rm ［J］.Math Balkanica：N S，1998，12：109－118.［7] REN Guangbin，CHEN Yingwei.Gradient estimates and Jackson's theorem in Qμspaces related to measures［J］.J Approx Theory，2008，155：97－110.［8] ZYGMUND A.Trigonometric series［M］.Cambridge：Cambridge Univ Press，1959.［9] REN Guangbin，WANG Mingzhi.Holomorphic Jackson's theorems in polydiscs［J］.J Approx Theory，2005，134：175－198.［10] RUDIN W.Function theory in polydiscs［M］.New York：Benjamin，1969.［11] 陈英伟，王志军，刘玉军.Hardy型空间Aμ中Jackson定理［J］.河北师范大学学报：自然科学版，2013，37（2）：113－118.CHEN Yingwei，WANG Zhijun，LIU Yujun.Jackson's theorem in Hardy type Aμspaces［J］.Journal of Hebei Normal University：Natural Science Edition，2013，37（2）：113－118.［12] CHEN Yingwei，WANG Zhijun，DONG Wenlei.Jackson's theorem in Hardy－Sobolev type space in the unit polydiscs［J］.ICICA2012，CCIS，2012，307：360－367.［13] SEWELL W E.Degree of approximation by polynomials in the complex domain［M］.Princeton：Prinston Univ.Press，1942.［14] GARNETT J B.Bounded analytic functions［M］.New York：Springer，2007.［15] STOROŽENKOÈA.Approximation of functions of class Hp，0＜p≤1［J］.Math USSR－Sb，1978，34：527－545.。

ORTHOGONAL CURVILINEAR COORDINATES

Ar
r Aθ r Aϕ sin θ +
1 ∂ r 2 sin θ ∂θ
+
∂2φ ∂ y2
1 ∂2φ r 2 ∂ϕ 2
∂ ∇ 2 φ = 1 ∂r r 2 ∂φ ∂r r2 2φ + 2 12 ∂ 2 r sin θ ∂ϕ
sin θ ∂φ ∂θ
∇φ =
+
+
∇ ·A=
∂ Ax ∂x
+
∂ ∂x
∂ Ay ∂y
+
∂ Az ∂z
∇ ·A= A + ∂∂zz ∇ ×A= ∇2φ =
2φ +∂ 2 ∂z
1 ∂(r Ar ) r ∂r
1 ∂ Aϕ r ∂ϕ
i
j
Curl. . .
∇ ×A=
Ax Laplacian. . . ∇2φ =
∂2φ ∂ x2
Ay +
Consider an infinitesimal volume enclosed by the surfaces u, v, w, u + du, v + dv, and w + dw (Figure 5).
Figure 5. The surface P RS ≡ u = constant, and the face of the curvilinear figure immediately opposite this is u + du = constant, etc. In terms of these surface constants P = P(u, v, w) Q = Q(u + du, v, w) R = R(u, v + dv, w) S = S(u, v, w + dw) where h1 , h2 , and h3 are functions of u, v, and w. • In rectangular Cartesians i, j, k h1 = 1, ˆ a ∂ ∂ =ˆ , i h1 ∂u ∂x ˆ ˆ ˆ • In cylindrical Cartesians r, θ, z h1 = 1, ˆ a ∂ ∂ ˆ =r , h1 ∂u ∂r ˆ ˆ ˆ • In spherical coordinates r, θ , ψ h1 = 1, ˆ ∂ a ∂ ˆ =r , h1 ∂u ∂r h2 = r, ˆ ˆ θ ∂ b ∂ = , h2 ∂v r ∂θ h3 = r sin θ ˆ ˆ c ∂ ψ ∂ = h3 ∂w r sin θ ∂ψ h2 = 1, ˆ b ∂ ∂ =ˆ , j h2 ∂v ∂y h2 = r, ˆ ˆ θ ∂ b ∂ = , h2 ∂v r ∂θ h3 = 1. ˆ c ∂ ˆ ∂ =k . h3 ∂w ∂z h3 = 1. ˆ c ∂ ∂ ˆ =z . h3 ∂w ∂z and and and P Q = h1 du P R = h2 dv P S = h3 dw

Isolation and Crystal Structure of 2—Bromoaldisin

Isolation and Crystal Structure of 2—Bromoaldisin 徐效华; 陈晓; 等【期刊名称】《《结构化学》》【年(卷),期】2001(020)003【摘要】The crystal structure of the title compound (C8H7BrN2O2,Mr=243.07) was isolated from the marine sponge Phacellia fusca Schmidt collected from the South China Sea. Its crystal structure was determined by single-crystal X-ray diffraction. The crystal is orthorhombic with space group Pbca, a=12.9952(8), b=7.4479(5), c=18.598(1) ?, V=1800.1(2) ?3, Z=8, Dc=1.794g/cm3, (=0.71073 ?, ( (MoK()=4.533mm-1, F(000)=960. The structrue was refined to R=0.0349, wR(F2)=0.0925 for 1589 reflections with I > 2((I). X-ray diffraction analysis reveals that the title compound has one five-membered pyrrole ring and one seven-membered azepin ring. There are two intermolecular hydrogen bonds between two molecules.【总页数】3页(P173-175)【作者】徐效华; 陈晓; 等【作者单位】InstituteandStateKeyLaboratoryofElemento－OrganicChemistry NankaiUniversity Tianjin300071 China【正文语种】中文【中图分类】O626【相关文献】1.Isolation, Crystal Structure and Antitussive Activity of 9S,9aS-neotuberostemonine [J], WU Yi;YE Qing-Mei;LIU Jing;XU Wei;ZHU Zi-Rong;JIANG Ren-Wang2.Isolation, Crystal Structure and Na+/K+-ATPase Inhibitory Activity of 1β-Hydroxydigitoxigenin [J], XU Yun-Hui;XU Jian;JIANG Xue-Yang;CHEN Zhi-Hua;XIE Zi-Jian;JIANG Ren-Wang;FENG Feng3.Isolation and Crystal Structure of Ent-kaurane Diterpenes from Rubus corchorifolius L.f. [J], CHEN Xue-Xiang;HUANG Jian-Xi;OU Yang-Wen;LIU Xiao-Juan;ZHOU Li-Ping;CAO Yong4.Isolation, Crystal Structure,and Anti-inflammatory Activity of Sakuranetin from Populus tomentosa [J], LIU Hai-Ping;CHAO Zhi-Mao;TAN Zhi-Gao;WU Xiao-Yi;WANG Chun;SUN Wen5.Isolation and Crystal Structure of 2-Bromoaldisin [J], 徐效化; 陈晓; 廖仁安; 谢庆兰因版权原因，仅展示原文概要，查看原文内容请购买。

台湾国立交通大学

台湾国⽴交通⼤学数学视频数学视频Calculus I 台湾国⽴交通⼤学 Michael Fuchs⽼師 36集（点击进⼊我的淘宝店）Calculus II 台湾国⽴交通⼤学 Michael Fuchs⽼師 29集（点击进⼊我的淘宝店）Chapter1　Functions and Model1-5　Exponential Functions1-6　Inverse Functions and LogarithmsChapter2　Limits and Derivatives2-2　The Limit of a Function2-4　The Precise Definition of a Limit2-3　Calculating Limits Using the Limit Laws2-6　Limits at Infinity; Horizontal Asymptotes2-5　Continuity2-8　Derivatives2-9　The Derivative as a FunctionChapter3　Differentiation Rules3-1　Derivatives of Polynomials and Exponential Functions3-2　The Product and Quotient Rules3-4　Derivatives of Trigonometric Functions3-5　The Chain Rule3-6 Implicit Differentiation3-8 Derivatives of Logarithmic Functions3-10　Related Rates3-7 Higher Derivatives3-11　Linear Approximations and DifferentialsChapter4　Applications of Differation4-1　Maximum and Minimum Values4-2　The Mean Value Theorem4-3 How Derivatives Affect the Shape of a Graph4-4 Indeterminate Forms a nd L’Hospital’s Rule4-7　Optimization Problems4-5　Summary of Curve Sketching4-10　AntiderivativesChapter5　Integrals5-1　Areas and Distances5-2　The Definite Integral5-3　The Fundamental Theorem of Calculus5-4　Indefinite Integrals and the Total Change Theorem5-5　The Substitution Rule5-6　The Logarithm Defined as an IntegralChapter6　Applications of Integration6-1　Areas between Curves6-2　Volumes6-3　Volumes be Cylindrical ShellsChapter7　Techniques of Integration7-1　Integration by Parts7-2　Trigonometric Integrals7-3　Trigonometric Substitution7-4　Integration of Rational Functions by Partial Fractions7-8　Improper Integrals7-7　Approximate IntegrationChapter8　Further Applications of Integration8-1　Arc Length8-2　Area of a Surface of RevolutionChapter10　Parametric Equations and Polar Coordinates10-1　Curves Defined by Parametric Equations10-2　Calculus with Parametric Curves10-3　Polar Coordinates10-4　Areas and Lengths in Polar Coordinates微积分(⼀) 台湾国⽴交通⼤学莊重⽼師　24集（点击进⼊我的淘宝店）微积分(⼆)台湾国⽴交通⼤学莊重⽼師 24集（点击进⼊我的淘宝店）課程章節第⼀章Functions and Model第⼆章Limits and derivatives第三章Differentiation Rules第四章The Properties of Gases第五章Integrals第六章Applications of Integration第七章Techniques of Integration第⼋章Further Applications of Integration第⼗章Parametric Equations and Polar Coordinates第⼗⼀章Infinite Sequences and Series第⼗⼆章Vectors and the Geometry of Space第⼗三章Vector Functions第⼗四章Partial Derivatives第⼗五章Multiple Integrals⾼等微积分(⼀)台湾国⽴交通⼤学⽩啟光⽼師　29集（点击进⼊我的淘宝店）⾼等微积分(⼆) 台湾国⽴交通⼤学　⽩啟光⽼師　27集（点击进⼊我的淘宝店）第⼀章The Real and Complex Number SystemsFields Axioms, Order Axioms　Completeness Axioms第⼆章Basic TopologyCardinality of SetsMetric SpacesCompact SetsConnected Sets第三章Numerical Sequences and SeriesConvergent SequencesCauchy SequencesUpper and Lower LimitsSeries of Nonnegative TermsThe Root and Ratio TestAbsolute Convergence, Rearrangements第四章ContinuityLimits of Functions and Continuous FunctionsContinuity and CompactnessContinuity and Connectednessdiscontinuities, Infinite Limits and Limits at Infinity第五章Differentiation　The Derivative of a Real Function, Mean Value TheoremL’Hopital’s RuleTaylor’s TheoremDifferentiation of Vector-valued Functions第六章The Riemann-Stieltjes Integral　Definition and Existence of the IntegralProperties of the IntegralIntegration and DifferentiationIntegration and Differentiation第六章The Riemann-Stieltjes IntegralIntegration and Differentiation第七章Sequence and Series of FunctionsSequence and Series of Functions --- the Main ProblemUniform Convergence and ContinuityUniform Convergence and IntegrationUniform Convergence and DifferentiationEquicontinuous Family of FunctionsThe Stone-Weierstrass Theorem第⼋章Some Special FunctionsPower seriesSome Special FunctionsFourier SeriesThe Gamma Function第九章Functions of several variablesFunction of Several VariablesFunction of Several Variables：DifferentiationFunction of Several Variables：DifferentiationThe Inverse Function TheoremThe Implicit Function TheoremThe Rank TheoremDeterminantsDifferentiation of Integrals偏微分⽅程(⼀) 台湾国⽴交通⼤学林琦焜⽼师 3.8GB （点击进⼊我的淘宝店）偏微分⽅程(⼆) 台湾国⽴交通⼤学林琦焜⽼师 3.4GB （点击进⼊我的淘宝店）内容纲要第⼀章 The Single First-Order Equation1-1 Introduction Partial differential equations occur throughout mathematics. In this part we will give some examples1-2 Examples1-3 Analytic Solution and Approximation methods in a simple example 1-st order linear example1-4 Quasilinear Equation The concept of characteristic1-5 The Cauchy Problem for the Quasilinear-linear Equations1-6 Examples Solved problems1-7 The general first-order equation for a function of two variables characteristic curves, envelope1-8 The Cauchy Problem characteristic curves, envelope1-9 Solutions generated as envelopes第⼆章Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables2-1 Characteristics for Linear and Quasilinear Second-Order Equations Characteristic2-2 Propagation of Singularity Characteristic curve and singularity2-3 The Linear Second-Order Equation classification of 2nd order equation2-4 The One-Dimensional Wave Equation dAlembert formula, dimond law, Fourier series2-5 System of First-Order Equations Canonical form, Characteristic polynominal2-6 A Quasi-linear System and Simple Waves Concept of simple wave第三章 Characteristic Manifolds and Cauchy Problem3-1 Natation of Laurent Schwartz Multi-index notation3-2 The Cauchy Problem Characteristic matrix, characteristic form3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem Local existence of solutions of the non-characteristic 3-4 The Lagrange-Green Identity Gauss divergence theorem3-5 The Uniqueness Theorem of Ho ren Uniqueness of analytic partial differential equations3-6 Distribution Solutions Introdution of Laurent Schwartzs theory of distribution (generalized function)第四章 The Laplace Equation4-1 Greens Identity, Fundamental Solutions, and Poissons Equation Dirichlet problem, Neumann problem, spherical symmetry, mean value theorem, Poisson formula4-2 The Maximal Principle harmonic and subharmonic functions4-3 The Dirichlet Problem, Greens Function, and Poisson Formula Symmetric point, Poisson kernel4-4 Perrons method Existence proof of the Dirichlet problem4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods Functional analysis, Riesz representation theorem, Dirichlet integra第五章 Hyperbolic Equations in Higher Dimensions5-1 The Wave Equation in n-Dimensional Space(1) The method of sphereical means(2) Hadmards method of descent(3) Duhamels principle and the general Cauchy problem(4) mixed problem5-2 Higher-Order Hyperbolic Equations with Constant Coefficients(1) Standard form of the initial-value problem(2) solution by Fourier transform,(3) solution of a mixed problem by Fourier transform5-3 Symmetric Hyperbolic System(1) The basic energy inequality(2)Finite difference method(3) Schauder method第六章 Higher-Order Elliptic Equations with Constant Coefficients6-1 The Fundamental Solution for Odd n Travelling wave6-2 The Dirichlet Problem Lax-Milgram theorem, Garding inequality6-3 Sobolev Space Weak solution and Hibert space第七章 Parabolic Equations7-1 The Heat Equation Self-Similarity, Heat kernel, maximum principle7-2 The Initial-Value Problem for General Second-Order Parabolic Equations(1) Finite difference and maximum principle(2) Existence of Initial Value Problem第⼋章 H. Lewys Example of a Linear Equation without Solutions8-1 Brief introduction of Functional Analysis Hilbert and Banach spaces, projection theorem, Leray-Schauder theorem8-2 Semigroups of linear operator Generation, representation and spectral properties8-3 Perturbations and Approximations The Trotter theorem8-4 The abstract Cauchy Problem Basic theory8-5 Application to linear partial differential equations Parabolic equation, Wave equation and Schrodinger equation8-6 Applications to nonlinear partial differential equations KdV equation, nonlinear heat equation, nonmlinear Schrodinger equation变分学导论应⽤数学系林琦焜⽼师台湾国⽴交通⼤学 2GB （点击进⼊我的淘宝店）内容纲要第⼀章变分学之历史名题1.1 Bernoulli 最速下降曲线1.2 最⼩表⾯积的迴转体1.3 Plateau问题(最⼩曲⾯)1.4 等周长问题1.5 古典⼒学之问题第⼆章 Euler- Lagrange⽅程2.1 变分之原理2.2 折射定律与最速下降曲线2.3 ⼴义座标2.4 Dirichlet 原理与最⼩曲⾯2.5 Lagrange乘⼦与等周问题2.6 Euler-Lagrage ⽅程之不变量2.7 Sturm-Liouville问题2.8 极值(积分)问题第三章 Hamilton系统3.1 Legendre变换3.2 Hamilton⽅程3.3 座标变换与守恒律3.4 Noether定理3.5 Poisson括号第四章数学物理⽅程4.1 波动⽅程4.2 Laplace与Poisson⽅程4.3 Schrodinger ⽅程4.4 Klein-Gordon ⽅程4.5 KdV ⽅程4.6 流体⼒学⽅程　课程书⽬变分学导论 (Lecture note by Chi-Kun Lin).向量分析台湾国⽴交通⼤学林琦焜 3.3GB （点击进⼊我的淘宝店）向量分析主要是要谈”梯度、散度与旋度”这三个重要观念，⽽对应的则是⽅向导数、散度定理、与Stokes定理因此重⼼就在於如何釐清线积分、曲⾯积分以及他们所代表的物理意义。

D-甘露醇与D-山梨醇在氯化钠水溶液中的稀释焓

D ⁃甘露醇与D ⁃山梨醇在氯化钠水溶液中的稀释焓李玲邱晓梅孙德志∗刘峰邸友莹尹宝霖(聊城大学化学化工学院,山东聊城252059)摘要应用等温流动微量热法测定了298.15K 时互为旋光异构体的D ⁃甘露醇与D ⁃山梨醇在不同浓度的氯化钠水溶液中的稀释焓,利用McMillan ⁃Mayer 理论计算了D ⁃甘露醇与D ⁃山梨醇在不同浓度的氯化钠水溶液中的焓对相互作用系数.结果表明,D ⁃甘露醇和D ⁃山梨醇在氯化钠水溶液中的焓对相互作用系数h 2均为正值,h 2的值随着氯化钠浓度的增加皆逐渐增大,但D ⁃山梨醇的焓对相互作用系数h 2增大的速率[d h 2/d m (NaCl)]比D ⁃甘露醇的要大.根据两多元醇分子构象结构的差异，溶质⁃溶质相互作用和溶质⁃溶剂相互作用对结果进行了解释.关键词：D ⁃甘露醇,D ⁃山梨醇,氯化钠,稀释焓,焓对相互作用系数中图分类号：O641,O642Dilution Enthalpies of D ⁃mannitol and D ⁃sorbitol in Aqueous SodiumChloride SolutionLI,LingQIU,Xiao ⁃MeiSUN,De ⁃Zhi ∗LIU,FengDI,You ⁃YingYIN,Bao ⁃Lin(College of Chemistry and Chemical Engineering,Liaocheng University,Liaocheng 252059,P.R.China)AbstractThe dilution enthalpies of D ⁃mannitol and D ⁃sorbitol in aqueous sodium chloride solution have beendetermined by using flow ⁃mix ⁃isothermal microcalorimetry at 298.15K.The enthalpic pairwise interaction coefficients (h 2)in the range of sodium chloride concentration (0~1.2mol ·kg -1)have been calculated according to the McMillan ⁃Mayer theory.It is found that h 2of coefficients D ⁃mannitol and D ⁃sorbitol are all positive in aqueous sodium chloride solution and become more positive with increase of the concentration of sodium chloride.Further analysis indicates that the increasing rate of the h 2coefficient [d h 2/d m (NaCl)]for D ⁃sorbitol is larger than that of D ⁃mannitol.The results are discussed in terms of the different conformations of the two polylols,interactions of solute with solute and solute with solvent.Keywords :D ⁃mannitol,D ⁃sorbitol,Sodium chloride,Dilution enthalpy,Enthalpic pairwise interactioncoefficients[Note]/whxb物理化学学报(Wuli Huaxue Xuebao )February Acta Phys.鄄Chim.Sin .,2006,22(2)：215~220Received:July 6,2005;Revised:August 22,2005.∗Correspondent,E ⁃mail:sundezhisdz@;Tel :0635⁃8230614;Fax:0635⁃8239121.山东省自然科学基金(2004zx15)及聊城大学科技发展计划项目(x031015)资助ⒸEditorial office of Acta Physico ⁃Chimica Sinica有关糖类和多羟基化合物水化特性及其在水溶液中与电解质或非电解质的相互作用的研究对生物学,医学,催化和环境科学都具有重要的意义[1⁃5].D ⁃甘露醇又名甘露糖醇(mannitol 或mannite),分子式为C 6H 14O 6,系常用药物,在人体内参与代谢过程,是一种较强的自由基清除剂[6⁃8].因其渗透性脱水和利尿作用,在临床上主要用作利尿剂和脱水剂,也适用于治疗脑水肿,防治肾功能衰竭、腹水等[9⁃11].D ⁃山梨醇,作为D ⁃甘露醇的一种重要的旋光异构体(仅仅构象不同),是葡萄糖经醛糖还原酶催化而还原成的多元醇,存在于人和动物体内许多细胞中.作用与临床应用类似D ⁃甘露醇,但作用较弱,它在调节细胞内外的渗透平衡以及糖尿病、肝脏功能性疾病和某些老年性疾病的病理学研究和诊治等方面发挥着重要215Acta Phys.鄄Chim.Sin.(Wuli Huaxue Xuebao),2006Vol.22作用[12⁃14].两多元醇分子结构的差别微小,但其生理作用却差别较大.此外,氯化钠对于生命体系是具有非常重要作用的电解质,因此对于氯化钠和D⁃甘露醇或D⁃山梨醇在稀水溶液中的相互作用的研究具有重要的理论和实际意义.再者,稀释焓是基本的溶液热力学参数,常用来表征溶液中溶剂化溶质分子之间的相互作用强弱,尤其是对与超额热力学性质相关的焓对相互作用维里系数的研究,可以获得溶质⁃溶质和溶质⁃溶剂相互作用的重要信息.在稍前的工作中,我们测定了313.15K下D⁃甘露醇和D⁃山梨醇在纯水和卤化钠水溶液中的稀释焓[15],获得了卤素阴离子对被稀释组分焓对相互作用系数的影响规律,为了工作的继续和深入,本文测定了298.15K 下D⁃甘露醇与D⁃山梨醇在不同浓度NaCl溶液中的稀释焓,并同其在313.15K下纯水中的稀释焓进行了比较.1实验部分D⁃山梨醇和D⁃甘露醇均为Aldrich公司(美国)产品,纯度≥99%.氯化钠,为分析纯试剂,上海试剂公司产品,在纯水中重结晶提纯后,393K下真空干燥72h.溶液配制采用称量法.实验用水为新蒸二次水.所有溶液在配制后的保存期不超过12h,以防止细菌污染的影响.稀释焓的测定采用瑞典产Thermal Activity Monitor2277精密微热量计的流动混合单元完成.测定原理和方法见文献[16],仪器的性能及其标定方法见文献[17].2结果与讨论实验数据参照文献[18]处理,采用McMillan⁃Mayer展开式进行拟合.摩尔稀释焓Δdil H m(m i→m f)与稀释前后溶液的质量摩尔浓度m i、m f间关系为Δdil H m(m i→m f)=h2(m f-m i)+h3(m2f-m2i)+ (1)式中h2、h3……为各级焓相互作用系数,可通过实验数据经最小二乘法获得.式(1)中m f=m i f2/[f1(m i M+1)+f2](2)Δdil H m=-W/(f2c i)(3)式(2)、(3)中,f2、f1分别为溶液和溶剂的流速(mg·s-1), c i=m i/(m i M+1)是以mol·kg-1(溶液)表示的被稀释组分的稀释前浓度,M为其摩尔质量(kg·mol-1),W为稀释过程的热功率(μW),Δdil H m表示摩尔稀释焓(J·mol-1).将实验所得的Δdil H m(m i→m f)、m i、m f代入式(1),用Origin软件的多元线性回归程序拟合得到Δdil H m 关于(m f-m i)的经验方程,同时得各级焓相互作用系数.在实验温度为298.15K下,D⁃山梨醇与D⁃甘露醇在氯化钠水溶液(0.1~1.2mol·kg-1)中的稀释焓结果分别列于表1和表2中,利用上式关联实验数据得到相应的各级焓相互作用系数分别列于表3和表4中.由于三分子以上的相互作用比较复杂,这里仅讨论作为溶液中被稀释组分两分子之间相互作用能之量度的二阶焓相互作用系数h2.D⁃山梨醇与D⁃甘露醇在纯水中的h2分别为28.41J·kg·mol-2、106.57J·kg·mol-2,与文献[15]相比(313.15K下D⁃山梨醇与D⁃甘露醇在纯水中的h2分别为104.6J·kg·mol-2、153.2J·kg·mol-2)明显要小,这说明温度升高,两多元醇分子相互靠近时需要吸收更多的能量.图1表示D⁃山梨醇与D⁃甘露醇在水和卤化钠水溶液中的二阶焓对相互作用系数h2随溶剂中所含氯化钠浓度的不同而变化的趋势.从图1中可以看出,首先,无论是在以纯水为溶剂还是在以氯化钠溶液为溶剂的溶液中,D⁃山梨醇与D⁃甘露醇的二阶焓对相互作用系数h2的值均为正值.这表明当两个D⁃山梨醇分子或两个D⁃甘露醇分子相互接近时需要吸收能量[19],主要是因为两个醇分子彼此充分靠近时需要能量去(部分)破坏各自周围的水化层.值得注意的是,水溶液中D⁃甘露醇的h2值比D⁃山梨醇的明显要大.说明两种多元醇的分子量和构造式相同仅构象不同[20⁃21],这种微观上的差异就导致了宏观热力学性质的明显差别.因为在D⁃甘露醇的稳定构象中四个仲羟基(—OH)向不同的方向伸展,有较多机会和水分子形成分子间氢键,而D⁃山梨醇的2⁃OH和3⁃OH伸向同一方向,更有机会形成分子内氢键,所以两个D⁃甘露醇分子相互靠近时需较多能量克服水分子的束缚.其次,在氯化钠水溶液中D⁃山梨醇和D⁃甘露醇的二阶焓对相互作用系数h2随着氯化钠溶液浓度的增大而逐渐增大.这反映了溶剂中氯化钠浓度的变化对D⁃甘露醇和D⁃山梨醇分子间的焓对作用有显著影响.在这种水⁃盐⁃多元醇的三元体系中,已溶解的溶质分子或离子周围都已形成水化层[22].当溶质质点相互靠近时,这些分子或盐离子之间的相互作用会引起各自溶剂共球的扰动,导致各水化层中水分子向本体相溶剂水的释放[23],需要能量克服水分子的束缚,是个吸热过程,因此溶质分子的部分去溶剂化作用对h2产生正的贡献.随着氯化钠浓度的增大,溶液中的离子或多元醇分子216No.2孙德志等：D⁃甘露醇与D⁃山梨醇在氯化钠水溶液中的稀释焓217表1D鄄甘露醇在水和氯化钠水溶液为溶剂时的稀释焓(298.15K)Table1Enthalpies of dilution of D⁃mannitol in water and aqueous sodium chloride solutions at298.15Km(NaCl)/(mol·kg-1)m i/(mol·kg-1)m f/(mol·kg-1)Δdil H m/(J·mol-1)m i/(mol·kg-1)m f/(mol·kg-1)Δdil H m/(J·mol-1)0.00000.09610.0428-3.370.32100.1410-15.260.12340.0549-4.900.35010.1533-16.930.15450.0683-6.550.38200.1669-18.650.18290.0811-8.190.40960.1790-20.160.21230.0940-9.530.43860.1911-21.750.24910.1099-11.590.46620.2016-23.380.28480.1255-13.320.09980.09500.0447-2.980.30610.1432-13.670.12440.0588-4.520.33870.1584-15.130.15230.0719-5.950.36040.1687-16.160.18230.0860-7.430.39190.1833-17.930.21460.1011-9.070.42690.1990-19.740.24110.1136-10.390.50220.2328-24.080.27410.1287-12.000.30000.05380.0252-1.600.28940.1337-15.850.08620.0403-3.830.32630.1501-18.000.11560.0540-5.610.36010.1658-19.830.14940.0698-7.620.39080.1792-21.880.18270.0848-9.650.41860.1917-23.430.22030.1023-11.700.45790.2086-26.080.25200.1166-13.630.50030.10800.0524-6.470.30690.1439-20.450.14480.0682-9.530.33960.1590-22.640.17270.0815-11.470.37120.1735-24.710.19900.0938-13.200.40880.1905-27.280.22480.1056-14.810.43520.2024-29.020.24950.1174-16.680.49360.2290-32.830.27870.1310-18.580.69220.08810.0403-5.600.33460.1511-25.260.12700.0580-9.150.37350.1683-28.090.16800.0766-12.430.41280.1855-31.380.19670.0894-14.600.44860.2008-34.070.22780.1035-16.880.48170.2152-36.540.26720.1213-20.000.49760.2223-37.680.30050.1359-22.680.82620.07160.0327-6.230.33900.1522-26.830.10920.0497-9.950.38210.1707-30.530.14580.0659-12.790.41790.1865-33.080.18340.0830-15.390.45410.2019-36.020.22190.1001-18.250.48210.2139-38.340.25940.1169-20.870.51070.2273-40.790.29690.1333-23.911.0000.06650.0293-5.990.30560.1322-26.760.09700.0426-9.130.33720.1458-29.000.12730.0554-11.760.40260.1726-34.870.16660.0732-15.020.44170.1900-37.900.19630.0856-17.420.48140.2058-41.910.23220.1011-20.380.50980.2183-44.620.26630.1157-23.37Acta Phys.鄄Chim.Sin.(Wuli Huaxue Xuebao ),2006Vol.220.11640.0507-11.000.34620.1486-34.250.14300.0622-13.980.38390.1641-38.040.17330.0752-16.690.42120.1796-41.530.20670.0894-20.630.43780.1867-43.340.24100.1041-23.690.49430.2090-49.160.27440.1182-26.881.20340.07980.0348-7.010.30920.1328-30.62m (NaCl)/(mol ·kg -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)Continued Table 1表2D 鄄山梨醇在水和氯化钠水溶液为溶剂时的稀释焓(298.15K)Table 2Enthalpies of dilution of D 鄄sorbitol in water and aqueous sodium chloride solutions at 298.15K0.00000.13370.0651 1.120.27980.13690.260.15800.07730.940.30500.14930.140.18320.08950.780.32860.16120.030.20600.10080.640.35720.1751-0.120.23370.11430.500.39080.1914-0.300.25580.12500.370.42530.2084-0.520.10000.11530.04860.520.36410.1506-4.060.15280.0641-0.200.39640.1636-4.680.19000.0796-0.880.43020.1771-5.430.22390.0933-1.440.46940.1928-6.200.25650.1061-2.060.51580.2109-7.320.29280.1216-2.800.56150.2277-8.620.33060.1370-3.420.29920.06820.0293-0.870.33090.1400-9.090.10090.0432-1.920.36710.1546-10.310.12280.0526-2.790.40760.1713-11.720.17360.0740-4.140.45050.1885-13.270.21240.0902-5.400.49530.2073-14.950.24900.1059-6.530.52850.2195-16.420.28740.1220-7.730.49960.09020.0379-3.380.27560.1141-11.960.12040.0503-4.860.30580.1284-13.100.15170.0635-6.320.34100.1413-14.880.18550.0790-7.600.36930.1519-16.420.21640.0905-9.150.40180.1650-18.070.24520.1031-10.370.44650.1832-20.160.69710.10260.0432-6.140.35270.1461-21.450.13820.0582-8.140.39600.1632-23.960.16950.0707-10.280.43920.1805-26.970.20690.0869-12.340.48180.1975-29.390.24230.1011-14.580.53260.2164-32.90m (NaCl)/(mol ·kg -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)218No.2孙德志等：D ⁃甘露醇与D ⁃山梨醇在氯化钠水溶液中的稀释焓亲水力会越强,那么离子或多元醇分子的去水化作用更难,去溶剂化作用对h 2的贡献将会更正.随着盐浓度的增大,上述作用会有所加强,所以两多元醇分子在氯化钠水溶液中的焓对相互作用系数h 2随着氯化钠浓度的增大而增大.从图1还可看出,随盐浓度的增大,D ⁃山梨醇的焓对相互作用系数h 2增大的速率[d h 2/d m (NaCl)]比D ⁃甘露醇的要大.同样这种h 2随NaCl 浓度变化率的差异也主要是由两个多元醇分子的构象差异引起的.这是由于在盐溶液中,当两个D ⁃山梨醇分子Continued Table 20.31680.1314-19.220.89960.09430.0399-7.100.35250.1460-25.500.13260.0560-9.800.38870.1608-28.330.16730.0704-12.570.42090.1733-30.570.20500.0863-15.200.46350.1909-33.720.23900.1001-17.490.51850.2127-37.510.27350.1143-19.970.56030.2280-40.890.31080.1296-22.401.10000.09300.0392-7.360.33990.1406-27.160.12880.0540-10.470.37910.1565-30.330.16740.0701-13.710.42360.1742-34.070.19660.0821-15.920.46400.1904-37.040.22730.0949-18.270.50100.2045-40.160.26530.1103-21.280.55190.2242-44.370.29700.1232-23.810.27920.1159-17.000.57660.2348-35.79m (NaCl)/(mol ·kg -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)m i /(mol ·kg -1)m f /(mol ·kg -1)Δdil H m /(J ·mol -1)表3D ⁃甘露醇在氯化钠水溶液中的焓相互作用系数Table 3Enthalpic interaction coefficients of D ⁃mannitol in aqueous sodium chloride solutionR is the correlation coefficients for equation (1).SD:standard derivationm (NaCl)/(mol ·kg -1)h 2/(J ·kg ·mol -2)h 3/(J ·kg 2·mol -3)h 4/(J ·kg 3·mol -4)R 2SD m i /(mol ·kg -1)0.00000.09980.30000.50030.69920.82621.00001.2034106.57117.74134.23151.77166.11176.28189.11206.57-35.84-67.17-71.91-50.72-51.33-108.6-117.2-70.9438.7471.7675.8435.6236.43111.3121.956.840.99990.99990.99990.99990.99990.99980.99990.99980.0730.0550.0700.0890.1240.1990.1510.1790.10~0.470.10~0.500.05~0.460.11~0.500.09~0.500.07~0.510.07~0.510.08~0.51表4D ⁃山梨醇在氯化钠水溶液中的焓相互作用系数Table 4Enthalpic interaction coefficients of D ⁃sorbitol in aqueous sodium chloride solution0.00000.10000.29920.49960.69710.89961.100028.4140.8462.2984.45111.7132.3150.7-43.15-29.32-30.49-24.82-20.56-26.46-35.7842.0434.2741.0935.6520.6422.8428.860.99990.99980.99990.99990.99990.99980.99990.00530.04110.05890.06750.10980.13640.10620.13~0.430.12~0.560.07~0.530.10~0.450.10~0.580.10~0.560.10~0.55m (NaCl)/(mol ·kg -1)h 2/(J ·kg ·mol -2)h 3/(J ·kg 2·mol -3)h 4/(J ·kg 3·mol -4)R 2SD m i /(mol ·kg -1)219Acta Phys.⁃Chim.Sin.(Wuli Huaxue Xuebao ),2006Vol.22相互靠近时,需要更多的能量去部分破坏分子内氢键.D ⁃山梨醇焓对相互作用系数增大趋势更大,表明盐的浓度越大,D ⁃山梨醇破坏分子内氢键更难.3结论利用流动式等温微量热技术在298.15K 下分别测定了D ⁃山梨醇与D ⁃甘露醇在纯水及氯化钠水溶液中的稀释焓,根据McMillan ⁃Mayer 理论对所测数据进行关联,得到了稀释焓对浓度变化的经验方程和各阶焓对相互作用系数h 2,h 3,h 4.结果表明:(1)298.15K 时纯水中D ⁃山梨醇与D ⁃甘露醇的h 2值比313.15K 时相应两多元醇在纯水中的h 2值明显要小,说明温度升高对h 2值有正的影响;(2)无论是纯水还是不同浓度的氯化钠水溶液为溶剂,D ⁃山梨醇与D ⁃甘露醇的二阶焓对相互作用系数h 2均为正值,但D ⁃甘露醇的h 2值比D ⁃山梨醇的明显要大.这是由于两种多元醇分子的构象不同,水溶液中h 2表现出明显差异;(3)在D ⁃山梨醇和D ⁃甘露醇的氯化钠水溶液中,随着盐浓度的增大,被稀释组分二阶焓对相互作用系数依次增大,而且D ⁃山梨醇的增大趋势比D ⁃甘露醇的要强.这种变化趋势差异主要是由于两种不同构象的多元醇分子在水溶液中的不同水化能力所致.References1Zhuo,K.L.;Wang,J.J.;Bai,G.Y.;Yan,H.K.;Wang,H.Q.Science in China B,2003,33(5):377[卓克垒,王键吉,白光月,阎海科,汪汉卿.中国科学B(Zhongguo Kexue B ),2003,33(5):377]2Barone,G.;Castronuovo,G.;Doucas,D.;Elia,L.;Matlia,C .J.phys.Chem.,1983,87(11):19313Morel,J.P.;Lhermet,C.;Desnoyers,J.E.Can.J.Chem.,1986,64(5):9964Barone,G.Thermochim.Acta,1990,162(1):175Back,J.F.;Oakenfull,D.;Smith,M.B.Biochemistry,1979,18(23):51916Cao,Z.F.;Zhang,S.M.;Wang,Q.J.;Luo,L.;Hu,W.M.;Cui,X.Z.Chinese Journal of Pharmacology and Toxicology,1994,8(3):237[曹兆丰,张尚明,王秋菊,罗玲,户万秘,崔学增.中国药理学与毒理学杂志(Zhongguo Yaolixue yu Dulixue Zazhi ),1994,8(3):237]7Shao,C.L.;Saito,M.;Yu,Z.L.Acta.Phys.⁃Chim.Sin.,2000,16(2):184[邵春林,齐藤真弘,余赠亮.物理化学学报(WuliHuaxue Xuebao ),2000,16(2):184]8Shao,C.L.;Saito,M.;Yu,Z.L.Acta Laser Biology Sinica,1999,8(2):97[邵春林,齐藤真弘,余赠亮.激光生物学报(Jiguang Shengwu Xuebao 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44128:The Chemical Rubber Co.,1974⁃1975:C ⁃49122Humphrey,R.S.;Hedwig,G.R.;Watson,I.D.;Malcolm,G.N.J.Chem.Thermodyn.,1980,12(6):59523Palecz,B.;Diekarski,H.Fluid Phase Equilib.,1999,164(2):257图1298.15K 时D ⁃甘露醇(a)和D ⁃山梨醇(b)的焓对相互作用系数h 2随氯化钠质量摩尔浓度的变化Fig.1Variation in enthalpic pairwise interactioncoefficients (h 2)of D ⁃mannitol(a)and D ⁃sorbitol (b)with the molality of sodium chloride in aqueous solutions at 298.15K220。

Polythieno[3,4-b]thiophene as an Optically Transparent Ion-Storage Layer

3332Chem.Mater.2009,21,3332–3336DOI:10.1021/cm900843bPolythieno[3,4-b]thiophene as an Optically TransparentIon-Storage LayerMichael A.Invernale,Venkataramanan Seshadri,Donna Marie D.Mamangun,Yujie Ding,James Filloramo,and Gregory A Sotzing*University of Connecticut,Department of Chemistry and the Polymer Program,97N.Eagleville Road,Storrs,Connecticut06269-3136Received March26,2009.Revised Manuscript Received May4,2009Ion storage layers have been employed in the construction of electrochromic devices to enhance device lifetimes through balanced ion shuttling.This has led to a search for a material which has a high charge capacity as well as optical transparency.Poly(thieno[3,4-b]thiophene)(PT34bT)exhibits high transparency in the visible region in both its neutral and oxidized states,in addition to having a high charge capacity,making it an ideal candidate for an ion storage layer.Herein we report devices fabricated by electrodeposition of several common chromophores,such as PEDOT,PDiBz-ProDOT,and PProDOT-Me2.Devices were made with and without a balanced layer of PT34bT on the counter electrode and were probed for coloration and contrast.It was found that the addition of the ion storage layer did not alter the color of any of the devices and resulted in a minimal,predicted loss of contrast corresponding to the thickness of the ion storage layer.IntroductionThe switching of conjugated polymers between an insulating and a conducting state is usually accompanied by a change in their absorption/transmittance and ion diffusion,forming the basis of their applications in elec-trochromics.In general,switching of conjugated poly-mers to their various oxidation states can also lead to multiple color transitions owing to their broad spectra.1 Both anodically and cathodically coloring conjugated polymers that switch between a colored state and a trans-missive,colorless state are well-known.2-5Examples of green to colorless and black to colorless materials have been demonstrated,as well as various IR attenuation devices.6-10All of these electrochromic(EC)cells are basically redox cells;that is,an oxidation process in one of the electrodes must be compensated by a reduction in the auxiliary electrode.Thus,a charge compensating layer on the second,auxiliary electrode is indispensable for prolong-ing the lifetime of the device.A dual polymer approach using an anodically coloring polymer and a cathodically coloring polymer has been demonstrated earlier.11In these devices,the anodically and cathodically coloring polymers are complementarily coloring in nature and hence have a synergistic effect in the bleached and colored states.The most commonly used anodically coloring polymers,which are bleached in their neutral state and colored in their oxidized state,are high band gap polymers.For electro-chromic applications,theλmax of the neutral high band gap polymers is expected to lay completely within the UV region;in most cases the low energy absorption of the neutral polymer tails into the visible region yielding a yellow color in the bleached state.Yellow is known to distort the perception of color by the human eye,which would be undesirable for applications such as windows or sunglasses.This can be detrimental for some applications, including variable transmittance smart windows and dis-plays.Thus,any ion storage layer which presents any yellowing effects is undesirable.Dual polymer complemen-tary coloring electrochromic windows are effective in vari-able transmission devices since they synergistically bleach. However,for full color polymeric displays,there is a strong need for one of the conjugated polymers not to exhibit any coloring effect in the visible region in any of the oxidation states.In essence,the second polymer must act as an ion-storage layer alone.Reynolds and co-workers reported that the use of PProDOP-NPrS(Figure1),a high band gap polymer*Corresponding author.E-mail:sotzing@.(1)Sotzing,G.A.;Reddinger,J.L.;Katritzky,A.R.;Soloducho,J.;Musgrave,R.;Reynolds,J.R.Chem.Mater.1997,9,1578–1587.(2)Unur,E.;Jung,J.H.;Mortimer,R.J.;Reynolds,J.R.Chem.Mater.2008,20(6),2328–2334.(3)Tehrani,P.;Hennerdal,L.O.;Dyer,A.L.;Reynolds,J.R.;Berggren,M.J.Mater.Chem.2009,19,1799–1802.(4)Sonmez,G.;Shen,C.K.F.;Rubin,Y.;Wudl,F.Angew.Chem.,Int.Ed.2004,43,1498–1502.(5)Andersson,P.;Forchheimer,R.;Tehrani,P.;Berggren,M.Adv.Funct.Mater.2007,17,3074–3082.(6)Beaujuge,P.M.;Ellinger,S.;Reynolds,J.R.Nat.Mater.2008,7,795–799.(7)Gunbas,G.E.;Durmus,A.;Toppare,L.Adv.Mater.2008,20,691–695.(8)Durmus,A.;Gunbas,G.E.;Toppare,L.Chem.Mater.2007,19(25),6247–6251.(9)Beaujuge,P.M.;Ellinger,S.;Reynolds,J.R.Adv.Mater.2008,20,2772–2776.(10)Dyer,A.L.;Grenier,C.R.G.;Reynolds,J.R.Adv.Funct.Mater.2007,17,1480–1486.(11)Sapp,S.A.;Sotzing,G.A.;Reynolds,J.R.Chem.Mater.1998,10,2101./cm Published on Web05/22/2009r2009American Chemical SocietyArticle Chem.Mater.,Vol.21,No.14,20093333with reduced low energy absorption tail in the visibleregion,can partially eliminate the yellow coloration.12This polymer was shown to exhibit a drastic shift in the λmax going from the UV region in the neutral state to the NIR in the oxidized state.Thus,it exhibits very littleabsorbance throughout the visible region in either ofthe oxidation states,acting as a neutral density filter.Reported values of contrast losses for the use of thismaterial were on the order of10%.Wudl and co-workershave suggested the use of a very low band gap conjugatedpolymer based on1,3-bis(20-[30,40-ethylenedioxy]thienyl)-benzo[c]thiophene-N-200-ethylhexyl-4,5-dicarboximide(DEDOT-ITNIm,Figure1),as the ion-storage layer indual polymer electrochromics,although no examples ofthese have been reported.13Herein,we demonstrate the use of poly(thieno[3,4-b]-thiophene)(PT34bT,Figure1),a low band gap polymer(E g=0.85eV),as the ion-storage layer for a dual polymerelectrochromic window.Electrochemically synthesizedpoly(thieno[3,4-b]thiophene)has earlier been reportedto be highly transmissive and colorless in the oxidizedstate and a transmissive sky blue in the neutral state.14PT34bT in the neutral state does not exhibit peaks withinthe visible region,like poly(DEDOT-ITNIm),and hencewill not significantly contribute to any coloration effects.Therefore,from an optical point of view,polymers suchas PT34bT will be more useful as ion-storage layers.Wehave also reported a higher doping level for poly(thieno-[3,4-b]thiophene)compared to that of poly(3,4-ethylene-dioxythiophene)(PEDOT).15The doping ratio remainshigh when compared to other electrochromic polymers,such as poly(3,4-propylenedioxythiophene)(PProDOT)and its derivatives.These two properties,namely,opticaltransmissivity and higher charge capacity,are essential inmaking PT34bT a good candidate for an ion-storagelayer in EC windows.Solid-state electrochromic deviceswere constructed and were comprised of PDiBz-Pro-DOT,PProDOT-Me2,and PEDOT(Figure2)each withPT34bT.These windows were characterized using elec-trochemical and optical techniques.A maximization of contrast for each of the four polymers was carried out with respect to film thickness and charge per unit area. Furthermore,the color transitions of the single polymers as well as the dual polymer systems are compared using 1976CIE color space plots,and it was found that PT34bT did not alter the color of any of these devices.Experimental SectionMaterials.DiBz-ProDOT was a generous gift from Prof.Anil Kumar.3,4-Dihydro-3,3-dimethyl-2H-thieno[3,4-b][1,4]dioxe-pine or2,2-dimethyl-3,4-propylenedioxythiophene(ProDOT-Me2)was synthesized using a transetherification ring closure starting with commercially available3,4-dimethoxythiophene (Organic Electronics Chemicals)and2,2-dimethylpropane-1,3-diol(Sigma-Aldrich)according to the literature procedure.16 EDOT was purchased from Sigma-Aldrich and distilled before use.Thieno[3,4-b]thiophene was prepared via a literature pro-cedure.13Acetonitrile(Fisher Scientific)was freshly distilled over calcium hydride(ACROS).Lithium trifluoromethanesul-fonate(LITRIF)was purchased from Aldrich and used as received.One sided indium doped tin oxide(ITO)coated SiO2 passivated unpolished float glass with a nominal resistance of 8-12Ωpurchased from Delta Technologies Inc.was used as received.Instrumentation.ΑCHI660A potentiostat was used for all the electrochemical polymerization and characterizations.A Per-kin-Elmer UV Lambda900spectrophotometer and a Varian Cary5000i with a150mm integrating sphere DRA were both used for the optical studies.Color calculations were carried out using Varian’s Color Software(1976CIE standards,a10°observer angle,and a D65illuminant).Film thicknesses were measured using a Veeco DekTak150v9Mechanical Profil-ometer.Electrochemistry.A nonaqueous Ag/Ag+(silver wire in 10mM AgNO3in0.1M LITRIF/ACN)was used as the reference electrode for all electrochemical polymerizations and spectroelectrochemistry.The electrode was calibrated to be 0.456V versus NHE using a ferrocene/ferrocenium couple. Electrochemical deposition of PT34bT,PEDOT,PProDOT-Me2,and PDiBz-ProDOT onto ITO were carried out from 0.1M LITRIF/ACN containing10mM monomer.Charge capacities were measured by switching the deposited polymer in0.1M LITRIF/ACN between-0.6and0.4V for each of the electrochromic polymers and-0.7and0.5V for the ion storage layer.Solid State EC Windows.The devices were constructed by sandwiching a photocured gel-electrolyte17in between two 1.5in.Â2in.ITO coated glass pieces electrodeposited with the oxidized electrochromic film and a neutral film ofPT34bT.(12)(a)Schwendeman,I.;Hickman,R.;Sonmez,G.;Schottland,P.;Zong,K.;Welsh,D.M.;Reynolds,J.R.Chem.Mater.2002,14,3118.(b)Reynolds,J.R.,Zong,K.,Schwendemann,I.,Sonmez,G.,Schottland,P.,Argun,A.A.,Aubert,P.H.U.S.Patent6,791,738,2003.(13)Sonmez,G.;Meng,H.;Wudl,F.Chem.Mater.2003,15,4923.(14)Sotzing,G.A.;Lee,K.Macromolecules2002,35,7281.(15)Seshadri,V.;Wu,L.;Sotzing,ngmuir2003,19,9479.(16)Agarwal,N.;Mishra,S.P.;Hung,C.-H.;Kumar,A.;Ravikanth,M.Bull.Chem.Soc.Jpn.2004,77(6),1173–1180.(17)Seshadri,V.;Padilla,J.;Bircan,H.;Radmard,B.;Draper,R.;Wood,M.;Otero,T.F.;Sotzing,.Electron.2007,8(4),367–381.3334Chem.Mater.,Vol.21,No.14,2009Invernale et al. The solid-state EC windows shown in Figure5are held atconstant potentials of-1and1V for the bleached and coloredstates,respectively.Results and DiscussionDual Polymer Electrochromic Devices Using PT34bT.The u0and v0values(1976CIE standards)for PT34bT inboth the bleached and colored states was calculated to beabout0.19-0.20and0.47-0.48,respectively.The u0andv0values for the6000K blackbody radiator(referencepoint)are0.2033and0.4712.Both the neutral andoxidized PT34bT does not exhibit any coloration of itsown and is close to that of the white reference point.Thefollowing half-reactions and overall redox reaction at thetwo electrodes coated with PDiBz-ProDOT and PT34bTdepicts the concept of using very low band gap conjugatedpolymers with little or no visible electrochromism as theion-storage layer.The reactions in Table1are represen-tative for the electrochromic polymers used in this study.Operating Potential Window.PT34bT and PDiBz-Pro-DOT were electrochemically deposited at1.3V(vs Ag/ Ag+).Dual polymer liquid cells comprised of PT34bT/ PDiBz-ProDOT with matched charge capacities were constructed with PT34bT in the neutral state and PDiBz-ProDOT in the oxidized state using0.1M LITRIF/ACN as the electrolyte.Figure3shows the cyclic voltammogram for a PT34bT/PDiBz-ProDOT EC win-dow for the first three cycles.After equilibrating at0V for 5s,the scanning was started in the positive direction initially.The devices were found to exhibit a full switch between a potential of-0.8and a potential of0.8V as compared to-1.5and0.9V for a PDiBz-ProDOT dual polymer EC window using PBEDOT-NMCz as an ano-dically coloring polymer.18Optical Properties and Colorimetric Analysis.Much study has been carried out on materials such as PEDOT, PDiBz-ProDOT,and PProDOT-Me2,and their devices are well documented.18-24To assess the use of PT34bT as a complementary electrode material for ECDs,we have synthesized our own chromophores.Dual polymer EC windows were constructed with oxidized PDiBz-ProDOT,PProDOT-Me2,and PEDOT and neutral PT34bT.EQCM data from previous studies has deter-mined a doping level for PT34bT to be near27%.The levels for the chromophores were calculated to be14%, 15%,and18%for PEDOT,PProDOT-Me2,and PDi-BzProDOT,respectively.Thus,the constructed devices were made in such a way that the film thickness or charge per unit area of each layer was balanced.For every charge injected to the gel layer by one electrode,another charge should be absorbed by the layer on the opposite electrode.For comparative purposes,film thicknesses for each electrochromic polymer were optimized with respect to maximum photopic contrast.The optimum photopic contrast for PDiBz-ProDOT,PProDOT-Me2, and PEDOT was achieved at a deposited charge capacity of approximately0.91mC/cm2(425nm film),1.39mC/ cm2(356nm film),and1.73mC/cm2(340nm film), respectively.Their resultant maximum photopic con-trasts follow respectively54%,70%,and57%.The films were prepared by solution state oxidative polymeriza-tion,and the contrast for a series of films was calculated from its absorption curves.The Supporting Information contains plots of contrast versus film thickness.Note that,for the noncoloring ion storage layer,PT34bT,the contrasts remain fairly low due to the lack of significant coloration.The film thicknesses of ion storage layer used were210nm,194nm,and220nm for PDiBz-ProDOT, PProDOT-Me2,and PEDOT,respectively.The maxi-mum expected loss of photopic contrast for these thick-nesses is9%(see Supporting Information).This furthers the applicability of the material,as even the thickest necessary depositions of the ion storage layer will not manifest as large color or contrast changes in the finaldevice.Figure3.Cyclic voltammogram of a PT34bT/PDiBz-ProDOT dual polymer EC window.Potentials are referenced against the PDiBz-ProDOT coated electrodes;scan rate=100mV/s.Table1.Representative Reactions for the Electrochromic Polymers Used in This Studybleached state colored StatePDiBz-ProDOT n+(nA-)+n e-f PDiBz-ProDOT0PT34bT0f PT34bT n+(nA-)+n e-PDiBz-ProDOT n+(n A-)+PT34bT0f PT34bT n+(n A-)PDiBz-ProDOT0(18)Padilla,J.;Seshadri,V.;Filloramo,J.;Mino,W.K.;Mishra,S.P.;Radmard,B.;Kumar,A.;Sotzing,G.A.;Otero,T.F.Synth.Met.2007,157(6-7),261–268.(19)Ma,L.J.;Li,Y.X.;Yu,X.F.;Yang,Q.B.;Noh,C.-H.Sol.EnergyMater.Sol.Cells2009,93(5),564–570.(20)Jain,V.;Sahoo,R.;Mishra,S.P.;Sinha,J.;Montazami,R.;Yochum,H.M.;Heflin,J.R.;Kumar,A.Macromolecules.2009,42(1),135–140.(21)Hsu,C.-Y.;Lee,K.-M.;Huang,J.-H.;Thomas,K.R.J.;Lin,J.T.;Ho,K.-C.J.Power Sources2008,185(2),1505–1508.(22)Ma,C.;Taya,M.;Xu,C.Electrochim.Acta2008,54(2),598–605.(23)Reynolds,J.R.Angew.Chem.,Int.Ed.2008,47(37),6945–6946.(24)Deepa,M.;Awadhia,A.;Bhandari,S.;Agrawal,S.L.Electrochim.Acta2008,53(24),7266–7275.Article Chem.Mater.,Vol.21,No.14,20093335The optical memory for all devices prepared was eval-uated over a 30min period using UV -vis analysis,and in all cases the devices were found to be stable (no loss of absorbance or contrast).The devices were stored over-night in a desiccator and retained their oxidation states over this time,as well (less than 1%change in transmis-sion).Switching speeds were calculated by UV -vis ana-lysi,s and it was found that the times were 1.2s,1.7s,and 1.5s with no PT34bT layer versus 0.9s,1.0s,and 1.0s with the ion storage layer,for PProDOT-Me 2,PEDOT,and PDiBz-ProDOT,respectively.The moderate en-hancement of switching speeds with the addition of an ion storage layer can be attributed to proper ion shuttling.The long-term switching stability for each device has also been initially evaluated over the course of 50cycles.It has been found that there is no significant difference between the devices over the first 50cycles,as there was no observed loss of charge or contrast over this period.Figure 4s hows the change in the visible spectrum for a representative PT34bT/PDiBz-ProDOT device as a func-tion of potential.The spectra shown herein are for the polymers alone with the absorbance of the other materials zeroed out as a blank,including the gel electrolyte.The dual polymer EC cell was a deep blue when the PDiBz-ProDOT layer is neutral (i.e.,when PT34bT is in its oxidized state)and a transmissive sky blue when the PT34bT layer is neutral (i.e.,when PDiBz-ProDOT is in its oxidized state).Hereafter,the terms “neutral state”and “oxidized state,”when applied to devices,will refer to the chromophore layer and not the ion storage layer.The bleached state,colored state,and contrast values for the devices prepared with and without PT34bT layers are summarized in Table 2.The color for the PEDOT/PT34bT device (λmax =593nm,maximum contrast at 515nm)ranges from a moderate blue colored neutral state to a transmissive sky blue colored oxidized state,while the colors of the PProDOT-Me 2/PT34bT device (λmax =593nm,peaks at 544and 623nm,shoulder at 496nm)goes from a deep purple in its neutral state to a similar transmissive sky blue in the oxidized form.Although the spectral sensitivity of the eyes is very broad,rangingbetween 400and 800nm,they are most sensitive at 555nm.Again,it is not appropriate to report the values at single wavelengths for eye-wear applications and hence a photopically weighted measurement is needed.The photopic values in the bleached and colored states were calculated using the following equation and shown in Table 2.T photopic ¼R 720380T ðλÞS ðλÞP ðλÞd λR 380S ðλÞP ðλÞd λwhere T (λ)is the spectral transmittance of the device under test,S (λ)is the normalized spectral emission of a 6000K blackbody,and P (λ)is the normalized spectral response of the eye.For translating the same or similar contrast within a device,a high transmissivity of the substrate and the gel electrolyte is required since the bleached state of the dual polymer EC window is not as high as that of the electro-chromic polymers alone.Nevertheless,the novel feature of this EC window configuration is that the color of the active electrochro-mic material is not significantly affected by the absor-bance of PT34bT,both in neutral and oxidized states.This could be very useful in fabricating dual polymer full color displays,wherein very low band gap conjugated polymers can be used as the common ion-storage layer on the auxiliary electrode irrespective of the colors of the active polymer layer.PT34bT will act like a neutral density filter if held in front of the coloring polymer.Figure 5shows the color trace of PDiBz-ProDOT in comparison with that of PT34bT/PDiBz-ProDOT dual polymer EC window as a function of potential,as well as images of the device and its control in their extreme color states.The comparisons of the color paths for PEDOT and PProDOT-Me 2are in the Supporting Information.The u 0and v 0values were calculated using a 6000K blackbody radiator as the reference point.This color space was chosen because of its relevance to transmis-sion-based devices,such as windows.For PDiBz-Pro-DOT,in the neutral blue state (-0.6V,vs Ag/Ag +)the u 0and v 0values are 0.1675and 0.3982,respectively.In the oxidized transmissive sky blue state (0.4V,vs Ag/Ag +)Figure 4.Representative visible transmittance spectra as a function of potential for a PT34bT/PDiBz-ProDOT EC window device.Table 2.Bleached,Colored State Transmittance Values and Contrast (%ΔT )at the Peaks and the Photopically Weighted Transmittance Valuesof the PT34bT/PDiBz-ProDOT,PT34bT/PProDOT-Me 2,andPT34bT/PEDOT Dual Polymer EC Windowselectrochromic materialλ(nm)bleached state (%)colored state (%)contrast (%ΔT )PT34bT/PDiBz-ProDOT 579772156photopic783741PT34bT/PProDOT-Me 257765263photopic66759PT34bT/PEDOT 51535431photopic29227PDiBz-ProDOT 579781563photopic 813843PProDOT-Me 257783578photopic821567PEDOT 51539336photopic373343336Chem.Mater.,Vol.21,No.14,2009Invernale et al.the u 0and v 0values are 0.2015and 0.4721,respectively.For PProDOT-Me 2,the neutral purple state had u 0v 0values of u 0=0.1802and v 0=0.3730and the oxidized transmissive blue state had the pairing u 0=0.1950,v 0=0.4533.The PEDOT system had neutral state colors of u 0=0.1913,v 0=0.4552and oxidized state colors of u 0=0.1925,v 0=0.4615.The luminance of the colors,as well as the photopic contrast of the devices,changed predic-tably with the introduction of the ion storage layer.This can be seen clearly by the values shown in Table 2,where a negative change in photopic contrast (going from devices without an ion storage layer to devices with an ion storage layer)of 2%,8%,and 7%was observed for PDiBz-ProDOT,PProDOT-Me 2,and PEDOT devices,respec-tively.This is consistent with the expected loss of contrast due to the PT34bT itself,where a maximum loss of 11%(9%for the thicknesses used in these experiments)was predicted based on the contrast for various film thicknesses of the ion storage layer (see Supporting Information).ConclusionsWe have demonstrated the use of a low band gap conjugated polymer,PT34bT,as an ion-storage layer in electrochromic windows.PT34bT was found to exhibit a neutral sky blue in the undoped form and a colorless transmissive state in the oxidized form.Dual polymer EC windows comprised of PDiBz-ProDOT,PProDOT-Me 2,and PEDOT with PT34bT were constructed and char-acterized electrochemically,and their optical properties were evaluated.As was expected,the contrast of this EC window in the transmission mode was lowered but the yellow tinge associated with other reported ion storage layers,with yellow being known to distort color percep-tion,was eliminated by the use of PT34bT.Furthermore,the CIE color plots and images indicate that in all cases studied the color of the active electrochromic poly-mer does not change significantly in the presence of PT34bT as the ion-storage layer on the auxiliary elec-trode.Thus,this dual EC window configuration using a very low band gap conjugated polymer,which is nearly colorless,could be useful for the fabrication of electro-chromic eyewear,windows,or the blue component of a full color display.Acknowledgment.We would like to thank NSF-Career (CHE-0349121)for their support of this work.Supporting Information Available:Plots for the contrast versus film thickness of the materials used in this paper,as well as the spectroelectrochemistry and color coordinate plots for the secondary chromophores used (PDF).This material is available free of charge via the Internet at.Figure 5.Transition of colors of electrochromic devices shown as CIE color space plots for PT34bT/PDiBz-ProDOT between -0.8V and +0.8V (filled circles),PDiBz-ProDOT between -0.6V and +0.4V,(unfilled circles/line);values shows are using 1976CIE color standards;all voltages are versus nonaqueous Ag/Ag +reference electrode.Color boxes are photographs of the devices in the extreme states (under the curve are with ion storage layer;over the curve are without ion storage layer).。

Orthogonal Polynomials

Polynomials 7 Some questions leading to classical orthogonal Electrostatics . . . . . . . . . . . . . . . . . . . Polynomial solutions of eigenvalue problems . . Harmonic analysis on spheres and balls . . . . Approximation theory . . . . . . . . . . . . . . 8 Heuristics polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Totik
72
with special functions, combinatorics and algebra, and it is mainly devoted to concrete orthogonal systems or hierarchies of systems such as the Jacobi, Hahn, Askey-Wilson, . . . polynomials. All the discrete polynomials and the q analogues of classical ones belong to this theory. We will not treat this part; the interested reader can consult the three recent excellent monographs [39] by M. E. H. Ismail, [28] by W. Gautschi and [6] by G. E. Andrews, R. Askey and R. Roy. Much of the present state of the theory of orthogonal polynomials of several variables lies also close to this algebraic part of the theory. To discuss them would take us too far from our main direction; rather we refer the reader to the recent book [24] by C. F. Dunkl and Y. Xu. The other part is the analytical aspect of the theory. Its methods are analytical, and it deals with questions that are typical in analysis, or questions that have emerged in and related to other parts of mathematical analysis. General properties ﬁll a smaller part of the analytic theory, and the greater part falls into two main and extremely rich branches: orthogonal polynomials on the real line and on the circle. The richness is due to some special features of the real line and the circle. Classical real orthogonal polynomials, sometimes in other forms like continued fractions, can be traced back to the 18th century, but their rapid development occurred in the 19th and early 20th century. Orthogonal polynomials on the unit circle are much younger, and their existence is largely due to Szeg˝ o and Geronimus in the ﬁrst half of the 20th century. B. Simon’s recent treatise [80, 81] summarizes and greatly extends what has happened since then. The connection of orthogonal polynomials with other branches of mathematics is truly impressive. Without even trying to be complete, we mention continued fractions, operator theory (Jacobi operators), moment problems, analytic functions (Bieberbach’s conjecture), interpolation, Pad´ e approximation, quadrature, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory (irrationality and transcendence), graph theory (matching numbers), combinatorics, random matrices, stochastic processes (birth and death processes; prediction theory), data sorting and compression, Radon transform and computer tomography. This work is a survey on orthogonal polynomials that do not lie on the unit circle. Orthogonal polynomials on the unit circle—both the classical theory and recent contributions—will be hopefully dealt with in a companion article. This work is meant for non-experts, and it therefore contains introductory materials. We have tried to list most of the actively researched ﬁelds not directly connected with orthogonal polynomials on the unit circle, but because of space limitation we have only one or two pages on areas where dozens of papers and several books had been published. As a result, our account is necessarily incomplete. Also, the author’s personal taste and interest is reﬂected in the survey, and the omission of a particular direction or a set of results reﬂects in no way on the importance or quality of the omitted works. For further backgound on orthogonal polynomials, the reader can consult

steepest descent in weight sobolev space

(3) and (4)
(t2 y ) − u = 0 u(0) = 0 u(1) = 1
((1 − t2 )y ) + 2u = 0 u(0) = 0 u(1) = 1.
2. Theory and Motivational Example. Consider 2ty = y on I = [0, 1] with ﬁnal condition y (1) = 1 and cast the equation as a least squares minimization problem, setting J (u) =
Hw
= u, v
L
w w + D1 (u), D1 (v ) L .
w by D1 and Hw by H . It is known, In the unweighted case (w ≡ 1), we denote D1 1, 2 [N3] that D1 is a function and H is the Hilbert space, H[0 ,1] . An argument follows, w showing that D1 is a function and Hw is a Hilbert space for appropriate w. 1 Theorem 2.1. If w ∈ CI is positive almost everywhere and w vanishes at some w w ˆ, then D1 is a function in the sense that no two elements of D1 have the same point, t ﬁrst coordinates and distinct second coordinates. Proof. Let

Biosci. Biotechnol. Biochem.杂志要求

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《2024年Sobolev方程的POD降维H~1-Galerkin混合有限元格式》范文

《Sobolev方程的POD降维H~1-Galerkin混合有限元格式》篇一Sobolev方程的POD降维H^1-Galerkin混合有限元格式的高质量范文一、引言Sobolev方程是偏微分方程中一种重要的模型，广泛运用于流体动力学、热传导等众多领域。

然而，随着问题规模的扩大，传统的数值解法在计算效率和精度上均面临挑战。

因此，降维技术成为了研究的热点。

本文将探讨Sobolev方程的POD（Proper Orthogonal Decomposition）降维H^1-Galerkin混合有限元格式的构造及性质。

二、Sobolev方程及背景知识Sobolev方程是一种二阶偏微分方程，描述了物理现象的时空演化过程。

其形式为：u_t + Au = f，其中u为未知函数，A为微分算子，f为已知源项。

本文将针对此类方程进行降维处理。

三、POD降维方法简介POD是一种基于数据驱动的降维方法，它通过对系统动态数据进行投影，提取出主要的模式和变化趋势，从而达到降维的目的。

在流体力学、气象学等领域有广泛的应用。

四、H^1-Galerkin混合有限元格式的构建混合有限元法是结合了有限元法和Galerkin方法的一种数值方法，特别适用于解决复杂的偏微分方程问题。

H^1空间是一种Sobolev空间，它具有优良的数学性质。

本文将构建Sobolev方程的H^1-Galerkin混合有限元格式。

五、POD降维H^1-Galerkin混合有限元格式的构造本部分将详细阐述POD降维H^1-Galerkin混合有限元格式的构造过程。

首先，通过POD方法对系统动态数据进行降维处理，提取出主要模式。

然后，将这些模式作为基函数，构建降维的H^1-Galerkin混合有限元格式。

六、格式的性质及数值实验本部分将分析降维H^1-Galerkin混合有限元格式的性质，如稳定性、收敛性等。

并通过数值实验，验证该格式在求解Sobolev方程时的精度和效率。

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J.Math.Anal.Appl.322(2006)1001–1017/locate/jmaa Sobolev orthogonal polynomials in two variablesand second order partial differential equationsJeong Keun Lee a,∗,1,L.L.Littlejohn ba Department of Mathematics,SunMoon University,Asan-si,ChoongNam336-708,Republic of Koreab Department of Mathematics and Statistics,Utah State University,Logan,UT84322-3900,USAReceived14July2005Available online3November2005Submitted by H.M.SrivastavaAbstractWe consider polynomials in two variables which satisfy an admissible second order partial differential equation of the formAu xx+2Bu xy+Cu yy+Du x+Eu y=λu,(∗) and are orthogonal relative to a symmetric bilinear form deﬁned byϕ(p,q)= σ,pq + τ,p x q x ,where A,...,E are polynomials in x and y,λis an eigenvalue parameter,σandτare linear functionals on polynomials.Weﬁnd a condition for the partial differential equation(∗)to have polynomial solutions which are orthogonal relative to a symmetric bilinear formϕ(·,·).Also examples are provided.©2005Elsevier Inc.All rights reserved.Keywords:Orthogonal polynomials in two variables;Bilinear symmetric form;Sobolev orthogonal polynomials in two variables;Second order partial differential equations*Corresponding author.E-mail addresses:jklee@sunmoon.ac.kr(J.K.Lee),lance@(L.L.Littlejohn).1This work was supported by Korea Research Foundation Grant(KRF-2004-015-C00017).0022-247X/$–see front matter©2005Elsevier Inc.All rights reserved.doi:10.1016/j.jmaa.2005.09.0621002J.K.Lee,L.L.Littlejohn/J.Math.Anal.Appl.322(2006)1001–10171.IntroductionIn1967,Krall and Sheffer[6]investigated a second order partial differential equation of the formA(x,y)u xx+2B(x,y)u xy+C(x,y)u yy+D(x)u x+E(y)u y=λu(1.1) and classiﬁed all weak orthogonal polynomials satisfying the partial differential equation(1.1), where A(x,y),...,E(y)are polynomials in x and y,andλis an eigenvalue parameter.As a generalization,we consider polynomial solutions to the partial differential equation(1.1) which are orthogonal relative to a symmetric bilinear formφ(·,·)on polynomials deﬁned by φ(p,q)= σ,pq + τ,p x q x ,(1.2) whereσandτare moment functionals,and p,q are polynomials in x and y.The caseτ=0was investigated by Krall and Sheffer.For the partial differential equation(1.1)considered by Krall and Sheffer,we know that:(i)C x=0(up to a linear change of independent variables),and(ii)partial derivatives with respect to x satisfy the partial differential equation of the same type as the partial differential equation(1.1)(see[4]).These facts remind us of the Hahn–Sonnine characterization theorem for classical orthogonalpolynomials[2,5,10]which states that:The only polynomial sequences{P n(x)}∞n=0(up to acomplex change of variable)which are simultaneously orthogonal with respect to bilinear forms of the form(p,q)0=Rp(x)q(x)dμ0,(p,q)1=R p(x)q(x)dμ0+Rp (x)q (x)dμ1,withμi(i=0,1)real-valued signed Borel measures,are the classical orthogonal polynomials of Jacobi,Laguerre,Hermite and Bessel polynomials.Naturally they lead us to the problem of investigating polynomials orthogonal relative toφ(·,·)in(1.2).But contrary to the classical orthogonal polynomials in one variable,orthogonal polynomials in two variables whose partial derivatives with respect to x or y are orthogonal does not satisfy the partial differential equation of the form(1.1)(see[3]for these materials).Instead,we consider polynomials in two variables which are orthogonal relative to a symmetric bilinear formφ(·,·)in(1.2)and satisfy the partial differential equation(1.1).In this paper,we give some basic facts on Sobolev orthogonal polynomials and the relationship between Sobolev orthogonal polynomials relative to a symmetric bilinear formφ(·,·)in(1.2) and the partial differential equation(1.1).Also,we give some examples of the partial differential equation having Sobolev orthogonal polynomials as solutions.2.Preliminaries:Basic theory of orthogonal polynomials in two variablesLet P n be the space of all polynomials in x and y of degree n.The set of all polynomials in two variables is denoted by P.By a polynomial system(in short,PS),we mean a sequenceJ.K.Lee,L.L.Littlejohn/J.Math.Anal.Appl.322(2006)1001–10171003 {φm,n(x,y)}∞m,n=0of polynomials such that degφm,n=m+n for each m,n 0and{φn−j,j}n j=0 is linearly independent modulo P n−1.We denote{φn−j,j(x,y)}n j=0by an(n+1)-dimensionalcolumn vectorΦn and a PS{φm,n(x,y)}∞m,n=0by{Φn}∞n=0.We say that a PS{Φn}∞n=0is monic ifφm,n(x,y)=x m y n modulo P m+n−1for each m,n 0. To a given PS{Φn}∞n=0,there corresponds a unique monic PS{P n}∞n=0which is deﬁned byP n=A−1nΦn,where A n=(a n j,k)n j,k=0andφn−j,j(x,y)= nk=0a n j,k x n−k y k modulo P n−1.It will be called thenormalization of{Φn}∞n=0.A linear functional on P is called a moment functional.We denote the action of a moment functionalσon polynomialπby σ,π instead of the customaryφ(π).Similarly,for a matrix Q=(Q i,j)with Q i,j being a polynomial, σ,Q is deﬁned to be the matrix( σ,Q i,j ).We see that σ,AB T = σ,BA T T for any column vectors A and B of polynomials.For a moment functionalσand any polynomialφ,we deﬁne the partial derivatives ofσby the formulas∂xσ,φ =− σ,∂xφ , ∂yσ,φ =− σ,∂yφ forφ∈P,(2.1) and deﬁne the multiplication onσby a polynomialψthrough the formulaψσ,φ = σ,ψφ forφ∈P.(2.2)Deﬁnition2.1.A PS{Φn}∞n=0is called an orthogonal basis(OB)relative toσif there is a nonzero moment functionalσsuch that for all n 0σ,φn−k,kπ =0,π∈P n−1,0 k n.And{Φn}∞n=0is called a weak orthogonal polynomial set(WOPS)relative toσif there is a nonzero moment functionalσsuch thatσ,φm,nφk,l =K m,nδm,kδn,l(K m,n∈R)if m+n=k+l.If K m,n=0(respectively K m,n>0)for each m,n 0,we say that{Φn}∞n=0is an orthogonal polynomial set(in short,OPS)(respectively a positive-deﬁnite OPS)relative toσ.It is obvious that there is an OB relative toσif and only if there is a WOPS relative toσ.Deﬁnition2.2.A moment functionalσis quasi-deﬁnite(respectively weakly quasi-deﬁnite)if there is an OPS(respectively a WOPS)relative toσ.From Deﬁnitions2.1and2.2,we see that a PS{Φn}∞n=0is an OPS(respectively a positive-deﬁnite OPS)relative toσif and only if σ,ΦmΦT n =H nδm,n and H n:= σ,ΦnΦT n is a nonsingular(respectively a positive-deﬁnite)diagonal matrix.For any PS{Φn}∞n=0,there is a unique moment functionalσ,which is called the canonical moment functional of{Φn}∞n=0,deﬁned by the conditionsσ,1 =1, σ,φm,n =0,m+n 1.Note that if a PS{Φn}∞n=0is an OB relative toσ,thenσis a constant multiple of canonical moment functional of{Φn}∞n=0.Although{Φn}∞n=0is an OPS relative toσ,its normalization {P n}∞n=0need not be an OPS relative toσbut{P n}∞n=0is an OB relative toσ.It is not easy to produce an OPS relative toσ,if any,from an OB relative toσ.1004J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–1017Theorem 2.1.[4,6]For any moment functional σthe following statements are equivalent.(i)σis quasi-deﬁnite.(ii)There is a unique monic OB {P n }∞n =0relative to σ.(iii)There is a monic OB {P n }∞n =0such that H n := σ,P n P T n is nonsingular for all n 0.Theorem 2.2(Favard’s theorem [11]).Let {Φn }∞n =0be a PS.Then the following statements are equivalent.(i){Φn }∞n =0is a WOPS relative to a quasi-deﬁnite moment functional σ.(ii)For n 0and i =1,2,there are matrices A n,i of order (n +1)×(n +2),B n,i of order(n +1)×(n +1),and C n,i of order (n +1)×n such that(a)x i Φn =A n,i Φn +1+B n,i Φn +C n,i Φn −1(here x 1=x,x 2=y),(b)rank C n =n +1,where C n =(C n,1,C n,2).Lemma 2.3.Let σbe a moment functional and ψbe a polynomial.Then we have(i)σ=0if and only if σx =0or σy =0,(ii)(ψσ)x =ψx σ+ψσx and (ψσ)y =ψy σ+ψσy .Proof.(i)The proof is obvious.(ii)A computation shows that for any p ∈P ,we have (ψσ)x ,p = ψσ,−p x = σ,−ψp x = σ,−(ψp)x +ψx p = σx ,ψp + σ,ψx p= ψσx ,p + ψx σ,ψx p = ψx σ+ψσx ,p ,which means (ψσ)x =ψx σ+ψσx .By a similar calculation,we have (ψσ)y =ψy σ+ψσy .2If the partial differential equation (1.1)has a PS {Φn }∞n =0as solutions,then it must be of the formL [u ]:=Au xx +2Bu xy +Cu yy +Du x +Eu y = ax 2+d 1x +e 1y +f 1 u xx +(2axy +d 2x +e 2y +f 2)u xy + ay 2+d 3x +e 3y +f 3 u yy +(gx +h 1)u x +(gy +h 2)u y=λn u,(2.3)where λn =an(n −1)+dn.We say that the partial differential equation (2.3)is admissible if λm =λn for m =n.Equa-tion (2.3)has a unique monic PS as solutions if and only if it is admissible.Theorem 2.4.[4]Let σbe the canonical moment functional of a PS {Φn }∞n =0.If {Φn }∞n =0satisﬁesthe partial differential equation (2.3),then σsatisﬁes the equationL ∗[σ]=(Aσ)xx +2(Bσ)xy +(Cσ)yy −(Dσ)x −(Eσ)y =0,(2.4)where L ∗[u ]:=(Au)xx +2(Bu)xy +(Cu)yy −(Du)x −(Eu)y is the formal Lagrange adjoint operator of L [·].Furthermore,(2.4)has a unique solution up to a multiplication constant if the partial differ-ential equation (2.3)is admissible.J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–10171005Theorem 2.5.[4,6]Let {Φn }∞n =0be an OB relative to σ.Then the following statements are all equivalent.(i){Φn }∞n =0satisﬁes the partial differential equation (2.3).(ii)σsatisﬁes the moment equations M 1[σ]:=(Aσ)x +(Bσ)y −Dσ=0,M 2[σ]:=(Bσ)x +(Cσ)y −Eσ=0.(2.5)Remark 2.1.For any moment functional τ,L ∗[τ]is written in the following form:L ∗[τ]= M 1[τ] x + M 2[τ] y .This formula will be used in Section 4.Theorem 2.6.[4]Let {Φn }∞n =0be a PS satisfying the admissible partial differential equation (2.3)and σthe canonical moment functional of {Φn }∞n =0.Then the following statements are equivalent.(i){Φn }∞n =0is a WOPS relative to σ.(ii)M 1[σ]=0.(iii)M 2[σ]=0.3.Theory of Sobolev orthogonal polynomials in two variablesWe know that any moment functional σdeﬁnes a symmetric bilinear form ϕ(·,·)on P ×P through the formulaϕ(p,q)= σ,pq .Conversely,a symmetric bilinear form can be generated by a moment functional provided some conditions are fulﬁlled.Theorem 3.1.Let ϕ(·,·)be a symmetric bilinear form on P ×P .Then the following statements are equivalent.(i)There is a moment functional σsuch that ϕ(p,q)= σ,pq for any p,q ∈P .(ii)ϕ(xp,q)=ϕ(p,xq)and ϕ(yp,q)=ϕ(p,yq)for any p,q ∈P .Proof.(⇒)It is obvious.(⇐)Deﬁne a moment functional σbyσ,p =ϕ(p,1),p ∈P .Then we have for any p,q ∈P ϕ(p,q)=ϕ p,deg q i +j =0a i,j x i y i =deg q i +j =0a i,j ϕ p,x i y j =deg q i +j =0a i,j ϕ x i y j p,1=ϕp deg q i +j =0a i,j x i y i ,1 =ϕ(pq,1)= σ,pq .21006J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–1017For any symmetric bilinear form ϕ(·,·)on P ×P ,we call ϕm,n k,l :=ϕ x k y l ,x m y nthe k,l m,n th moment of ϕ(·,·)and the matrixD n (ϕ):=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ϕ0,00,0ϕ0,01,0ϕ0,00,1···ϕ0,0n,0···ϕ0,00,n ϕ1,00,0ϕ1,01,0ϕ1,00,1···ϕ1,0n,0···ϕ1,00,n ϕ0,10,0ϕ0,11,0ϕ0,10,1···ϕ0,1n,0···ϕ0,10,n .....................ϕn,00,0ϕn,01,0ϕn,00,1···ϕn,0n,0···ϕn,00,n .....................ϕ0,n 0,0ϕ0,n 1,0ϕ0,n 0,1···ϕ0,n n,0···ϕ0,n 0,n ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠the n th Hankel matrix,and Δn (ϕ):=det D n (ϕ)the n th Hankel determinant of ϕ(·,·).We deﬁne the value of ϕ(·,·)on a pair (u ,v )of column vectors of polynomials.Let u =(u 1,u 2,...,u m )T and v T =(v 1,v 2,...,v n ).We deﬁne ϕ u ,v T = ϕ(u i ,v j ) m i =1,n j =1.Then we see that for any matrices A and B (when the matrix multiplication can be deﬁned)ϕ A u ,(B v )T =Aϕ(u ,v T )B T .(3.1)Deﬁnition 3.1.Let {Φn }∞n =0be a PS.(i){Φn }∞n =0is a Sobolev orthogonal basis (SOB)if there is a nonzero symmetric bilinear formϕ(·,·)such that for all n 0ϕ(φn −k,k ,π)=0,0 k n,π∈P n −1.(ii){Φn }∞n =0is a weak Sobolev orthogonal polynomial set (WSOPS)if there is a nonzero sym-metric bilinear form ϕ(·,·)such thatϕ(φm,n φk,l )=K m,n δm,k δn,l ,K m,n ∈R .If K m,n =0,then we say that {Φn }∞n =0is a Sobolev orthogonal polynomial set (SOPS).In this case,we say that {Φn }∞n =0is a WSOPS or SOPS relative to ϕ(·,·).It is obvious that if {Φn }∞n =0is a WSOPS relative to ϕ(·,·),then ϕ(Φn ,ΦT n )is a diagonalmatrix for n 0.Deﬁnition 3.2.A symmetric bilinear form ϕ(·,·)is quasi-deﬁnite (respectively weakly quasi-deﬁnite)if there is a SOPS (respectively a WSOPS)relative to ϕ(·,·).Theorem 3.2.For any symmetric bilinear form ϕ(·,·),the following statements are all equiva-lent.(i)ϕ(·,·)is weakly quasi-deﬁnite.(ii)There is a SOB relative to ϕ(·,·).J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–10171007(iii)There is a monic SOB relative to ϕ(·,·).Proof.(i)⇒(ii).Let {Φn }∞n =0be a WSOPS relative to ϕ(·,·).Then {Φn }∞n =0itself is a SOBrelative to ϕ(·,·).(ii)⇒(iii).Let {Φn }∞n =0be a SOB relative to ϕ(·,·)and {P n }∞n =0be the normalization of {Φn }∞n =0.Then {P n }∞n =0is a monic SOB relative to ϕ(·,·).(iii)⇒(i).Let {P n }∞n =0be a monic SOB relative to ϕ(·,·)and H n :=ϕ(P n ,P T n ).Then H n is asymmetric matrix so that there is a nonsingular matrix A n such that A n H n A T n :=D n is diagonal.Then Φn :=A n P n is a WSOPS relative to ϕ(·,·)since ϕ Φn ,ΦT n =ϕ A n P n ,(A n P n )T =A n ϕ P n ,P T n A T n =D n .2Lemma 3.3.For any homogeneous polynomial H (x,y)=Σn i =0a i xn −i y i ,there exists a unique polynomial R n −1(x,y)∈P n −1such that ϕ(H +R n −1,π)=0for all π∈P n −1if and only if Δn −1(ϕ)=0.Proof.Let R n −1(x,y)=Σn i +j =0r i,j x i y i .Thenϕ(H +R n −1,π)=0,π∈P n −1⇐⇒ϕ H +R n −1,x r y s =0,0 r +s n −1⇐⇒n −1i +j =0r i,j ϕ x i y j ,x r y s =−ϕ H,x r y s ,0 r +s n −1.This is a linear system for the unknowns r i,j whose coefﬁcient matrix is D n −1(ϕ).If Δn −1(ϕ)=0,then all r i,j ’s are uniquely determined.Conversely,if this linear system has a unique solution,we must have Δn −1(ϕ)=0.2Theorem 3.4.For any symmetric bilinear form ϕ(·,·),the following statements are all equiva-lent.(i)Δn (ϕ)=0for n 0.(ii)ϕ(·,·)is quasi-deﬁnite.(iii)There is a unique monic SOB relative to ϕ(·,·).(iv)There is a monic SOB {P n }∞n =0relative to ϕ(·,·)such thatH n :=ϕ P n ,P T n ,n 0,is nonsingular.Proof.(i)⇔(iii).It is obvious by Lemma 3.3.(iv)⇒(ii).Since ϕ(P n ,P T n )is a symmetric nonsingular matrix,there is a nonsingular sym-metric matrix A n of order (n +1)×(n +1)such that A n ϕ(P n ,P T n )A T n :=D n is diagonal.ThenA n ϕ P n ,P T n A T n=ϕ A n P n ,P T n A T n =ϕ A n P n ,(A n P n )T =D n .Thus {A n P n }∞n =0is a SOPS relative to ϕ(·,·).This proves that ϕ(·,·)is quasi-deﬁnite.1008J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–1017(ii)⇒(iii).Let {P n }∞n =0and {Q n }∞n =0be monic SOB’s relative to ϕ(·,·).Let R(x,y)=P m,n (x,y)−Q m,n (x,y)for m +n 1since P 0,0=Q 0,0.Then R(x,y)∈P m +n −1and is or-thogonal to P m +n −1.Thus R(x,y)≡0and so P m,n (x,y)=Q m,n (x,y)for all m +n 0.(iii)⇒(iv).Assume that det ϕ(P n ,P T n )=0for n 0.Then there is a nonzero (n +1)-dimensional row vector C such that 0=Cϕ P n ,P T n=ϕ C P n ,P T n .This implies thatP n +1,0,P n +1,0+C P n ⊥P n ,which is a contradiction to the assumption that there is a unique monic SOB relative to ϕ(·,·).2Theorem 3.5.Let {Φn }∞n =0be a SOPS relative to ϕ(·,·).If there is a polynomial α(x,y)of degree t such thatϕ(αp,q)=ϕ(p,αq)for all p,q ∈P ,then {Φn }∞n =0satisﬁes the (2t +1)term recurrence relationα(x,y)Φn =n +ti =n −t C n,i Φi ,C n,n −t =0,where C n,i is a constant matrix of order (n +1)×(i +1)for n −t i n +t.Proof.Let α(x,y)Φn =Σn +t i =0C n,i Φi .Then for 0 k <n −t C n,k ϕ Φk ,ΦT k =ϕ n +t i =n −tC n,i Φi ,ΦT k =ϕ α(x,y)Φn ,ΦT k=ϕ Φn ,α(x,y)ΦT k =0since deg α(x,y)ΦT k <n .Thus we have C n,k =0for 0 k <n −t.For k =n −t,we see that C n,n −t =ϕ Φn ,ΦT n ˜C T n ϕ Φn −t ,ΦT n −t −1=0since if we write α(x,y)Φn −t = n i =0˜C i Φi (˜C n =0),we have C n,n −t ϕ Φn −t ,ΦT n −t =ϕ α(x,y)Φn ,ΦT n −t =ϕ Φn ,α(x,y)ΦT n −t =ϕ Φn ,n i =0ΦT i ˜C T i =ϕ Φn ,ΦT n ˜C T n .24.Second order partial differential equations and Sobolev orthogonal polynomials in two variablesIn this section,we are concerned with polynomials in two variables which satisfy an admissi-ble second order partial differential equationL [u ]:=Au xx +2Bu xy +Cu yy +Du x +Eu y = ax 2+d 1x +e 1y +f 1 u xx +(2axy +d 2x +e 2y +f 2)u xy + ay 2+d 3x +e 3y +f 3 u yy +(gx +h 1)u x +(gy +h 2)u y=λn u,(4.1)J.K.Lee,L.L.Littlejohn /J.Math.Anal.Appl.322(2006)1001–10171009and are orthogonal relative to a symmetric bilinear form ϕ(·,·)of the formϕ(p,q)= σ,pq + τ,p x q x ,(4.2)where σand τare moment functionals.If {Φn }∞n =0is a SOPS relative to ϕ(·,·)in (4.2),then σis a constant multiple of the canon-ical moment functional of {Φn }∞n =0since ϕ(φi,j ,1)= σ,φi,j + τ,∂x φi,j ∂x 1 = σ,φi,j for i +j 1.Theorem 4.1.Let ϕ(·,·)be a symmetric bilinear form in (4.2).The following statements (i)and (ii)are equivalent.(i)The partial differential operator L [·]in (4.1)is symmetric on polynomials in the sense that ϕ L [p ],q =ϕ p,L [q ] for all p,q ∈P ,(4.3)(ii)σand τsatisfy the relations M 1[σ]:=(Aσ)x +(Bσ)y −Dσ=0,M 2[σ]:=(Bσ)x +(Cσ)y −Eσ=0,(4.4) M (x)1[τ]:=(Aτ)x +(Bτ)y −(D +A x )τ=0,M (x)2[τ]:=(Bτ)x +(Cτ)y −(E +2B x )τ=0,(4.5)C x τ=0.(4.6)Furthermore,if {Φn }∞n =0is a SOPS relative to ϕ(·,·),the statements (i)and (ii)are equiva-lent to(iii){Φn }∞n =0satisﬁes the partial differential equation (4.1).Proof.(i)⇔(ii).Since we have,by (2.1),(2.2)and Lemma 2.3,for all p,q ∈Pϕ L [p ],q = L ∗[qσ]−L ∗[q xx τ]−L ∗[q x τx ],p ,ϕ p,L [q ] = L [q ]σ− L [q ] xx τ− L [q ] x τx ,p ,we can see that (4.3)is equivalent to L ∗[qσ]−L [q ]σ+ L [q ] x τx −L ∗[q x τx ]+ L [q ] xx τ−L ∗[q xx τ]=0for all q ∈P ,which can be written asqL ∗[σ]+q x 2M 1[σ]+D x τx −L ∗[τx ] +q y M 2[σ]+q xx A x τx −2M 1[τx ]+(A xx +2D x )τ−L ∗[τ] +q xy 2B x τx −2M 2[τx ] +q yy C x τx +q xxx 2A x τ−2M 1[τ] +q xxy 4B x τ−2M 2[τ] +2q xyy C x τ=0.Thus we have the following set of equations for σand τ:L ∗[σ]=0,(4.7)2M 1[σ]+D x τx −L ∗[τx ]=0,(4.8)1010J.K.Lee,L.L.Littlejohn/J.Math.Anal.Appl.322(2006)1001–1017M2[σ]=0,(4.9)A xτx−2M1[τx]+(A xx+2D x)τ−L∗[τ]=0,(4.10)B xτx−M2[τx]=0,(4.11)C xτx=0,(4.12)A xτ−M1[τ]=0,(4.13)2B xτ−M2[τ]=0,(4.14)C xτ=0.(4.15)Here,we observe thatL∗[σ]=M1[σ]x+M2[σ]y,M1[τx]=M(x)1[τ]x+D xτ−(B xτ)y,M2[τx]=M(x)2[τ]x+B xτx−C xτy,M1[τ]=M(x)1[τ]+A xτ,M2[τ]=M(x)2[τ]+2B xτ.(4.16) By using our observations(4.16),Lemma2.3and C x=d3,after the tedious calculations,wecan write(4.8),(4.10),(4.11)and(4.14)in a simpler form as the followings:2M1[σ]+D xτx−L∗[τx]=2M1[σ]−M(x)1[τ]xx−M(x)2[τ]xy+(C xτ)xy,A xτx−2M1[τx]+(A xx+2D x)τ−L∗[τ]=−3M(x)1[τ]x−M(x)1[τ]y,B xτx−M2[τx]=−M(x)2[τ]x−(C xτ)y,A xτ−M1[τ]=−M(x)1[τ],2B xτ−M2[τ]=−M(x)2[τ].Note that all the relations(4.7)–(4.15)are expressed in terms of M i[σ],M(x)i[τ]and C xτ.If M i[σ]=0,M(x)i[τ]=0for i=1,2and C xτ=0,we have(4.7)–(4.15),which implies (4.3).Conversely,if(4.7)–(4.15)hold true,then we can easily see that(4.4)–(4.6)hold.Now assume that{Φn}∞n=0is a SOPS relative toϕ(·,·)and{P n}∞n=0be the normalization of {Φn}∞n=0.(i)⇒(iii).Since L[P n]is a vector of polynomials of degree n,we may writeL[P n]=nk=0C n,k P k(4.17)for some constant matrices C n,k of order(n+1)×(k+1)for0 k n.Then for0 j<n,C n,jϕP j,P T j=ϕnk=0C n,k P j,P T j=ϕL[P n],P T j=ϕP n,LP T j=0.Hence C n,j=0for0 j<n and L[P n]=λn P n by comparing the coefﬁcients in both sides of(4.17).(iii)⇒(i).Since L [P n ]=λn P n for n 0,we have for m =nϕ L [P n ],P T m −ϕ P n ,L P T m =(λn −λm )ϕ P n ,P T m =0.Thus we have (i)by linearity.2As we remarked in introduction,all partial differential equations investigated by Krall andSheffer satisfy C x =0up to a linear change of variables.And if C x =0,we have no new result because we have τ=0.Thus it is natural to assume that C x =0.Then we have the following result.Theorem 4.2.Let {Φn }∞n =0a SOPS with respect to a symmetric bilinear form ϕ(·,·)in (4.2).If C x =0,then the followings are equivalent.(i){Φn }∞n =0satisﬁes the partial differential equation (4.1).(ii)σand τsatisfy moment equations (4.4)and (4.5),respectively.If a SOPS {Φn }∞n =0satisﬁes the partial differential equation (4.1)with C x =0,we can seethat by Theorems 2.5and 4.1,{Φn }∞n =0is a WOPS {Φn }∞n =0relative to σ.Similarly,we knowthat any PS is a WOPS relative to τif it satisﬁes the partial differential equationAv xx +2Bv xy +Cv yy +(D +A x )v x +(E +2B x )v y =μn u,(4.18)where μn =an(n +1)+gn.In fact,the PS consisting of partial derivatives of {Φn }∞n =0is a WOPS relative to τsince it satisﬁes the differential equation (4.18)(see [9,Theorem 3.8]).Theorem 4.3.If L [p ]=λp and L [q ]=μq for λ=μ,then polynomials p and q are orthogonal with respect to a symmetric bilinear form (4.2),i.e.,ϕ(p,q)= σ,pq + τ,p x q x =0for any solutions σand τof (4.4)and (4.5).Proof.It sufﬁces to observe that(λ−μ)ϕ(p,q)=ϕ(λp,q)−ϕ(p,μq)=ϕ L [p ],q −ϕ p,L [q ]=0by Theorem 4.1(i).2Theorem 4.4.Let {Φn }∞n =0be a SOPS relative to ϕ(·,·)and satisfy the admissible partial differ-ential equation (4.1)with C x =0.Suppose that there is a polynomial f (x,y)of degree 2such thatAf x +Bf y −A x f =0,Bf x +Cf y −2B x f =0.Then(i)if σis quasi-deﬁnite,then {Φn }∞n =0is an OB relative to σ.Moreover,there is a polynomial f (x,y)such thatτ=kf (x,y)σfor some constant k.If τ=0,then {P (x)n }∞n =0is a monic OB relative to τ,where {P (x)n }∞n =0is a monic PS obtained from the normalization {P n }∞n =0of {Φn }∞n =0through the partial differ-entiation with respect to x ,deﬁned byP (x)n −k,k =1n +1−k∂x P n +1−k,k for 0 k n.(4.19)(ii)If τis quasi-deﬁnite,then {Φn }∞n =0is an OB relative to σ.Moreover,there is a polynomialf (x,y)such thatf (x,y)σ=kτfor some constant k.Hence f (x,y)σ=0or f (x,y)σis quasi-deﬁnite.Proof.(i)Let {Q n }∞n =0be a monic OB relative to σ.Since σsatisﬁes (4.4),{Q n }∞n =0satisfy thepartial differential equation (4.1).Then Q n =P n for all n 0by the uniqueness of monic PSsolutions to the partial differential equation (4.1)where {P n }∞n =0is the normalization of {Φn }∞n =0.Thus {Φn }∞n =0is an OB relative to σ.On the other hand,τand f (x,y)σsatisfy the same Eq.(4.5),which are the moment equations corresponding to the partial differential equation (4.18).Since the partial differential equation (4.18)is admissible,the moment equations (4.5)have the unique solution by Theorem 2.4.Thus there is a constant k such thatτ=kf (x,y)σsince {P (x)n }∞n =0is a unique monic PS satisfying the partial differential equation (4.18)and it is amonic OB relative to f (x,y)σsatisfying the partial differential equation (4.18)if τ=0.(ii)Since σis a constant multiple of the canonical moment functional of {Φn }∞n =0and satisﬁes the moment equations (4.4),{Φn }∞n =0is an OB relative to σby Theorem 2.5.Next,we see that there is a constant k such thatf (x,y)σ=kτsince τand f (x,y)σsatisfy the same moment equation (4.5)corresponding to the admissible partial differential equation (4.18).Hence f (x,y)σ=0or f (x,y)σis quasi-deﬁnite.2Remark 4.1.The assumption in Theorem 4.4holds for almost OPS’s investigated by Krall and Sheffer [6]and Kwon et al.[7].Further,we refer [9]for interesting properties of polynomial solutions satisfying the differential equation (4.1)with A y =0and C x =0.5.ExamplesIn this section,we provide examples of SOPS’s which satisfy the partial differential equa-tion (4.1)with C x =0and are orthogonal with respect to a symmetric bilinear form (4.2).All differential equations were dealt by Krall and Sheffer [6].Example 5.1.Consider the differential equationxu xx +u yy +(1+α−x)u x −yu y +nu =0.(5.1)We know that (5.1)has a PS {Φn }∞n =0as solutions,whereφn −k,k (x,y)=L (α)n −k (x)H k 1√y ,{L (α)n (x)}∞n =0are Laguerre polynomials and {H n (y)}∞n =0are Hermite polynomials given by L (α)n (x)=n k =0n +αn −k (−x)kk !,H n (y)=[n/2] k =0(−1)k k !(n −2k)!y n −2k4k.We note that for Laguerre polynomials,we have the relations d dxL (α)n (x)=(−1)L (α+1)n −1(x),L (−1)n (x)=(−x)L (0)n −1(x),n 1.(5.2)Since C x =0,by Theorem 4.1,σand τsatisfy the equations(xσ)x =(1+α−x)σ,σy =−yσ, (xτ)x =(2+α−x)τ,τy =−yτ.(5.3)Case 1.α>−1.By solving (5.3),we have the distributional representations for σand τσ=x αe −x e −12y 2dx dy,τ=x α+1e −x e −12y 2dx dy,and {Φn }∞n =0is an OPS relative to σ.Furthermore,{Φn }∞n =0is a SOPS relative to the Sobolevinner productϕ(p,q)=∞ 0∞−∞p(x,y)q(x,y)x αe −x e −12y 2dx dy+∞ 0∞−∞p x (x,y)q x (x,y)x α+1e −x e −12y 2dx dy.In fact,we have the orthogonality relationϕ(φn −k,k φm −j,j )= σ,φn −k,k φm −j,j + τ,∂x φn −k,k ∂x φm −j,j=σ,L (α)n −k (x)L (α)m −j (x)H k y √ H j y √+τ,L (α) n −k (x)L (α)m −j (x)H k y √2 H j y √2= H (x)x αe −x ,L (α)n −k (x)L (α)m −j (x)e −12y 2,H k y √2 H j y √2 + H (x)x α+1e −x ,L (α) n −k (x)L (α)m −j (x)e −12y 2,H k y √2 H j y √2 =δm,n δk,j L (α)n −k 2+ L (α+1)n −k −1 2 e −12y 2,H 2k y √2 .Case 2.α=−1.Then we haveσ=δ(x)dx ⊗e −12y 2dy,τ=e −x e −12y 2dx dy,(⊗means the tensor product)and we know that {L (−1)n −k (x)H k (y/√)}∞n =0,n k =0is a WOPS relative to σ.Moreover,{L (−1)n −k (x)H k (y/√2)}∞n =0,n k =0is a SOPS relative to the Sobolev inner productϕ(p,q)=∞−∞p(0,y)q(0,y)e−12y2dy +∞ 0∞ −∞p x (x,y)q x (x,y)e −x e −12y 2dx dy.We can see that ϕ(φn −k,k φm −j,j )=K n,k δm,n δk,j (K n,k =0for each n,k 0)from the followingcalculation:ϕ(φn −k,k φm −j,j )= σ,φn −k,k φm −j,j + τ,∂x φn −k,k ∂x φm −j,j = σ,L (−1)n −k (x)L (−1)m −j (x)H k y √ H j y √+ τ,L (−1) n −k (x)L (−1)m −j (x)H k y √ H j y √= δ(x),L (−1)n −k (x)L (−1)m −j (x)e −12y 2,H k y √ H j y √+ H (x)e −x ,L (0)n −k −1(x)L (0)m −j −1(x)e −12y 2,H k y √2 H j y √2=δk,j δn −k −1,0δm −j −1,0+δm,n H (x)e −x ,L (0)n −k −1 2 e −12y 2,H 2k y √=⎧⎪⎨⎪⎩0,k =j, e −12y 2,H 20 y √ ,m =n =0,[δn −k,0+ L (0)n −k −1(x) 2] e −12y 2,H 2ky √2,m =n 1,k =j.Here we used the relations (5.2).This example was motivated by Kwon and Littlejohn(see [8]),who dealt with the Sobolev orthogonality of {L (−k)n (x)}∞n =0(k a positive integer).Example 5.2.Consider the differential equationx 2−1 u xx +2xyu xy + y 2−1u yy +gxu x +gyu y =n(n +g −1)u.(5.4)In [7],we showed that the partial differential equation (5.4)has an OPS {Φn }∞n =0as solutions if g =1,0,−1,...,whereφn −k,k (x,y)=P (g +2k −22,g +2k −22)n −k (1−x 2)k 2P (g −32,g −32)k y √2 (0 k n),(5.5)and {P (α,β)n(x)}∞n =0are Jacobi polynomials given byP (α,β)n(x)=n k =0n +αk n +βn −k(x −1)n −k (x +1)k .。