Solution of the D-dimensional Klein-Gordon equation with equal scalar and vector ring-shape
Advanced determinant calculus
C. KRATTENTHALER†
math.CO/9902004 v3 31 May 1999
Institut f¨ ur Mathematik der Universit¨ at Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: kratt@pap.univie.ac.at WWW: http://radon.mat.univie.ac.at/People/kratt
1
2
C. KRATTENTHALER
or maybe
1≤i,j ≤n
det
x+y+j x − i + 2j
−
x+y+j x + i + 2j
.
ideas would you have? (Just to tell you that I do not ask for something impossible: Each of these four determinants can be evaluated in “closed form”. If you want to see the solutions immediately, plus information where these determinants come from, then go to (2.7), (2.17)/(3.12), (2.19)/(3.30), respectively (3.47).) Okay, let us try some row and column manipulations. Indeed, although it is not completely trivial (actually, it is quite a challenge), that would work for the first two determinants, (1.1) and (1.2), although I do not recommend that. However, I do not recommend at all that you try this with the latter two determinants, (1.3) and (1.4). I promise that you will fail. (The determinant (1.3) does not look much more complicated than (1.2). Yet, it is.) So, what should we do instead? Of course, let us look in the literature! Excellent idea. We may have the problem of not knowing where to start looking. Good starting points are certainly classics like [119], [120], [121], [127] and [178]1. This will lead to the first success, as (1.1) does indeed turn up there (see [119, vol. III, p. 311]). Yes, you will also find evaluations for (1.2) (see e.g. [126]) and (1.3) (see [112, Theorem 7]) in the existing literature. But at the time of the writing you will not, to the best of my knowledge, find an evaluation of (1.4) in the literature. The purpose of this article is threefold. First, I want to describe a few useful and efficient tools which should enable you to evaluate nontrivial determinants (see Section 2). Second, I provide a list containing a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared (see Section 3). Third, even if you should not find your determinant in this list, I point out references where further such determinant evaluations can be found, maybe your determinant is there. Most important of all is that I want to convince you that, today, Evaluating determinants is not (okay: may not be) difficult! When George Andrews, who must be rightly called the pioneer of determinant evaluations, in the seventies astounded the combinatorial community by his highly nontrivial determinant evaluations (solving difficult enumeration problems on plane partitions), it was really difficult. His method (see Section 2.6 for a description) required a good “guesser” and an excellent “hypergeometer” (both of which he was and is). While at that time especially to be the latter was quite a task, in the meantime both guessing and evaluating binomial and hypergeometric sums has been largely trivialized, as both can be done (most of the time) completely automatically. For guessing (see Appendix A)
低维拓扑简介
考研论坛»数学»低维拓扑knight51发表于2005-7-28 08:34低维拓扑<P>下面说说低维拓扑的内容：低维拓扑是微分拓扑的一部分，主要研究3，4维流形与纽结理论。
又叫几何拓扑。
主要以代数拓扑与微分拓扑为工具。
它与微分几何和动力系统关系密切。
国外搞这个方向的也几乎都搞微分几何和动力系统。
我国这个方向北大最牛，美国是伯克利和普林斯顿最牛。
比起代数几何来，它比较好入门。
初学者只需要代数拓扑，微分拓扑，黎曼几何的知识就行了。
美国这方面比较牛，几乎每个搞基础数学研究的都会低维拓扑。
</P><DIV class=postcolor>纠正一下上面的错误，美国也不是每个搞基础数的都精通低维拓扑，而是懂一些低维拓扑的知识。
如果入门后还想更加深入了解它，那还需要读一些双曲几何和拓扑动力系统的书。
</DIV><!-- THE POST --><!-- THE POST --><DIV class=postcolor>下面介绍一下这方面的牛人：Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplaybetween analysis, topology and geometry.Wall [8] goes on to describe Thurston's work in more detail:-The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).</DIV><!-- THE POST -->第2个牛人：Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaréconjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.Besides these results about closed simply connected four manifolds, Freedman also proved:(a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R.(b) There is a nonsmoothable closed four manifold.© The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:-My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.<DIV class=postcolor>介绍第3个牛人：Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case.In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:-Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold.Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-(1) Differential geometry of holomorphic vector bundles.(2) Applications of gauge theory to 4-manifold topology.and he relates his contribution to that of many others in the field.Donaldson's work in summed up by R Stern in [6]:-In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds</DIV><DIV class=postcolor>下面continue介绍第4个牛人：Robion Kirby。
TheKlein-Gordonequation：克莱因戈登方程
where the Lagrangian density satisfies the Euler-Lagrange equations of motions
(25)
such that the Euler-Lagrange equations of motion just give the Klein-Gordon equation (12) and its complex conjugate.
as the basic field equation of the scalar field.
The plane waves (10) are basic solutions and the field (9) is constructed by
a general superposition of the basic states.
Quantization
The challenge is to find operator solutions of the Klein-Gordon equation (12) which satisfy eq. (28). In analogy to the Lagrange density (24) , the hamiltonian is
Lecture 8
The Klein-Gordon equation
WS2010/11: ‚Introduction to Nuclear and Particle Physics‘
The bosons in field theory
Bosons with spin 0
scalar (or pseudo-scalar) meson fields
(23)
改进求解 Woods －Saxon 势和 Poschl －Teller 势的方法
改进求解 Woods －Saxon 势和 Poschl －Teller 势的方法郝海玲;闫景富【期刊名称】《实验科学与技术》【年(卷),期】2015(000)004【摘要】用 Wang 的 Obrechkoff 数值方法来求解常见的 Schr dinger 方程，即两步高阶微商。
该方法的特点是采用增加奇数高阶微商使得数值结果 P 稳定。
Schr dinger 方程中，例如一维的 Woods －Saxon 势和 P schl －Teller 势，使用该方法计算后，不仅提高了计算效率，也提高了数值结果的精度。
%In this paper,we focus on the new kind of P -stable two -step Obrechkoff method for the ultra -high -accurate solution of a one -dimensional Schr dinger equation.Through improving the Wang's method,we develop a new kind of P -stable two -step Ob-rechkoff method by adding the odd higher -order derivatives.This proposed method is very effective but has very high local truncation error.We apply our new method to the one -dimensional Schr dinger equation such as the well -know Woods -Saxon potential and P schl -Teller potential.Their numerical solution testified that the new method is very reliability.【总页数】4页(P16-18,91)【作者】郝海玲;闫景富【作者单位】晋中职业技术学院基础部，山西晋中 030600;中国石油大学北京信息学院，北京 102249【正文语种】中文【中图分类】O411【相关文献】1.点正则变换下Woods-Saxon势映射P(o) schl-Teller I势 [J], 汪菁;郭建友2.近似解析求解Woods-Saxon势场Klein-Gordon方程 [J], 陈文利;史艳维;冯晶晶3.一维修正Poschl—Teller型势的Dirac方程的束缚态 [J], 陈昌远;孙国耀4.利用形状不变性求解变形Woods-Saxon势的能量本征值 [J], 陈文利;冯晶晶;樊亚云5.运用形状不变性技术计算Poschl—Teller Ⅰ势的能量本征值 [J], 姚小科;贾春生因版权原因，仅展示原文概要，查看原文内容请购买。
非线性Klein-Gordon方程的最低阶混合元超收敛分析新模式
非线性Klein-Gordon方程的最低阶混合元超收敛分析新模式樊明智;王芬玲【摘要】针对一类非线性Klein-Gordon方程利用最简单的双线性元Q11及Q01×Q10元建立了最低阶且自然满足Brezzi-Babuska条件的混合元逼近格式.基于双线性元的积分恒等式结果,建立了插值与Riesz投影之间的超收敛估计,再结合Q01×Q10元的高精度分析结果和插值后处理技术,在半离散和全离散格式下,导出了关于原始变量u和流量p分别在H1模和L2模意义下单独利用插值或Riesz投影所无法得到的超逼近性和超收敛结果.【期刊名称】《许昌学院学报》【年(卷),期】2016(035)005【总页数】9页(P1-9)【关键词】非线性Klein-Gordon方程;超逼近性和超收敛结果;混合有限元新模式;半离散和全离散格式【作者】樊明智;王芬玲【作者单位】许昌学院数学与统计学院,河南许昌461000;许昌学院数学与统计学院,河南许昌461000【正文语种】中文【中图分类】O242.21本文考虑如下的非线性Klein-Gordon方程其中Ω⊂R2为有界矩形区域，∂Ω为Ω的边界，X=(x,y)，f(X,t)∈L2(Ω)，γ是正常数，u0(X)，u1(X)是已知充分光滑的函数，假设a(u)，g(u)满足如下条件：(i)a(u)关于u一致有界，即存在正常数a0,a1满足,a0≤a(u)≤a1；(ii)a(u)和g(u)对变量满足Lipschitz条件，即存在正常数L使得Klein-Gordon方程具有丰富的实际背景和物理意义，它用于描述相对论量子力学和量子场论中自旋为零的粒子的最基本方程和Schrödinger方程的相对论形式，对于它的研究值得物理学家和数学家的高度关注.关于Klein-Gordon方程已有研究, 例如文献[1]对无界区域上一维Klein-Gordon方程建立一个显式差分格式, 并给出该格式的稳定性和收敛性结果；文献[2]和[3]研究了一维情形下的数值解；文献[4]讨论了二维Klein-Gordon方程存在唯一的整体解.不难看出,当a(u)=1和g(u)=sinu时，问题(1)变成了sine-Gordon方程, 因此sine-Gordon方程是问题(1)的特殊情况, 并得到一些有价值的成果[5～8], 由于Klein-Gordon方程比sine- Gordon方程复杂, 使问题的处理更为困难. 因此据我们所知到目前为止尚未见到有关问题(1)混合元格式的高精度分析.混合有限元方法与传统Galerkin有限元方法相比具有对空间的光滑度要求较低、并能同时得到原始变量和流量的误差估计等突出优势, 已成为一种常用的数值逼近方法.对于经典的混合有限元格式来说,混合元空间要满足Brezzi-Babuška条件[9，10] (简记为B-B条件),给构造合适的空间对带来一定的困难. 最近文献[11、12]给出了二阶椭圆问题新的混合有限元逼近格式, 具有自由度小且当逼近空间对满足包含关系时, B-B条件成立,同时又能避开散度算子带来的困扰等特点,文献[13]将此方法应用到线性抛物方程, 给出关于时间半离散混合格式和全离散化混合有限元格式,但仅仅得到了最优误差估计,文献[14]及[15]进一步研究了二阶椭圆问题和线弹性问题在新格式下的超收敛性,文献 [16]将其推广到线性Sobolev方程得到了非协调混合元格式半离散格式下的超收敛性和向后欧拉全离散格式下关于空间步长的超逼近性.该文目的是将Riesz投影的优势和插值的高精度分析方法的特色有机的结合起来, 利用Q11及Q01×Q10元针对方程(1)给出了最低阶混合元超收敛分析新模式. 借助于双线性元的高精度分析结果[17],得到了精确解的双线性插值与其Riesz投影之间的超收敛估计. 进一步的结合插值后处理技巧, 在半离散和全离散格式下,分别导出了原始变量u和分别在H1模和L2模意义下单独利用Riesz投影和插值技巧所无法得到的超逼近性和超收敛结果.设Th为Ω上的一族矩形剖分，∀K∈Th定义单元K的中心为(xK,yK)，其边长分别为2hx,K，2hy,K，取}.单元K的顶点为a1(xK-hx,K,yK-hy,K)，a2(xK+hx,K,yK-hy,K)，a3(xK+hx,K,yK+hy,K)，a4(xK-hx,K,yK+hy,K)，单元K的边为(mod4). 定义有限元空间：其中Qij=span{xrys,0≤r≤i，0≤s≤j}.设和分别为和所诱导的插值算子,且满足这里是对应边∂K的单位切向量.引理1[17] 若u∈H3(Ω)，则进一步地，若u∈H4(Ω)，则若，则利用文献[17]中类似的方法可得如下的结论：引理2 若u∈H3(Ω)则证明为了得到高精度估计引进误差函数[17]：令u-Ihu=φ，对ω1∈Q01做Taylor展开有于是注意到‴，根据分部积分公式得∫Kφxdxdy=∫KF″(y)φxdxdy=(∫l3-∫l1)F′(y)φxdxdy-∫KF′(y)φxydx= F′(yK+hy,K)(φ(xK+hx,K,yK+hy,K)-φ(xK-hx,K,yK+hy,K))-F′(yK-hhy,K)(φ(xK+hx,K,yK-hy,K)-φ(xK-hx,K,yK-hy,K))-(F(yK+hy,K)∫l3φxydx-F(yK-hy,K)∫l1φxydx)+∫KF(y)φxyy=∫KF(y)uxyy,∫Kφx(y-yK)dxdy=‴y.根据式(6)～(8)和逆不等式可知∫Kφxω1dxdy=.同理可证利用式(9)和(10)该引理2得证.设的Riesz(或椭圆)投影算子,即对，满足并且Rh具有性质：设，有接下来，我们给出插值和投影之间的超收敛估计.引理3 若,则.证明根据式(2)和(11)得((Rhu-Ihu)，(Rhu-Ihu))=((Rhu-u)，(Rhu-Ihu))+((u-Ihu)，(Rhu-Ihu))≤ch2|u|3|Rhu-Ihu|1.从而引理3得证.令=-u，则方程(1)可改写为令则问题(13)的变分形式为：求,使得其满足变分问题(14)的有限元逼近方程为：求满足定理1 设和分别是(14)和(15)的解,则有其中.证明首先令ξ.∀，根据式(14)和(15)有如下误差方程在(19a)和(19b)中分别令vh=ξt和ξt，并将(19a)+γ(19b)得(ξ,(η,.由Cauchy和Young不等式及式(12)可知基于式(11)得由假设(i)和(ii)可知,,根据式(21)～(24)将(20)变形为对(25)从0到t积分,并注意到ξ(X,0)=ξt(X,0)=0得将Gronwall引理应用于(26)可知借助(27)和引理3有，从而(16)式得证.另一方面，利用引理2和引理3得在 (19b)中利用引理2和式(4)、(26)及(27)得,即，从而式(17)成立，定理1证毕.注1 若将式(15)和(18)中的Rh换成插值Ih时可得如下结论：(1)结合式(3)可得半离散超逼近结果此时，u∈H4(Ω)的要求比本文定理1中的u∈H3(Ω)的光滑度要高.(Ⅱ)借助于文献[5～7]中的导数转移技巧有显然与定理1相比对ut的光滑度要求稍高.注2 若仅用投影时虽然可以得到关于空间步长的超逼近性，但如何构造关于投影的后处理算子仍然是悬而未决的问题.因此，到目前为止无法直接得到关于投影的超收敛结果.为了得到整体超收敛，我们先把Th相邻的四个小单元合并构成一个大单元(如图1).并设Zi(i=1,2,…,9)为四个小单元的所有顶点.在上根据文献[17]构造具有如下性质的插值后处理算子:其中)为上的连续函数空间，则算子分别满足如下性质：定理2 在定理1的条件下，有如下的整体超收敛结果证明利用定理1和(29)可知，).即式(31)得证.同理借助定理1和式(30)可证式(32)，定理2证毕.在本节中我们将主要讨论全离散格式下的误差估计，仅讨论a(u)=a(X)的情形.设0=t0≤t1≤…tN-1≤tN=T是上步长为τ=T/N的剖分，tn=nτ, n=0,1,2,…,N,Un 代表t=tn时u(tn)在Vh中的逼近.为了方便起见，我们引入下面一些记号：定义(14)全离散逼近格式为：求满足其中utt(X,0)=-a(X)u1(X)+γΔu0(X)-g(u0(X))+f(X,0).定理和分别是(14)和(33)的解，则有其中.证明为了进行误差估计，记∀由(14)和(33)可导出如下误差方程其中根据(37b)可得在(37a)和(38)中分别取,并将(37a)+γ(38)有,不难看出，(39)的右端各项分别变形为接下来我们给出(39)式右端的估计，注意到借助于(43)有根据(11)可知利用假设(i)和插值理论得G3=.利用假设(i)和(43)将G4估计为.借助泰勒展开式直接计算有利用式(48)有.综合式(40)～(42)、(44)～(47)及(49)并取得其中.对式(50)关于j从1到n-1求和得).由初始条件和泰勒展开式得因此，再结合U1的定义可知再借助于式(52)并注意到ξ0=0得利用式(53)将式(51)变形为其中(ξn，ξn-1).根据Young不等式得，(ξn，，得选择适当小的τ，使1-cτ>0，再根据离散的Gronwall引理得.利用引理3和三角不等式有即式(34)得证.在式(37b)中令h=θn，再利用(4)和引理2及1的估计结果得，定理3得证.注3 若将式(36)中的Rh换成插值Ih时，结合(3)可得如下结论：其中.此时式(55)和式(56)中对解的光滑度要求比定理3偏高.注4 本文方法对抛物方程、双曲方程、抛物积分微分方程、双曲积分微分方程均使用.【相关文献】[1] Han H D. Zhang Z W. An analysis of the finite difference method for one-dimensional Klein- Gordon equation on unbounded domain[J]. Appl Numer Math, 2009, 59(7): 1 568-1 583.[2] Khalifa M E, Elgamal M. A numerical solution to Klein-Gordon equation with Dirichlet doundary condition[J]. Applied Mathematics Computation, 2005, 160(2): 451-475.[3] Wang Q F, Cheng D Z. Numerical solution of damped-nonlinear Klein-Gordon equations using variational method and finite element approach[J]. Applied Mathematics Computation, 2005, 162(1): 381-401.[4] Nakao H, Pavel I N. Wave operators to a quadraticnon nonlinear Klein-Gordon equation in two space dimensions[J]. Nonlinear Analysis TMA, 2009, 71(9): 3 826-3 833. [5] Shi D Y, Zhang D. Approximation of nonconforming quasi-Wilson element for sine-Gordon equa-tion[J]. Journal of Computational Mathematics, 2013,31(3):271-282.[6] 石东洋,张斐然.Sine-Gordon方程的一类非协调有限元分析[J].计算数学,2011,33(3):289- 297.[7] 王芬玲,石东洋.非线性sine-Gordon方程Hermite型有限元新的超收敛分析及外推[J].应用数学学报,2012,35(5):777-788.[8] 石东洋,王芬玲,赵艳敏.非线性sine-Gordon方程的各向异性线性元高精度分析新模式[J].计算数学,2014,36(3):245-256.[9] Babuska I. Error-bounds for finite element method[J]. Numerical Mathematics, 1971,16: 322- 333.[10] Brezzi F.On the existence,uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[J]. SIAM Journal on Numerical Analysis, 1974, 13: 185-197.[11] 陈绍春,陈红如.二阶椭圆问题新的混合元格式[J].计算数学,2010,32(2):213-218.[12] 史峰,于佳平,李开泰.椭圆型方程的一种新型混合有限元格式[J].工程数学学报,2011,28(2):231-236.[13] 李磊,孙萍,罗振东.抛物方程一种新混合有限元格式及误差分析[J].数学物理学报,2012,32A(6):1 158-1 165.[14] 石东洋,李明浩.二阶椭圆问题一种新格式的高精度分析[J].应用数学学报,2014,37(1):45-58.[15] Shi D Y, Li M H. superconvergence anaalysis of a stable conforming rectangular mixed finite ele-ments for the linear elasticity problem[J]. Journal of Computational Mathematics, 2014, 32(2): 205-214.[16] Shi D Y, Zhang Y D. High accuracy analysis of a new nonconforming mixed finite element sch-eme for Sobolev equation[J]. Applied Mathematics Computation, 2011, 218(7): 3 176-3 186.[17] 林群,严宁宁.高效有限元构造与分析[M].保定:河北大学出版社,1996.。
The Quantum Zero Space Charge Model for Semiconductors
The energy functional E involves two dimensionless parameters, namely ; . In applications one frequently has 2] 2 1: One may expect that the smallness of 2 expresses some redundancy of the model equations. The question arises whether the thermal equilibrium solution of the QHD is close to the solution (if it exists at all) of a limiting model where is set to zero. Such models will be called "quantum zero-space-charge models" ("QZSCs"). Typical for QZSCs is the involvement of quantum mechanical terms and the replacement of Poisson's equation by the "zero-space-charge assumption" n ? p ? C = 0. Whether QZSCs are a good approximation or not depends not only on the order of magnitude of 2 but also on the ration of 2 and 2 which equals 2 2 sup jC j 2 = K h (T )2 ;
配位化学在工业中的应用
配位化学在工业中的应用配位化学又称络合物化学，配位化合物简称配合物或络合物。
配合物是由一个或几个中心原子或中心离子与围绕着它们并与它们键合的一定数量的离子或分子（这些称为配位体）所组成的。
配位化合物在化学工业和生活中起着重要的作用，1963年化学诺贝尔奖金联合授给德国M.普朗克学院的K.齐格勒博士和意大利米兰大学的G.纳塔教授。
他们的研究工作是发展了乙烯的低压聚合，这使数千种聚乙烯物品成为日常用品。
齐格勒-纳塔聚合催化剂是金属铝和钛的配合物。
而今，配位化学的研究已经有了很大的突破，现代配位化学理论在推进工业研究中得到了应用并成为工业设计原理的一个组成部分。
1、配位化学的前期发展历程配合物在自然界中普遍存在，历史上最早有记载的是1704 年斯巴赫（Diesbach）偶然制成的普鲁士蓝KCN·Fe(CN)2·Fe(CN) 3，其后1798 年塔斯赫特（Tassert）合成[Co(NH3)6]Cl3。
十九世纪末二十世纪初，创立了配位学说，成为化学历史中重要的里程碑。
二十世纪以来，配位化学作为一门独立的学科，以其蓬勃发展之势，使传统的无机化学和有机化学的人工壁垒逐渐消融，并不断与其他学科如物理化学、材料科学及生命科学交叉、渗透，孕育出许多富有生命力的新兴边缘学科，为化学学科的发展带来新的契机[1]。
2、配位化学新的发展及应用趋势本世纪60 年代初期，由于发现了一批具有金属- 金属化学键的配合物，配位化学的研究重点从单核配合物转向多配合物，从而开始了对多金属偶合体系的研究。
在此研究过程中，发现很早已为人们熟知利用的普鲁士蓝等一类混合价配合物，不仅可以用于传统的染料工业，还可以更广泛地应用于陶瓷、矿物、材料科学、高温超导等许多领域。
如可用于合成高导电率的分子金属和超导材料、磁性材料、优良的非线性光学材料以及非线性导电材料等。
因此，此类配合物引起各个学科研究者，如合成化学家、固体化学家、地质学家、生物学家、物理学家的极大兴趣，成为当前化学基础研究的前沿领域。
随机常微分方程的龙格库塔解法(英文)
随机常微分方程的龙格库塔解法(英文)The Runge-Kutta method, also known as the RK method, is a numerical technique used to solve ordinary differential equations (ODEs). It is a widely used method in the field of computational physics and engineering, as it is relatively simple to implement and can often provide good approximations of the solutions to ODEs.The basic idea behind the Runge-Kutta method is to divide the interval over which the ODE is to be solved into a series of smaller intervals, and to use the known values of the variables at the start of each interval to estimate their values at the end of the interval. This is done using a weighted average of the derivative of the variables at different points within the interval.There are several variations of the Runge-Kutta method, including the popular fourth-order method, which uses four estimates of the derivative to compute the final solution. The accuracy of the method can be improved by using higher-order versions, but at the cost of increased computational complexity.In summary, the Runge-Kutta method is a useful tool for solving ODEs,particularly when an analytical solution is not available. It is relatively easy to implement and can provide good approximations of the solutions to ODEs, making it a popular choice in the field of computational physics and engineering.。
顶点算子代数模范畴的grothendick群(英文)
顶点算子代数模范畴的grothendick群(英文)In mathematics, the Grothendieck group of a vertex operator algebra is a group whose elements are formal linear combinations of irreducible modules of the algebra, modulo a suitable notion of equivalence. It is named in honour of Alexander Grothendieck, the famous mathematician whopioneered algebraic geometry and founded the Grothendieck school.The Grothendieck group of a vertex operator algebra is related to the representation theory of the algebra; in particular, it captures the structure of its irreducible modules, as well as their tensor product decompositions. Itis also closely related to the K-theory of vector bundles, which is one of Grothendieck's most important contributionsto mathematics.The Grothendieck group of a vertex operator algebra can be defined in several ways. One definition is based on the notion of a vertex operator algebra as a special kind of commutative algebra, and on the notion of a generic module as a module which is both projective and injective. Thisdefinition is then used to define isomorphism classes of irreducible vertex operator algebra modules, and to definethe Grothendieck group of the algebra.Another definition of the Grothendieck group uses the notion of a vertex operator algebra as a module over itself. This approach defines isomorphism classes of irreducible modules of the algebra, and then uses those classes toconstruct the Grothendieck group of the algebra.Finally, the Grothendieck group of a vertex operator algebra can be defined using the representation theory of the algebra. In this approach, the Grothendieck group is constructed from the structure constants of the irreducible modules of the algebra.The Grothendieck group of a vertex operator algebra has several properties which make it a useful tool in algebraic geometry and representation theory. For example, it is a finitely generated abelian group, which means that it is amenable to computation. It also provides a useful way to classify and study the irreducible modules of the algebra. Finally, it is related to several other groups, such as theK-theory of vector bundles and the Burnside ring of a group.Overall, the Grothendieck group of a vertex operator algebra provides a fundamental tool in the representation theory and algebraic geometry of such algebras. It is an important concept which continues to be studied, and which has many applications in mathematics.。
Localized Solutions of the Non-Linear Klein-Gordon Equation in Many Dimensions
calculations of the solution (2) in successive times, see Fig.1 where the results
are presented for the parameters m = 5, c = 1, k = 2, b = 15 in the conventional
3
We use here the technique of integral equations. In conditions of the validity of (4) the inequality (9) can be written as follows
(z
− vgrt)2 ∆2
+
x2 ∆2⊥
arXiv:0712.3284v1 [hep-th] 19 Dec 2007
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2 each other as bic field interaction except for an additional change in the angular momentum barrier. In addition, the D-dimensional Schr¨ odinger and Klein-Gordon wave equations have been solved for some types of ring-shaped potentials using the NU method [22, 25, 32]. The aim of the present paper is to obtain the exact bound-state solutions of the D-dimensional Klein-Gordon with an oscillatory-type plus new ring-shaped potential. The radial and angular parts of the Klein-Gordon equation with this type of noncentral potential are solved using the NU method. This work is organized as follows: in section , we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle with an equal scalar and vector oscillatory-type ring-shaped potential. We separate the wave equation into radial and angular parts. Section is devoted to a brief description of the NU method. In section , we present the exact bound-state solutions to the radial and angular equations in D-dimensions. Finally, the relevant conclusions are given in section .
arXiv:0801.4857v3 [quant-ph] 14 Feb 2008
We present the exact solution of the Klein-Gordon equation in D-dimensions in the presence of the noncentral equal scalar and vector pseudoharmonic potential plus the new ring-shaped potential using the Nikiforov-Uvarov method. We obtain the exact bound-state energy levels and the corresponding eigen functions for a spin-zero particles. We also find that the solution for this noncentral ring-shaped pseudoharmonic potential can be reduced to the three-dimensional pseudoharmonic solution once the coupling constant of the noncentral part of the potential becomes zero. Keywords: Bound-states, energy eigenvalues and eigenfunctions, Klein-Gordon equation, pseudoharmonic potential, ring-shaped potential, non-central potentials, Nikiforov and Uvarov method. PACS numbers: 03.65.-w; 03.65.Fd; 03.65.Ge.
Solution of the D -dimensional Klein-Gordon equation with equal scalar and vector ring-shaped pseudoharmonic potential
Sameer M. Ikhdair and
1∗ †
Ramazan Sever1
PACS numbers: Keywords:
INTRODUCTION
In nuclear and high energy physics [1, 2], one of the interesting problems is to obtain exact solutions of the relativistic wave equations like Klein-Gordon, Dirac and Salpeter wave equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to understand the motion of a spin-0 particle in large class of potentials. In recent years, much works have been done to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the fourvector potential (V) and the space-time scalar potential (S). For the case S = ±V, the solution of these wave equations with physical potentials has been studied recently. The exact solutions of these equations are possible only for certain central potentials such as Morse potential [3], Hulth´ en potential [4], Woods-Saxon potential [5], P¨ oschl-Teller potential [6], reflectionless-type potential [7], pseudoharmonic oscillator [8], ring-shaped harmonic oscillator [9], V0 tanh2 (r/r0 ) potential [10], five-parameter exponential potential [11], Rosen-Morse potential [12], and generalized symmetrical double-well potential [13], etc by using different methods. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2, 14]. However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the existence of Klein-Gordon bound states) (i.e., S > V ) [15, 16, 17, 18, 19]. Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15]. On the other hand, the other exactly solvable potentials are the noncentral ring-shaped potentials [20]. These potentials involve an attractive Coulomb potential plus a repulsive inverse square potential, that is, one like the Coulombic ring-shaped potential [21, 22] revived in quantum chemistry by Hartmann et al [23]. The oscillatory ring-shaped potential studied by Quesne [24] have been investigated using various quantum mechanical approaches [25]. In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Yasuk et al [26] applied the Nikiforov-Uvarov method [27] to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [28] also applied the same method to solve the Klein-Gordon equation for the Kratzer-type potential. Recently, Chen and Dong [29] proposed a new ring-shaped potential and obtained the exact solution of the Schr¨ odinger equation for the Coulomb potential plus this new ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Ref. [29] appears to be very similar to the potential used by Ref. [26]. Additionally, Cheng and Dai [30], proposed a new potential consisting from the modified Kratzer’s potential [31] plus the new proposed ring-shaped potential in [29]. They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional Schr¨ odinger equation using the NU method. The two quantum systems solved by Refs [29, 30] are closely relevant to