Symbolic Computation and q-Deformed Function Solutions of (2+1)-Dimensional Breaking Soliton

小学上册英语第3单元测验卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The first person to swim across the English Channel was ______ (马修·温斯顿).2.The __________ (历史的情感连接) bridge gaps.3.The first Olympic champion was _______. (科罗比乌斯)4.The melting point of ice is _______ degrees Celsius.5.An element's reactivity depends on the arrangement of its _____ (electrons).6. A ________ (金鱼) swims gracefully in its bowl.7. (African) kingdoms were rich in resources and trade. The ____8. A chemical reaction can lead to a change in ______ state.9.The ______ is a skilled poet.10.We have a ______ (快乐的) gathering for special occasions.11.The ________ has a long tail and likes to climb.12.What is the name of the famous American holiday celebrated on the fourth Thursday in November?A. ChristmasB. ThanksgivingC. New Year's DayD. Independence Day答案:B13.The first successful flight of a powered aircraft was in _______.14.What is the name of the famous detective created by Arthur Conan Doyle?A. Hercule PoirotB. Sam SpadeC. Sherlock HolmesD. Philip Marlowe答案:C15.The process of neutralization results in the formation of ______.16.The chemical symbol for aluminum is ______.17. A _____ (盆栽) can be placed indoors.18.What is the name of the famous statue in New York Harbor?A. Christ the RedeemerB. Statue of LibertyC. DavidD. The Thinker 答案: B19.My family celebrates ______ together. (我的家人一起庆祝______。

【参考文档】物理所等在费米子负符号问题研究中取得进展-word范文本文部分内容来自网络整理，本司不为其真实性负责，如有异议或侵权请及时联系，本司将立即删除！== 本文为word格式，下载后可方便编辑和修改！ == 物理所等在费米子负符号问题研究中取得进展费米子负符号问题是量子蒙特卡罗模拟遇到的一个最困难的问题，也是多体量子理论研究的一个基本问题。

量子蒙特卡罗是一种精确的数值模拟方法，但在有费米子负符号问题的系统，量子蒙特卡罗模拟的计算误差，随着温度的降低或系统体积的增加呈指数增长，失去了这种方法的可靠性。

负符号问题起源于费米子交换的反对易性。

对于大多数相互作用费米子系统，负符号问题总是存在。

但在负u哈伯德模型或一些其它格点量子模型中，负符号问题可以被消除，使得精确的量子蒙特卡罗模拟成为可能，极大地促进了量子多体问题的研究。

寻找不存在负符号问题的相互作用费米子系统，是量子多体理论研究的一个重要目标，其中的一个重要进展，就是201X年发表在《物理评论快报》上的一项工作(prl115,250601(201X))。

在那项工作中，研究人员证明一个系统如果存在o(n,n)(splitorthogonalgroup)对称性，那么这个系统就不存在负符号问题。

最近，中国科学院物理研究所/北京凝聚态物理国家实验室（筹）t06组的博士生魏忠超在导师向涛指导下，与美国加州大学圣迭戈分校教授吴从军、美国威廉与玛丽学院教授张世伟等合作，在这个问题的研究上又取得了新的进展。

他们证明，相互作用费米子系统只要存在马约拉纳(majorana)反射正定性或kramers正定性，就没有负符号问题。

魏忠超等人的证明，包含了具有o(n,n)对称性的系统，涵盖了目前已知的所有不存在负符号问题的费米子格点模型，并使得甄别和发现新的无负符号问题的费米子格点模型变得更为便捷，提高了对费米子负符号问题的认识层次，对量子蒙特卡罗模拟的研究会起到一定的推动作用。

Package‘prWarp’October14,2022Title Warping Landmark ConfigurationsVersion1.0.0Date2020-10-07Description Compute bending energies,principal warps,partial warp scores,and the non-affine com-ponent of shape variation for2D landmark configurations,as well as Mardia-Dryden distribu-tions and self-similar distributions of landmarks,as described in Mit-teroecker et al.(2020)<doi:10.1093/sysbio/syaa007>.Working examples to decom-pose shape variation into small-scale and large-scale components,and to decompose the to-tal shape variation into outline and residual shape components are provided.Two land-mark datasets are provided,that quantify skull morphology in humans and papionin primates,re-spectively from Mitteroecker et al.(2020)<doi:10.5061/dryad.j6q573n8s>and Grun-stra et al.(2020)<doi:10.5061/dryad.zkh189373>.Depends R(>=3.6.0)Imports MorphoLicense GPL-3Encoding UTF-8LazyData trueSuggests knitr,rmarkdown,geomorphVignetteBuilder knitrRoxygenNote7.1.1NeedsCompilation noAuthor Anne Le Maitre[aut,cre](<https:///0000-0003-2690-7367>), Silvester Bartsch[aut],Nicole Grunstra[aut],Philipp Mitteroecker[aut]Maintainer Anne Le Maitre<************************.at>Repository CRANDate/Publication2020-11-0208:40:02UTC12array.to.xxyy R topics documented:array.to.xxyy (2)create.pw.be (3)HomoMidSag (4)md.distri (5)papionin (6)ssim.distri (7)tps.all (8)xxyy.to.array (9)Index11 array.to.xxyy Convert(p x k x n)data array into2D data matrixDescriptionConvert a three-dimensional array of landmark coordinates into a two-dimensional matrixUsagearray.to.xxyy(A)ArgumentsA A3D array(p x k x n)containing landmark coordinates for a set of specimensValueFunction returns a two-dimensional matrix of dimension(n x[p x k]),where rows represent speci-mens and columns represent variables.The pfirst columns correspond to X coordinates,etc.See Alsoxxyy.to.arrayExamplesA<-array(rnorm(40),c(5,2,4))#random2D coordinates of5landmarks for4specimens array.to.xxyy(A)create.pw.be3 create.pw.be Principal warpsDescriptionComputes the principal warps and the bending energy of a reference shape configuration,as well as the variance of the partial waprs,the partial warp scores and the non-affine component of shape vari-ation for2D landmark coordinates(3D not implemented).Small-scale and large-scale components of shape variation can also be computed.Usagecreate.pw.be(A,M_ref,d=NULL)ArgumentsA a k x2x n array,where k is the number of2D landmarks,and n is the samplesize.M_ref a k x2refence matrix(usually the sample mean shape),where k is the number of2D landmarksd(optional)an integer value comprised between1and(k-3)to compute small-scale shape components(between1and d)and large-scale shape components(between d+1and k-3)ValueA list containing the following named components:bendingEnergy the bending energy(the(k-3)eigenvalues of the bending energy matrix)principalWarps the k x(k-3)matrix of principal warps(the k eigenvectors of the bending energy matrix)partialWarpScoresthe n x(2k-6)matrix of partial warp(the projection of the vectors of shapecoordinates,expressed as deviations from the reference shape,onto the principalwarps)variancePW the variance of the(k-3)partial warpsXnonaf the n x2k matrix of the non-affine component of shape variationXsmall the n x2k matrix of the small-scale shape variation(if d is provided)Xlarge the n x2k matrix of the large-scale shape variation(if d is provided)ReferencesBookstein FL.(1989).Principal Warps:Thin-plate splines and the decomposition of deforma-tions.IEEE Transactions on pattern analysis and machine intelligence11(6):567–585.https: ///abstract/document/247924HomoMidSag See AlsoSee CreateL for the creation of the bending energy matrixExamples#2D landmark coordinateslibrary("geomorph")data("HomoMidSag")#datasetn_spec<-dim(HomoMidSag)[1]#number of specimensk<-dim(HomoMidSag)[2]/2#number of landmarkshomo_ar<-arrayspecs(HomoMidSag,k,2)#create an array#Procrustes registrationhomo_gpa<-Morpho::procSym(homo_ar)m_overall<-homo_gpa$rotated#Procrustes coordinatesm_mshape<-homo_gpa$mshape#average shape#Computation of bending energy,partial warp scores etc.homo_be_pw<-create.pw.be(m_overall,m_mshape)#Partial warp variance as a function of bending energylogInvBE<-log((homo_be_pw$bendingEnergy)^(-1))#inverse log bending energylogPWvar<-log(homo_be_pw$variancePW)#log variance of partial warpsmod<-lm(logPWvar~logInvBE)#linear regression#Plot log PW variance on log BE^-1with regression lineplot(logInvBE,logPWvar,col="white",asp=1,main="PW variance against inverse BE",xlab="log1/BE",ylab="log PW variance") text(logInvBE,logPWvar,labels=names(logPWvar),cex=0.5)abline(mod,col="blue")HomoMidSag HomoMidSag datasetDescription2D Cartesian coordinates of87landmarks quantifying the skull morphology along the midsagittal plane for24adult modern humans.Usagedata(HomoMidSag)FormatA data frame with24rows and184variablesmd.distri5ReferencesBartsch,Silvester(2019)The ontogeny of hominid cranial form:A geometric morphometric anal-ysis of coordinated and compensatory processes.Master’s thesis,University of Vienna.Mitteroecker,Philipp et al.(2020)Morphometric variation at different spatial scales:coordination and compensation in the emergence of organismal form.Systematic Biology,syaa007.https: ///10.1093/sysbio/syaa007Mitteroecker,Philipp et al.(2020)Data form:Morphometric variation at different spatial scales: coordination and compensation in the emergence of organismal form.Dryad Digital Repository.https:///10.5061/dryad.j6q573n8smd.distri Mardia-Dryden distributionDescriptionCreate a matrix of2D shape coordinates drawn from a Mardia-Dryden distribution(3D not imple-mented)Usagemd.distri(M_ref,n,sd=0.05)ArgumentsM_ref a k x2refence matrix(usually the sample mean shape),where k is the number of2D landmarksn the number of observationssd the standard deviation of the distribution(default=0.02)Valuethe n x2k matrix of shape coordinates drawn from a Mardia-Dryden distributionExamples#2D landmark coordinateslibrary("geomorph")data("HomoMidSag")#datasetn_spec<-dim(HomoMidSag)[1]#number of specimensk<-dim(HomoMidSag)[2]/2#number of landmarkshomo_ar<-arrayspecs(HomoMidSag,k,2)#create an array#Procrustes registrationhomo_gpa<-Morpho::procSym(homo_ar)m_mshape<-homo_gpa$mshape#average shape#Mardia-Dryden distribution6papionin Xmd<-md.distri(m_mshape,n=n_spec,sd=0.005)#Visualizationplot(Xmd[,1:k],Xmd[,(k+1):(2*k)],asp=1,las=1,cex=0.5,main="Mardia-Dryden distribution",xlab="X",ylab="Y")papionin papionin datasetDescription2D Cartesian coordinates of70landmarks quantifying the skull morphology along the midsagittal plane for67adult modern primates(mostly papionins).The data correspond to a list with the6 following elements:Usagedata(papionin)FormatA list of6elements.Details•coords The3D array of landamrk coordinates•species The vector of species names•semi_lm The vector of semilandmark numbers of the full dataset•curves The list of curves for sliding semilandmarks of the full dataset•links The matrix of links between landmarks for the full dataset•outline A list of4elements for the analysis of the outline shape:subset,the landmark numbers for the susbet;semi_lm,the vector of semilandmark numbers;curves,the list of curves for sliding semilandmarks;links,the matrix of links between landmarks for the full dataset. ReferencesGrunstra,Nicole D.S.et al.(in press)Detecting phylogenetic signal and adaptation in papionin cranial shape by decomposing variation at different spatial scales.Grunstra,Nicole D.S.et al.(2020)Data form:Detecting phylogenetic signal and adaptation in pa-pionin cranial shape by decomposing variation at different spatial scales.Dryad Digital Repository.https:///10.5061/dryad.zkh189373ssim.distri7 ssim.distri Self-similar distributionDescriptionCreate a matrix of2D shape coordinates drawn from a self-similar distribution(3D not imple-mented)Usagessim.distri(M_ref,n,sd=0.02,f=1)ArgumentsM_ref a k x2refence matrix(usually the sample mean shape),where k is the number of2D landmarksn the number of observationssd the standard deviation of the distribution(default=0.02)f a scaling factorValuethe n x2k matrix of shape coordinates drawn from a self-similar distributionExamples#2D landmark coordinateslibrary("geomorph")data("HomoMidSag")#datasetn_spec<-dim(HomoMidSag)[1]#number of specimensk<-dim(HomoMidSag)[2]/2#number of landmarkshomo_ar<-arrayspecs(HomoMidSag,k,2)#create an array#Procrustes registrationhomo_gpa<-Morpho::procSym(homo_ar)m_mshape<-homo_gpa$mshape#average shape#Self-similar distributionXdefl<-ssim.distri(m_mshape,n=n_spec,sd=0.05,f=1)#Visualizationplot(Xdefl[,1:k],Xdefl[,(k+1):(2*k)],asp=1,las=1,cex=0.5,main="Self-similar distribution",xlab="X",ylab="Y")8tps.all tps.all Thin plate spline mapping(2D and3D)for several sets of landmarkcoordinatesDescriptionMaps landmarks via thin plate spline based on a reference and a target configuration in2D and3D.This function is an extension of the tps3d function a set of specimens.Usagetps.all(X_array,REF_array,TAR_matrix)ArgumentsX_array original coordinates-a3D array(p x k x n)containing original landmark coor-dinates for a set of specimensREF_array reference coordinates(e.g.,outline landmarks for all specimens)-a3D array(p x k x n)containing reference landmark coordinates for a set of specimens TAR_matrix target coordinates(e.g.,average outline landmarks)-a matrix(n x k)containing target landmark coordinatesDetailsp is the number of landmark points,k is the number of landmark dimensions(2or3),and n is the number of specimens.ValueFunction returns a3D array(p x k x n)containing the deformed input(original landmark set warped onto the target matrix).ReferencesBookstein FL.(1989).Principal Warps:Thin-plate splines and the decomposition of deforma-tions.IEEE Transactions on pattern analysis and machine intelligence11(6):567–585.https: ///abstract/document/24792See AlsoSee tps3dExamplesdata("papionin")#load dataset#Full dataset:70landmarkspapionin_ar<-papionin$coords#Outline dataset:subset of54landmarksoutline_ar<-papionin_ar[papionin$outline$subset,,]#Subset:Macaca onlymac<-grep("Macaca",papionin$species)#genus Macacapapionin_macaca<-papionin_ar[,,mac]outline_macaca<-outline_ar[,,mac]#Landmark sliding by minimizing bending energy+superimposition(GPA)library("Morpho")papionin_gpa<-procSym(papionin_macaca,SMvector=papionin$semi_lm,outlines=papionin$curves)outline_gpa<-procSym(outline_macaca,SMvector=papionin$outline$semi_lm,outlines=papionin$outline$curves)#Warping the slid landmarks of the full landmark dataset to the average outline shape residual_shape<-tps.all(X_array=papionin_gpa$dataslide,REF_array=outline_gpa$dataslide,TAR_matrix=outline_gpa$mshape)xxyy.to.array Convert landmark data matrix into array(p x k x n)DescriptionConvert a matrix of landmark coordinates into a three-dimensional arrayUsagexxyy.to.array(M,p,k=2)ArgumentsM A matrix of dimension n x[p x k]containing landmark coordinates for a set of specimens.Each row contains all landmark coordinates for a single specimen.Thefirst columns correspond to the X coordinates for all landmarks,etc.p Number of landmarksk Number of dimensions(2or3)ValueFunction returns a3D array(p x k x n),where p is the number of landmark points,k is the number of landmark dimensions(2or3),and n is the number of specimens.The third dimension of this array contains names for each specimen if specified in the original input matrix.See Alsoarray.to.xxyyExamplesX<-matrix(rnorm(40),nrow=4)#Random2D coordinates of5landmarks for4specimens xxyy.to.array(X,5,2)Indexarray.to.xxyy,2,10create.pw.be,3CreateL,4HomoMidSag,4md.distri,5papionin,6ssim.distri,7tps.all,8tps3d,8xxyy.to.array,2,911。

极大前缀码的积
沈传龙;潘慧丽
【期刊名称】《杭州师范大学学报（自然科学版）》
【年(卷),期】2005(004)005
【摘要】主要给出关于极大前缀码的积的必要条件的一个结论:设X是字母表A上的一个稀疏码,Y是A*的一个非空稀疏子集,若XY是极大前缀码,则X和Y都是极大前缀码.同时给出该命题的一个推论.
【总页数】3页(P331-333)
【作者】沈传龙;潘慧丽
【作者单位】杭州师范学院,数学系,浙江,杭州,310036;杭州师范学院,数学系,浙江,杭州,310036
【正文语种】中文
【中图分类】O152.7
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Hybrid Symbolic-Numeric Computation*Erich KaltofenDepartment of MathematicsMassachusetts Institute of T echnology Cambridge,Massachusetts02139-4307,USA kaltofen@Lihong ZhiKey Laboratory of Mathematics Mechanization Academy of Mathematics and Systems ScienceBeijing100080,Chinalzhi@/~lzhiCategories and Subject Descriptors:I.2.1[Comput-ing Methodologies]:Symbolic and Algebraic Manipulation —Algorithms;G.1.2[Mathematics of Computing]:Numeri-cal Analysis—ApproximationGeneral Terms:algorithms,experimentation Keywords:symbolic/numeric hybrid methodsTUTORIAL ABSTRACTSeveral standard problems in symbolic computation,such as greatest common divisor and factorization of polynomials, sparse interpolation,or computing solutions to overdeter-mined systems of polynomial equations have non-trivial so-lutions only if the input coefficients satisfy certain algebraic constraints.Errors in the coefficients due tofloating point round-offor through phsical measurement thus render the exact symbolic algorithms unusable.By symbolic-numeric methods one computes minimal deformations of the coeffi-cients that yield non-trivial results.We will present hybrid algorithms and benchmark computations based on Gauss-Newton optimization,singular value decomposition(SVD) and structure-preserving total least squares(STLS)fitting for several of the above problems.A significant body of results to solve those“approximate computer algebra”problems has been discovered in the past 10years.In the Computer Algebra Handbook the section on “Hybrid Methods”concludes as follows[2]:“The challenge of hybrid symbolic-numeric algorithms is to explore the ef-fects of imprecision,discontinuity,and algorithmic complex-ity by applying mathematical optimization,perturbation theory,and inexact arithmetic and other tools in order to solve mathematical problems that today are not solvable by numerical or symbolic methods alone.”The focus of our tu-torial is on how to formulate several approximate symbolic computation problems as numerical problems in linear al-gebra and optimization and on software that realizes their solutions.∗This research was supported in part by the National Science Foun-dation of the USA under Grants CCR-0305314and CCF-0514585 (Kaltofen)and OISE-0456285(Kaltofen,Yang and Zhi).This research was partially supported by NKBRPC(2004CB318000) and the Chinese National Natural Science Foundation under Grant 10401035(Yang and Zhi).Kaltofen’s permanent address:Dept.of Mathematics,North Car-olina State University,Raleigh,North Carolina27695-8205,USA, kaltofen@.Copyright is held by the author/owner(s).ISSAC’06,July9–12,2006,Genova,Italy.ACM1-59593-276-3/06/0007.Approximate Greatest Common Divisors[3].Our pa-per at this conference presents a solution to the approximate GCD problem for several multivariate polynomials with real or complex coefficients.In addition,the coefficients of the minimally deformed input coefficients can be linearly con-strained.In our tutorial we will give a precise definition of the approximate polynomial GCD problem and we will present techniques based on parametric optimization(slow) and STLS or Gauss/Newton iteration(fast)for its numeri-cal solution.The fast methods can compute globally optimal solutions,but they cannot verify global optimality.We show how to apply the constrained approximate GCD problem to computing the nearest singular polynomial with a root of multiplicity at least k≥2.Approximate Factorization of Multivariate Polyno-mials[1].Our solution and implementation of the approx-imate factorization problem follows our approach for the approximate GCD problem.Our algorithms are based ona generalization of the differential forms introduced by W.Ruppert and S.Gao to many variables,and use SVD or STLS and Gauss/Newton optimization to numerically com-pute the approximate multivariate factors.Solutions of Zero-dimensional Polynomial Systems[4].We translate a system of polynomials into a systemof linear partial differential equations(PDEs)with constant coefficients.The PDEs are brought to an involutive form by symbolic prolongations and numeric projections via SVD.The solutions of the polynomial system are obtained by solv-ing an eigen-problem constructed from the null spaces of the involutive system and its geometric projections.1.REFERENCES[1]Gao,S.,Kaltofen,E.,May,J.P.,Yang,Z.,andZhi,L.Approximate factorization of multivariatepolynomials via differential equations.In Gutierrez,J.,Ed.ISSAC2004Proc.2004Internat.Symp.Symbolic Algebraic Comput.,pp.167–174.[2]Grabmeier,J.,Kaltofen,E.,and Weispfenning,puter Algebra Handbook Springer Verlag,2003,pp.109–124.[3]Kaltofen,E.,Yang,Z.,and Zhi,L.Approximategreatest common divisors of several polynomials withlinearly constrained coefficients and singularpolynomials.In Dumas,J-G.,Ed.ISSAC2006Proc.2006Internat.Symp.Symbolic Algebraic Comput..[4]Reid,G.,Tang,J.,and Zhi,L.A completesymbolic-numeric linear method for camera posedetermination.In Sendra,J.,Ed.ISSAC2003Proc.2003Internat.Symp.Symbolic Algebraic Comput.,pp.215–223.7。

收稿日期：2020-11-25基金项目：福建省自然科学基金(2016J01032)作者简介：程金发(1966-)，男，江西省乐平市人，博士，教授，博士生导师.*通信作者.E-mail ：***************.cn非一致格子上离散分数阶差分与分数阶和分程金发*（厦门大学数学科学学院福建厦门，361005）摘要：众所周知，一致格子上分数阶和分与分数阶差分的思想概念也是最近几年才兴起的，并且在该邻域得到了很大的发展.但是在非一致格子x ()z =c 1z 2+c 2z +c 3或者x ()z =c 1q z +c 2q -z +c 3上，分数阶和分与分数阶差分的定义是什么，这是一个十分复杂和有趣的问题.本文首次提出非一致格子上分数阶和分与Riemann-Liouville 分数阶差分、Caputo 分数阶差分的定义以及非一致格子上广义Abel 积分方程的求解等基础性结果.关键词：超几何差分方程；非一致格子；分数阶和分；分数阶差分；特殊函数中图分类号：33C45；33D45；26A33；34K37文献标志码：A文章编号：2095-7122（2021）01-0001-013On the fractional sum and fractional difference on nonuniform latticesCHENG Jinfa *(School of Mathematical Sciences,Xiamen University,Xiamen,Fujian 361005,China )Abstract:As is well known,the idea of a fractional sum and difference on uniform lattice is more current,and gets a lot of development in this field.But the definitions of fractional sum and fractional difference of f ()z on nonuniform lattices x ()z =c 1z 2+c 2z +c 3or x ()z =c 1q z +c 2q -z +c 3seem much more complicated andinteresting.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on nonuniform lattices.The solution of the generalized Abel equation is obtained etc.Key words:special function;orthogonal polynomials;adjoint difference equation;difference equation of hy-pergeometric type;nonuniform lattice第34卷第1期2021年3月闽南师范大学学报（自然科学版）Journal of Minnan Normal University （Natural Science ）Vol.34Ｎｏ．1Mar.20211背景回顾及问题提出正如我们在本文序言指出的，分数阶微积分的概念几乎与经典微积分同时起步，可以回溯到Euler 和Leibniz 时期.经过几代数学家的努力，特别是近几十年来，分数阶微积分已经取得了惊人的发展和广阔的应用，有关分数阶微积分的著作层出不穷，例如文献[1-4],但是在一致格子x ()z =z 和x ()z =q z 或者q -z ,z ∈C 上关于离散分数阶微积分的思想，仍然是最近才兴起的.虽然关于一致格子x ()z =z 和x ()z =q z 的离散分数微积分出现和建立相对较晚,但是该领域目前已经做出了大量的工作，且取得了很大的发展[5-8].在最近十年的学术著作中，程金发[9],Goodrich 和Peterson [10]相继出版了两本有关离散分数阶方程理论、离散分数微积分的著作，其中全面系统地介绍了离散分数微积分的基本定义和基本定理，以及最新的参考资料.有关q -分数阶微积分方面的著作可参见Annaby 和Mansour [11].非一致格子的定义回溯到超几何型微分方程[12-13]:σ()z y ′′()z +τ()z y ′()z +λy ()z =0，(1)的逼近，这里σ()z 和τ()z 分别是至多二阶和一阶多项式，λ是常数.Nikiforov 等[14-15]将式(1)推广到如下最一般的复超几何差分方程σˉ[]x ()s ΔΔx ()s -12éëêùûú∇y ()s ∇x ()s +12τˉ[]x ()s éëêùûúΔy ()s Δx ()s +∇y ()s ∇x ()s +λy ()s =0,(2)这里σˉ()x 和τˉ()x 分别是关于x ()s 的至多二阶和一阶多项式，λ是常数，Δy ()s =y ()s +1-y ()s ,∇y ()s =y ()s -y ()s -1，并且x ()s 必须是以下非一致格子.定义1[16-17]两类格子函数x ()s 称之为非一致格子，如果它们满足x ()s =-c 1s 2+-c 2s +-c 3,(3)x ()s =c 1q s +c 2q -s +c 3,(4)这里c i ,-c i 是任意常数，且c 1c 2≠0,-c 1-c 2≠0.当c 1=1,c 2=c 3=0，或c 2=1,c 1=c 3=0或者-c 2=1,-c 1=-c 3=0时，这两种格子函数x ()s ：x ()s =s ，(5)x ()s =q s 或x ()s =q -s(6)称之为一致格子.给定函数F ()s ，定义关于x γ()s 的差分或差商算子为∇γF ()s =∇F ()s ∇x γ()s ,且∇k γF ()z =∇∇x γ()z ()∇∇x γ+1()z ⋯()∇()F ()z ∇x γ+k -1()z .()k =1,2,⋯关于差商算子，命题1是常用的.命题1给定两个复函数f ()s ,g ()s ，成立恒等式Δυ()f ()s g ()s =f ()s +1Δυg ()s +g ()s Δυf ()s =g ()s +1Δυf ()s +f ()s Δυg ()s ,Δυ()f ()s g ()s =g ()s +1Δυf ()s -f ()s +1Δυg ()s g ()s g ()s +1=g ()s Δυf ()s -f ()s Δυg ()s g ()s g ()s +1,Δυ()f ()s g ()s =f ()s -1Δυg ()s +g ()s Δυf ()s =g ()s -1Δυf ()s +f ()s Δυg ()s ,(7)Δυ()f ()s g ()s =g ()s -1Δυf ()s -f ()s -1Δυg ()s g ()s g ()s -1=闽南师范大学学报（自然科学版）2021年2g ()s Δυf ()s -f ()s Δυg ()s g ()s g ()s -1.我们必须指出,在非一致格子式(3)或者式(4),即使当n ∈N ，如何建立非一致格子的n -差商公式，也是一件很不平凡的工作，因为它是十分复杂的，也是难度很大的.事实上,在文献[14-15]中,Nikiforov 等利用插值方法得到了如下n -阶差商∇()n 1[]f ()s 公式：定义2[12-13]对于非一致格子式(3)或式(4)，让n ∈N +，那么∇()n 1[]f ()s =∑k =0n ()-1n -k[]Γ()n +1q[]Γ()k +1q[]Γ()n -k +1q×∏l =0n∇x []s +k -()n -12∇x []s +()k -l +12f ()s -n +k =∑k =0n()-1n -k[]Γ()n +1q[]Γ()k +1q[]Γ()n -k +1q×∏l =0n ∇x n +1()s -k ∇x []s +()n -k -l +12f ()s -k ,(8)这里[]Γ()s q 是修正的q -Gamma 函数，它的定义是[]Γ()s q=q -()s -1()s -24Γq ()s ，并且函数Γq ()s 被称为q -Gamma 函数;它是经典Euler Gamma 函数Γ()s 的推广.其定义是Γq ()s =ìíîïïïï∏k =0∞(1-q k +1)()1-q s -1∏k =0∞(1-q s +k),当||q <1;q -()s -1()s -22Γ1q ()s ,当||q >1.(9)经过进一步化简后，Nikiforov 等在文献[14]中将n 阶差分∇()n 1[]f ()s 的公式重写成下列形式:定义3[14]对于非一致格子式(3)或式(4)，让n ∈N +，那么∇()n 1[]f ()s =∑k =0n ()[]-n qk[]k q ![]Γ()2s -k +c q[]Γ()2s -k +n +1+c qf ()s -k ∇x n +1()s -k ,这里[]μq=γ()μ=ìíîïïïïq u2-q -u 2q 12-q -12如果x ()s =c 1q s +c 2q -s +c 3;μ,如果x ()s =-c 1s 2+-c 2s +c 3,(10)且c =ìíîïïïïïïïïlog c 2c 1log q ,当x ()s =c 1q s +c 2q -s +c 3,-c 2-c 1,当x ()s =-c 1s 2+-c 2s +c 3.程金发：非一致格子上离散分数阶差分与分数阶和分第1期3现在存在两个十分重要且具有挑战性的问题需要进一步深入探讨:1）对于非一致格子上超几何差分方程式（2），在特定条件下存在关于x ()s 多项式形式的解，如果用Rodrigues 公式表示的话，它含有整数阶高阶差商.一个新的问题是：若该特定条件不满足，那么非一致格子上超几何差分方程式（2）的解就不存在关于x ()s 的多项式形式，这样高阶整数阶差商就不再起作用了.此时非一致格子超几何方程的解的表达形式是什么呢？这就需要我们引入一种非一致格子上分数阶差商的新概念和新理论.因此,关于非一致格子上α-阶分数阶差分及α-阶分数阶和分的定义是一个十分有趣和重要的问题.显而易见，它们肯定是比整数高阶差商更为难以处理的困难问题，自专著[14-15]出版以来，Nikiforov 等并没有给出有关α-阶分数阶差分及α-阶分数阶和分的定义，我们能够合理给出非一致格子上分数阶差分与分数阶和分的定义吗?2)另外，我们认为作为非一致格子上最一般性的离散分数微积分，它们也会有独立的意义，并可以导致许多有意义的结果和新理论.本文的目的是探讨非一致格子上离散分数阶和差分.受文章篇幅所限，本文我们仅合理给出非一致格上分数阶和分与分数阶差分的基本定义，其它更多结果例如：非一致格子离散分数阶微积分的一些基本定理，如:Euler Beta 公式，Cauchy Beta 积分公式，Taylor 公式、Leibniz 公式在非一致格子上的模拟形式，非一致格子上广义Abel 方程的解，以及非一致格子上中心分数差分方程的求解，离散分数阶差和分与非一致格子超几何方程之间联系等内容，请参见笔者新专著[16].2非一致格子上的整数和分与整数差分设x ()s 是非一致格子，这里s ∈ℂ.对任意实数γ，x γ()s =x ()s +γ2也是一个非一致格子.让∇γF ()s =f ()s .那么F ()s -F ()s -1=f ()s []x γ()s -x γ()s -1.选取z ,a ∈ℂ，和z -a ∈N .从s =a +1到z ，则有F ()z -F ()a =∑s =a +1zf ()s ∇x r()s .因此，我们定义∫a +1z f ()s d ∇x γ()s =∑s =a +1zf ()s ∇xγ()s .容易直接验证下列式子成立.命题2给定两个复变函数F ()z ,f ()z ，这里复变量z ,a ∈C 以及z -a ∈N ，那么成立1）∇γéëêùûú∫a +1zf ()s d ∇x γ()s =f ()z ;2）∫a +1z∇γF ()s d ∇x γ()s =F ()z -F ()a .现在让我们定义非一致格子上的广义n -阶幂函数[]x ()s -x ()z ()n 为[]x ()s -x ()z ()n =∏k =0n -1[]x ()s -x ()z -k ,()n ∈N +,当n 不是正整数时，需要将广义幂函数加以进一步推广，它的性质和作用是非常重要的，非一致格子上广义幂函数[]x γ()s -x γ()z ()α的定义如下：闽南师范大学学报（自然科学版）2021年4定义4[17-18]设α∈C ，广义幂函数[]x γ()s -x γ()z ()α定义为[]x γ()s -x γ()z ()α=ìíîïïïïïïïïïïïïïïïïïïïïΓ()s -z +a Γ()s -z ,如果x ()s =s ,Γ()s -z +a Γ()s +z +γ+1Γ()s -z Γ()s +z +γ-α+1,如果x ()s =s 2,()q -1αq α()γ-α+12Γq ()s -z +αΓq ()s -z ,如果x ()s =q s ,12α()q -12αq -α()s +γ2Γq ()s -z +αΓq ()s +z +γ+1Γq ()s -z Γq ()s +z +γ-α+1,如果x ()s =q s +q -s 2.(11)对于形如式(4)的二次格子，记c =-c 2-c 1，定义[]x γ()s -x γ()z ()α=-c 1αΓ()s -z +a Γ()s +z +γ+c +1Γ()s -z Γ()s +z +γ-α+c +1；(12)对于形如式(3)的二次格子，记c =logc 2c 1log q，定义[]xγ()s -x γ()z ()α=éëùûc 1()1-q 2αq -α()s +γ2Γq()s -z +a Γq()s +z +γ+c +1Γq()s -z Γq()s +z +γ-α+c +1,(13)这里Γ()s 是Euler Gamma 函数，且Γq ()s 是Euler q -Gamma 函数，其定义如式(9).命题3[17-18]对于x ()s =c 1q s +c 2q -s +c 3或者x ()s =-c 1s 2+-c 2s +-c 3，广义幂[]x γ()s -x γ()z ()α满足下列性质：[]x γ()s -x γ()z []x γ()s -x γ()z -1()μ=[]x γ()s -x γ()z ()μ[]xγ()s -x γ()z -μ=(14)[]xγ()s -x γ()z ()μ+1；(15)[]xγ-1()s +1-x γ-1()z ()μ[]xγ-μ()s -x γ-μ()z =[]x γ-μ()s +μ-x γ-μ()z []x γ-1()s -x γ-1()z ()μ=[]x γ()s -x γ()z ()μ+1；(16)ΔzΔx γ-μ+1()z []xγ()s -x γ()z ()μ=-∇s∇x γ+1()s []x γ+1()s -x γ+1()z ()μ=(17)-[]μq []x γ()s -x γ()z ()μ-1；(18)∇z∇x γ-μ+1()z {}1[]xγ()s -x γ()z ()μ=-ΔsΔx γ-1()s ìíîïïüýþïï1[]x γ-1()s -x γ-1()z ()μ=(19)[]μq[]xγ()s -x γ()z ()μ+1(20)这里[]μq 定义如式（10）.程金发：非一致格子上离散分数阶差分与分数阶和分第1期5现在让我们详细给出非一致格子x γ()s 上整数阶和分的定义，这对于我们进一步给出非一致格子x γ()s 上分数阶和分的定义是十分有帮助的.设γ∈R ，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的1-阶和分定义为y 1()z =∇-1γf ()z =∫a +1z f ()s d ∇x γ()s ，（21）这里y 1()z =∇-1γf ()z 定义在数集{}a +1,mod ()1中.那么由命题2，我们有∇1γ∇-1γf ()z =∇y 1()z ∇x γ()z =f ()z ，（22）并且对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的2-阶和分定义为y 2()z =∇-2γf ()z =∇-1γ+1[]∇-1γf ()z =∫a +1z y 1()s d ∇x γ+1()s =∫a +1z d ∇x γ+1()s ∫a +1s f ()t d ∇x γ()t =∫a +1z f ()t d ∇x γ()t ∫tz d ∇x γ+1()s =∫a +1z []x γ+1()z -x γ+1()t -1f ()s d ∇x γ()s .(23)这里y 2()z =∇-2γf ()z 定义在数集{}a +1,mod ()1中.同时，可得∇1γ+1∇1γ-1y 1()z =∇y 2()z ∇x γ+1()z =y 1()z ，∇2γ∇-2γf ()z =∇∇x γ()z ()∇y 2()z ∇x γ+1()z =∇y 1()z ∇x γ()z =f ()z .（24）更一般地，由数学归纳法，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中函数f ()z ，我们可以给出函数f ()z 的n -阶和分定义为y k ()z =∇-kγf ()z =∇-1γ+k -1[]∇-()k -1γf ()z =∫a +1z y k -1()s d ∇x γ+k -1()s =1[]Γ()k q∫a +1z []xγ+k -1()z -x γ+k -1()t -1()k -1f ()t d ∇x γ()t ,()k =1,2,⋯(25)这里[]Γ()k q=ìíîïïq -()k -1()k -2Γq ()k ,如果x ()s =c 1q s +c 2q -s +c 3;Γ()α,如果x ()s =-c 1s 2+-c 2s +c 3,这满足下式[]Γ()k +1q=[]k q []Γ()k q ,[]Γ()2q =[]1q []Γ()1q =1.那么成立∇kγ∇-k γf ()z =∇∇x γ()z ()∇∇x γ+1()z ⋯()∇y k ()z ∇x γ+k -1()z =f ()z .()k =1,2,⋯（26）需要指出的是，当k ∈C 时，式（25）右边仍然是有意义的，因此自然地，我们就可以对非一致格子x γ()s 闽南师范大学学报（自然科学版）2021年6给出函数f ()z 的分数阶和分定义如下：定义5（非一致格子分数阶和分）对任意Re α∈R +，对于非一致格子式（3）和式（4），数集{}a +1,a +2,⋯,z 中的函数f ()z ，我们定义它的α-阶分数阶和分为∇-αγf ()z =1[]Γ()αq∫a +1z []xγ+α-1()z -x γ+α-1()t -1()α-1f ()s d ∇x γ()s ，（27）这里[]Γ()αq=ìíîïïq -()s -1()s -2Γq ()α,如果x ()s =c 1q s +c 2q -s +c 3;Γ()α,如果x ()s =-c 1s 2+-c 2s +c 3,这满足下式[]Γ()α+1q=[]αq []Γ()αq .3非一致格子上的Abel 方程及分数阶差分非一致格子x γ()s 上f ()z 的分数阶差分定义相对似乎更困难和复杂一些.我们的思想是起源于非一致格子上广义Abel 方程的求解.具体来说，一个重要的问题是：让m -1<Re α≤m ，定义在数集{}a +1,a +2,⋯,z 的f ()z 是一给定函数，定义在数集{}a +1,a +2,⋯,z 的g ()z 是一未知函数，它们满足以下广义Abel 方程∇-αγg ()z =∫a +1z []x γ+α-1()z -x γ+α-1()t -1()α-1[]Γ()αqg ()t d ∇x γ()t =f ()t ，（28）怎样求解该广义Abel 方程式（28）？为了求解方程式（28），我们需要利用重要的Euler Beta 公式在非一致格子下的基本模拟.定理1[16]（非一致格子上Euler Beta 公式）对于任何α,β∈C ，那么对非一致格子x ()s ，我们有∫a +1z []x β()z -x β()t -1()β-1[]Γ()βq[]x ()t -x ()αα[]Γ()α+1qd ∇x 1()t =[]x β()z -x β()α()α+β[]Γ()α+β+1q.（29）定理2（Abel 方程的解）设定义在数集{}a +1,mod ()1中的函数f ()z 和函数g ()z 满足∇-αγg ()z =f ()z ,0<m -1<Re α≤m ，那么g ()z =∇m γ∇-m +αγ+αf ()z （30）成立.证明我们仅需证明∇-m γg ()z =∇-m +αγ+αf ()z ，即∇-()m -αγ+αf ()z =∇-()m -αγ+α∇-αγg ()z =∇-m γg ()z .事实上，由定义5可得程金发：非一致格子上离散分数阶差分与分数阶和分第1期7∇-()m -αγ+af ()z =∫a +1z []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αqf ()t d ∇x γ+α()t =∫a +1z []x γ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αqd ∇x γ+α()t ⋅∫a +1z []xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqg ()s d ∇x γ()s =∫a +1zg ()s ∇x γ()s ∫sz []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αq⋅[]xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqd ∇x γ+α()t .在定理1中，将α+1替换成s ；α替换成α-1；β替换成m -α，且将x ()t 替换成x γ+α-1()t ，那么x β()t 替换成x γ+m -1()t ，则我们能够得出下面的等式∫sz []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αq[]xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqd ∇x γ+α()t =[]xγ+m -1()z -x γ+m -1()s -1()-m -1[]Γ()m q,因此，我们有∇-()m -αγ+af ()z =∫a +1z []x γ+m -1()z -x γ+m -1()s -1()-m -1[]Γ()m qg ()s d ∇x γ()s =∇-mγg ()z ,这样就有∇m γ∇-()m -αγ+a f ()z =∇m γ∇-m γg ()z =g ()z .由定理2得到启示，很自然地我们给出关于f ()z 的Riemann-Liouville 型α-阶()0<m -1<Re α≤m 分数阶差分的定义如下：定义6（Riemann-Liouville 分数阶差分）让m 是超过Re α的最小正整数，对于非一致格子x γ()s ，数集{}α,mod ()1中f ()z 的Riemann-Liouville 型α-阶分数阶差分定义为∇αγf ()z =∇m γ()∇α-mγ+αf ()z .（31）形式上来说，在定义5中，如果α替换成-α，那么式（27）的右边将变为∫a +1z []xγ-α-1()z -x γ-α-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ()t =∇∇x γ-α()t ()∇∇x γ-α+1()t ⋯()∇∇x γ-α+n -1()t ⋅∫a +1z[]xγ+n -α-1()z -x γ+n -α-1()t -1()n -α-1[]Γ()n -αqf ()t d ∇x γ()t =∇n γ-α∇-n +αγf ()z =∇αγ-αf ()z .（33）闽南师范大学学报（自然科学版）2021年8从式（33），我们也可以得到f ()z 的Riemann-Liouville 型α-阶分数阶差分如下:定义7（Riemann-Liouville 型分数阶差分2）对任意Re α>0，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的Riemann-Liouville 型α-阶分数阶差分定义为∇αγ-αf ()z =∫a +1z x γ-α-1()z -x γ-α-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ()t ，（34）将∇γ-α()t 替换成∇γ()t ，那么∇αγf ()z =∫a +1z []x γ-1()z -x γ-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ+α()t ,（35）这里假定[]Γ()-αq ≠0.4非一致格子上Caputo 型分数阶差分在本节，我们将给出非一致格子上Caputo 型分数阶差分的合理定义.定理3（分部求和公式）给定两个复变函数f (s )，g (s )，那么∫a +1z g (s )∇γf (s )d ∇x γ(s )=f (z )g (z )-f (a )g (a )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s )，这里z ,a ∈C ，且假定z -a ∈N .证明应用命题1，可得g (s )∇γf (s )=∇γ[f (z )g (z )]-f (s -1)∇γg (s )，这样就有g (s )∇r f (s )=∇r [f (z )g (z )]-f (s -1)∇r g (s ).关于变量s ，从a +1到z 求和，那么可得∫a +1z g (s )∇γf (s )d ∇x γ(s )=∫a +1z ∇γ[f (z )g (z )]∇x γ(s )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s )=f (z )g (z )-f (a )g (a )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s ).与非一致格子上Riemann-Liouville 型分数阶差分定义的思想来源一样，对于非一致格子上Caputo 型分数阶差分定义思想，也是受启发于非一致格子上广义Abel 方程式（28）的解.在本文第3节，借助于非一致格子上的Euler Beta 公式，我们已经求出广义Abel 方程∇-αγg (z )=f (z )，0<m -1<α≤m ，是g (z )=∇αγf (z )=∇m γ∇-m +αγ+αf (z ).（36）现在我们将用分部求和公式，给出式（36）的另一种新的表达式.事实上，我们有∇a γf (z )=∇m γ∇-m +aγ+a f (z )=∇mγ∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]qf (s )d ∇x γ+α(s ).（37）应用恒等式∇(s )[x γ+m -1(z )-x γ+m -1(s )](m -α)∇x γ+α(s )=∇(s )[x γ+m -1(z )-x γ+m -1(s -1)](m -α)∇x γ+α(s -1)=-[m -α]q [x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)，那么以下表达式程金发：非一致格子上离散分数阶差分与分数阶和分第1期9∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]qf (s )d ∇x γ+α(s )，可被改写成∫a +1zf (s )∇(s ){-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇s =∫a +1z f (s )∇γ+α-1{-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇x γ+α-1(s ).应用分部求和公式，可得∫a +1zf (s )∇γ+α-1{-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇x γ+α-1(s )=f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α)[Γ(m -α+1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s ).因此，这可导出∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]q}f (s )d ∇x γ+α(s )=f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α)[Γ(m -α+1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s ).(38)进一步，考虑∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s )，(39)利用恒等式∇(s )[x γ+m -1(z )-x γ+m -1(s )](m -α+1)∇x γ+α-1(s )=∇(s )[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)∇x γ+α-1(s -1)=-[m -α+1]q [x γ+m -1(z )-x γ+m -1(s -1)](m -α)，表达式(39)能被改写成∫a +1z∇γ+α-1[f (s )]∇(s ){-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇s =∫a +1z∇γ+α-1[f (s )]∇γ+α-2{-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇x γ+α-2(s ).由分部求和公式，我们有∫a +1z ∇γ+α-1[f (s )]∇γ+α-2{-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇x γ+α-2(s )=∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q +∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q[∇γ+α-2∇γ+α-1]f (s )d ∇x γ+α-2(s )=闽南师范大学学报（自然科学版）2021年10∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q∇2γ+α-2f (s )d ∇x γ+α-2(s ).因此，我们得到∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s )=∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q∇2γ+α-2f (s )d ∇x γ+α-2(s ).(40)同理，用数学归纳法，我们可得∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+k -1)[Γ(m -α+k )]q∇kγ+α-k [f (s )]d ∇x γ+α-k (s )=∇kγ+α-kf (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+k )[Γ(m -α+k +1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+k )[Γ(m -α+k +1)]q∇k +1γ+α-(k +1)f (s )d ∇x γ+α-(k +1)(s ).(k =0,1,⋯,m -1)(41)将式（38）,（40）和（41）代入式（37），则有∇αγf ()z =∇m γìíîïïf ()a []x γ+m -1()z -x γ+m -1()a ()m -α[]Γ()m -α+1q +∇γ+α-1f ()a []xγ+m -1()z -x γ+m -1()a ()m -α+1[]Γ()m -α+2q+∇kγ+α-kf ()a []x γ+m -1()z -x γ+m -1()a ()m -α+k []Γ()m -α+k +1q+⋯+∇m -1γ+α-()m -1f ()a []x γ+m -1()z -x γ+m -1()a ()2m -α-1[]Γ()2m -αq+üýþïï∫a +1z []xγ+m -1()z -x γ+m -1()s -1()2m -α-1[]Γ()2m -αq∇m γ+α-mf ()s d ∇x γ+α-m ()s =∇m γ{}∑k =0m -1∇kγ+α-kf ()a []x γ+m -1()z -x γ+m -1()a ()m -α+k []Γ()m -α+k +1q+∇α-2m γ+α-m ∇mγ+α-m f ()z =∑k =0m -1∇kγ+α-kf ()a []x γ-1()z -x γ-1()a ()-α+k []Γ()-α+k +1q+∇α-m γ+α-m ∇mγ+α-m f ()z .总之，我们有下面的程金发：非一致格子上离散分数阶差分与分数阶和分第1期11定理4（广义Abel 方程解2）假设定义在数集{}a +1,a +2,⋯,z 上的函数f ()z 和g ()z 满足∇-αγg ()z =f ()z ,0<m -1<Re α≤m ,那么g ()z =∑k =0m -1∇k γ+α-kf ()a []xγ-1()z -x γ-1()a ()-α+k []Γ()-α+k +1q+∇α-m γ+α-m ∇mγ+α-m f ()z .受到定理4的启示，我们很自然地给出函数f ()z 的α-阶()0<m -1<Re α≤m Caputo 分数阶差分如下：定义8（Caputo 分数阶差分）让m 是Re α超过的最小整数，非一致格子上定义在数集{}a +1,a +2,⋯,z 函数f ()z 的α-阶Caputo 分数阶差分定义为C∇αγf ()z =∇α-m γ+α-m ∇mγ+α-m f ()z .最后，本文再强调指出：对于非一致格子上超几何差分方程式（2），在特定条件下存在关于x ()s 多项式形式的解，如果用Rodrigues 公式表示的话，它含有整数阶高阶差分.一个重要的问题是：若该特定条件不满足，那么非一致格子超几何差分方程的解就不存在关于x ()s 的多项式形式，这样高阶整数阶差分将不再起作用了，这就迫切需要我们引入一种非一致格子上分数阶差分的新概念和新理论.因此，关于非一致格子上阶分数阶差分及阶分数阶和分的定义是一个十分有趣和重要的问题.有关非一致格子超几何差分方程与离散分数阶差和分的联系，更深入的内容参见笔者著作[16]及文献[19-21].(42)(43)参考文献：[1]Kilbas A A,Srivastava H M,Trujillo J J.Theory and applications of fractional differential equations[M].Holland:North-Hol-land Mathatics Studies,Elsevier,2006.[2]Miller S,Ross B.An introduction to the fractional calculus and fractional differential equations[M].NewYork:JohnWiley andSons,1993.[3]Podlubny I.Fractional Differential Equations[M].San Diego,CA:Academic Press,1999.[4]Samko S G,Kilbas A A,Marichev O I.Fractional integrals and derivatives:theory and applications[M].London:Gordon andBreach,1993.[5]Anastassiou G A.Nabla discrete fractional calculus and nalba inequalities[J].Mathematical and Computer Modelling,2010,51:562-571.[6]Atici F M,Eloe P W.Discrete fractional calculus with the nable operator[J].Electronic Journal of Qualitative Theory of Differ-ential Equations,Spec.Ed.I,2009(3):1-12.[7]Atici F M,Eloe P W.Initialvalue problems in discrete fractional calulus[J].Pro.Amer.Math.Soc,2009,137:981-989.[8]Ferreira A C,Torres F M.Fractional h-differences arising from the calculus of variations[J].Appl Anal Discrete Math,2011(5):110-121.[9]程金发.分数阶差分方程理论[M].厦门:厦门大学出版社,2011.[10]Goodrich C,Peterson A C.Discrete fractional discrete fractional discrete fractional calculus[M].Switzerland:Springer Inter-national Publishing,2015.[11]Annaby M H,Mansour Z S.q-Fractional Calculus and Equations[M].NewYork:Springer-Verlag,2012.[12]Andrews G E,Askey R,Roy R.Special functions.Encyclopedia of Mathematics and its Applications[M].Cambridge:Cam-bridge University Press,1999.[13]Wang Z X,Guo D R.Special Functions[M].Singapore:World Scientific Publishing,1989.闽南师范大学学报（自然科学版）2021年12[14]Nikiforov A F,Suslov S K,Uvarov V B.Classical orthogonal polynomials of a discrete variable[M].Berlin:Springer-Verlag,1991.[15]Nikiforov A F,Uvarov V B.Special functions of mathematical physics:a unified introduction with applications[M].Basel:Birkhauser Verlag,1988.[16]程金发.非一致格子超几何方程与分数阶差和分[M].北京:科学出版社,2021.[17]Atakishiyev N M,Suslov S K.Difference hypergeometric functions,in:progress in approximation theory[M].New York:Springer-Verlag,1992:1-35.[18]Suslov S K.On the theory of difference analogues of special functions of hypergeo-metric type[J].Russian Math Surveys,1989,44:227-278.[19]Cheng J F,Jia L K.Generalizations of rodrigues type formulas for hypergeometric difference equations on nonuniform[J].Journal of Difference Equations and Applications,2020,26(4):435-457.[20]Cheng J F,Dai W Z.Adjoint difference equation for a Nikiforov-Uvarov-Suslov difference equation of hypergeometric typeon non-uniform Lattices[J].Ramanujan Journal,2020,53:285-318.[21]Cheng J F.On the complex difference equation of hypergeometric type on non-uniform lattices[J].Acta Mathematical Sinica,English Series,2020,36(5):487–511.[责任编辑：钟国翔]程金发：非一致格子上离散分数阶差分与分数阶和分第1期13。

离散数学中英⽂名词对照表离散数学中英⽂名词对照表外⽂中⽂AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作⽤addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平⾯algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent ⼏乎相等的alternating group 三次交代群annihilator 零化⼦antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合⼦asymmetric 不对称的(⾮对称的)atom 原⼦atomic formula 原⼦公式augmenting digeon hole principle 加强的鸽⼦笼原理augmenting path 可增路automorphism ⾃同构automorphism group of graph 图的⾃同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity ⽆穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation ⼆元关系binary symmetric channel ⼆进制对称信道binomial coefficient ⼆项式系数binomial theorem ⼆项式定理binomial transform ⼆项式变换bipartite graph ⼆分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对⾓线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡⼉积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中⼼characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number ⾊数chromatic polynomial ⾊多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着⾊coloring problem 着⾊问题combination number 组合数combination with repetation 可重组合common factor 公因⼦commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(⼦图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全⼆分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分⼦）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分⽀connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（⽆⽭盾性）continuum 连续统contraction of graph 图的收缩contradiction ⽭盾式（永假式）contravariant functor 反变函⼦coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通⽤映射countably infinite set 可列⽆限集（可列集）covariant functor (共变)函⼦covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对⾓线函⼦diameter 直径digraph 有向图dilemma ⼆难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选⾔三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring ⼦除环divison ring 除环divisor (factor) 因⼦domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离⼼率edge chromatic number 边⾊数edge coloring 边着⾊edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独⽴数eigenvalue of graph 图的特征值elementary divisor ideal 初等因⼦理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism ⾃同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧⼏⾥德算法Euclid domain 欧⽒整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多⾯体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推⼴规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number ⼴义 Fibonacci 数extended Lucas number ⼴义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface ⾯factor 因⼦factorable 可因⼦化的factorization 因⼦分解faithful (full) functor 忠实（完满）函⼦Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤⼦finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限⽣成模first order theory with equality 带符号的⼀阶系统five-color theorem 五⾊定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函⼦four-color theorem(conjecture) 四⾊定理（猜想）F-reduced product F-归纳积free element ⾃由元free monoid ⾃由单元半群free occurrence ⾃由出现free R-module ⾃由R-模free variable ⾃由变元free-?-algebra ⾃由?代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (G?del-Bernays-von Neumann system)GBN系统generalized petersen graph ⼴义 petersen 图generating function ⽣成函数generating procedure ⽣成过程generator ⽣成⼦（⽣成元）generator matrix ⽣成矩阵genus 亏格girth （腰）围长G?del completeness theorem G?del 完全性定理golden section number 黄⾦分割数（黄⾦分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第⼀类边⾊图graph of second class for edge coloring 第⼆类边⾊图graph rank 图秩graph sequence 图序列greatest common factor 最⼤公因⼦greatest element 最⼤元（素）Grelling paradox Grelling 悖论Gr?tzsch graph Gr?tzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函⼦homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超⼋⾯体图hypothelical syllogism 假⾔三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等⾃然变换imbedding 嵌⼊immediate predcessor 直接先⾏immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree ⼊次（⼊度）independent 独⽴的independent number 独⽴数independent set 独⽴集independent transcendental element 独⽴超越元素index 指数individual variable 个体变元induced subgraph 导出⼦图infinite extension ⽆限扩域infinite group ⽆限群infinite set ⽆限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函⼦injective (one to one mapping) 单射（内射）inner face 内⾯inner neighbour set 内（⼊）邻集integral domain 整环integral subdomain ⼦整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因⼦invariant factor ideal 不变因⼦理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆⾃然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换⼦）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-⾊图k-connected graph k-连通图k-critical graph k-⾊临界图k-edge chromatic graph k-边⾊图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman ⼥⽣问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin ⽅lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最⼩元least upper bound 上确界（最⼩上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因⼦left adjoint functor 左伴随函⼦left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻⽅many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表⽰maximal element 极⼤元maximal ideal 极⼤理想maximal outerplanar graph 极⼤外平⾯图maximal planar graph 极⼤平⾯图maximum matching 最⼤匹配maxterm 极⼤项（基本析取式）maxterm normal form(conjunctive normal form) 极⼤项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极⼩项minimal κ-connected graph 极⼩κ-连通图minimal polynomial 极⼩多项式Minimanoff paradox Minimanoff 悖论minimum distance 最⼩距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极⼩项（基本合取式）minterm normal form(disjunctive normal form)极⼩项范式（析取范式）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁⽅Nn-ary operation n-元运算n-ary product n-元积natural deduction system ⾃然推理系统natural isomorphism ⾃然同构natural transformation ⾃然变换neighbour set 邻集next state 下⼀个状态next state transition function 状态转移函数non-associative algebra ⾮结合代数non-standard logic ⾮标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规⼦群（不变⼦群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁⽅orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外⾯outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平⾯图Ppancycle graph 泛圈图parallelism 平⾏parallelism class 平⾏类parity-check code 奇偶校验码parity-check equation 奇偶校验⽅程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽⼦笼原理planar graph （可）平⾯图plane graph 平⾯图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表⽰法polynomial ring 多项式环possible world 可能世界power functor 幂函⼦power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着⾊proper factor 真正因⼦proper filter 真滤⼦proper subgroup 真⼦群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive ⾃反的regular graph 正则图regular representation 正则表⽰relation matrix 关系矩阵replacement theorem 替换定理representation 表⽰representation functor 可表⽰函⼦restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函⼦right cancellable 右可消的right factor 右因⼦right zero divison 右零因⼦ring 环ring of endomorphism ⾃同态环ring with unity element 有单元的环R-linear independence R-线性⽆关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满⾜的saturated 饱和的scope 辖域section 截⼝self-complement graph ⾃补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent ⽮列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原⼦）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category ⼩范畴smallest element 最⼩元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph ⽣成⼦图spanning tree ⽣成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系⼤集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra ⼦代数subcategory ⼦范畴subdirect product ⼦直积subdivison of graph 图的细分subfield ⼦域subformula ⼦公式subdivision of graph 图的细分subgraph ⼦图subgroup ⼦群sub-module ⼦模subrelation ⼦关系subring ⼦环sub-semigroup ⼦半群subset ⼦集substitution theorem 代⼊定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关⽹络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验⼦syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system ?3 , ?n , ??0 , ??系统?3 , ?n , ??0 , ??system L 公理系统 Lsystem ?公理系统?system L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system ?n 公理系统?nsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重⾔式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全⾊数total chromatic number conjecture 全⾊数猜想total coloring 全着⾊total graph 全图total matrix ring 全⽅阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅⾏商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡⼦群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤⼦ultrapower 超幂ultraproduct 超积unary operation ⼀元运算unary relation ⼀元关系underlying graph 基础图undesignated truth value ⾮特指值undirected graph ⽆向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯⼀分解unique factorization domain (Gauss domain) 唯⼀分解整域unique k-colorable graph 唯⼀k着⾊unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通⽤结构universal enveloping algebra 通⽤包络代数universal generalization 全称推⼴规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph ⽆标号图untorsion ⽆扭模upper (lower) bound 上（下）界useful equivalent 常⽤等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff) word 字Zzero divison 零因⼦zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。

变形Jaynes-Cummings模型中的Pancharatnam相位王发强;梁瑞生【摘要】关系.【期刊名称】《华南师范大学学报（自然科学版）》【年(卷),期】2009(000)003【总页数】4页(P39-42)【关键词】变形;Pancharatnam相位;几何相位【作者】王发强;梁瑞生【作者单位】华南师范大学信息光电子科技学院,广东广州,510631;华南师范大学信息光电子科技学院,广东广州,510631【正文语种】中文【中图分类】O431.2自从Berry几何相位于1984年被发现以来[1]，量子力学几何相位得到了深入且广泛的研究[2-4]，并在实验系统中观测到了相关现象[5-6]，其中文献[6]观测到了单光子水平上的几何相位. 另外，Berry关于几何相位的概念已经被推广，如Bhandari 和Samuel 将Pancharatnam 定义的非绝热非回复光路中光波相位的比较推广到量子力学中[5,7]，其比Berry相位更普遍. 在应用方面，几何相位可以用来实现量子门和量子计算[8]，Pancharatnam-Berry相位可以被用来控制光场波前形状[9].另一方面，变形的Jaynes-Cummings(J-C)模型得到了相当广泛的研究[10-12]，因为其在量子场论、量子引力、自旋链、核物理以及分子光谱等方面具有潜在的应用前景. 在变形J-C模型中，通常谐振子算符变为单参数或双参数变形谐振子算符. 众所周知，通常谐振子算符是满足Bose型对易关系的，当变形谐振子参数偏离1时，则变形场光子的统计特性将偏离传统的Bose统计，这将产生许多新的效应.另外，变形J-C模型中还包含了光场与原子耦合系数与光场强度相关的非线性效应，而Pancharatnam相位可以通过Mach-Zehnder干涉仪测量. 因此，本文将研究不同参数条件下变形J-C模型中的Pancharatnam相位随时间的演化规律.本文第1部分给出系统的微观模型及Pancharatnam相位的理论推导，第2部分是数值计算和结果分析，最后给出结论.当一两能级原子与单模变形量子光场作用时，系统Hamiltonian在旋波近似下，可写为[10-12]：其中，ω为原子基态1〉之间的跃迁频率，σZ、σ+和σ-是原子的赝自旋算符，g 是原子与光场的耦合参数，A†(A)是频率为ν变形光场的产生(湮灭)算符，N为变形光场的粒子数算符. 光场算符满足如下的变形谐振子对易关系：其中，结构函数Φ(x)与具体的变形方案有关，具体形式将在后面给出. 一般来讲，结构函数Φ(x)是满足Φ(0)=0的解析函数，且A†A=Φ(N)≡[N]，AA†=Φ(N+1)≡[N+1].假设系统的初态如下：其中，|n〉表示光场处于变形谐振子Fock态.则系统t时刻的态矢量为：).为简化计算，本文假定原子与光场处于共振状态，即ω=ν. 通过计算，可以得到：bn(t)=bCncos(gt)-iaCn+1sin(gt).当光场初始处于变形相干态时，上式中的系数Cn为[11-12]：其中，α为复数，变形的e指数定义为由此，可得到系统的Pancharatnam相位为[13]：φt=arg〈ψ(0)|ψ(t)〉=其中，X(t)、Y(t)为：Y(t)=(a*bα*+ab*α)×,X(t)=+下面讨论当系统为通常J-C模型以及变形J-C模型时，系统Pancharatnam相位随时间的演化.首先，讨论系统为通常的J-C模型时，Pancharatnam相位随时间的演化. 当光场产生和湮灭算符对易关系中的结构函数为Φ(N)=N时，系统即为通常J-C模型. 由图1的(a)和(b)可以看出，当光场的平均光子数为25和9时，系统Pancharatnam相位随归一化时间gt的演化出现坍塌与回复，即相位随时间振荡——消失——振荡. 众所周知，当光场处于相干态时，与其作用的原子布居数反转随时间也出现坍塌与反转，这是因为，光场的相干态可以看作是不同粒子数态(Fock态)的相干叠加，而不同粒子数态演化产生的相位是不同的，其相干叠加，导致了坍塌与回复的出现. 以上结果表明，系统的Pancharatnam相位反映了与光场、原子量子特性相关的信息. 另外，由图1(c)可以看出，当平均光子数较小时，系统的Pancharatnam相位将不出现坍塌与回复，而是一直作某种有规律的振荡. 这与原子的布居数反转随时间的演化规律是一致的.其次，讨论系统为q变形J-C模型时，Pancharatnam相位随时间的演化. 当光场产生和湮灭算符对易关系中的结构函数取Φ(N)≡[N]=(qN-q-N)/(q-q-1)(0lt;qlt;1)时，系统通常被称为q变形的J-C模型. 当q→1时，结构函数[N]=N, 系统将变为前面的通常J-C模型[10-12]. 当然，系统还存在其他的单参数变形，如[N]Q=(QN-1)/(Q-1). 但其与前面的q变形是相关联的，即当Q=q2时，[N]=q1-N[N]Q，所以，此处，我们只考虑q变形就可以了. 由图2可以发现，变形的J-C模型Pancharatnam相位演化，与前面通常的J-C模型一样，同样存在坍塌与回复. 对比图1与图2，可以看出，当平均光子数较大时，即|α|=25，变形J-C模型的Pancharatnam相位演化与通常的J-C模型的Pancharatnam相位演化存在较大差别，即振荡的时间、坍塌与回复的时间周期皆不同. 但当平均光子数减小时，这种差别也逐渐减小. 对比图3与图1、图2，可以发现，当q偏离1，进一步减小时，系统的Pancharatnam相位演化也将出现较大的变化，但这种差别在平均光子数为1时，变得非常细微.最后，我们将讨论双参数(p,q)变形J-C模型中的Pancharatnam相位演化. 当光场产生和湮灭算符对易关系中的结构函数取[N]=(qN-p-N)/(q-p-1) (0lt;qlt;1, pgt;1, pqlt;1)时，系统便被称为(p,q)变形的J-C模型. 当p→q时，系统即是前面的q变形J-C模型. 由图4可以发现，当平均光子数较大时，系统的Pancharatnam相位作周期性振荡，且当平均光子数减小时，其振荡波形将出现分裂，与q变形情况下的演化规律有较大差异，这是因为，在(p,q)变形模型中，结构函数[N]并不总是随N的增加而增加，当N大于某一确定值时，[N]将随N的增加而减小，这将导致系统的部分本征能谱变负或存在简并[12]. 当平均光子数为1时，其演化趋势与前面模型相对应的情况下的趋势一致. 研究发现，系统的Pancharatnam相位同样也反映了与光场、原子量子特性相关的信息.由上面的结果可以看出，当平均光子数较大时，变形J-C模型的Pancharatnam相位演化规律与通常J-C模型的Pancharatnam相位演化规律差异较大. 而当平均光子数较小时，此差异则较小. 这是由于光场算符的结构函数是光子数算符的函数，即系统具有非线性, 所导致的.本文首先解析计算了变形J-C模型的Pancharatnam相位，然后，具体计算和讨论了变形J-C模型的Pancharatnam相位随归一化时间的演化规律，并与通常J-C模型的Pancharatnam相位演化规律作了比较. 结果表明，Pancharatnam相位反映了与原子布居数反转以及光场量子特性相关的信息，且平均光子数较大时，变形J-C模型的Pancharatnam相位演化规律与通常J-C模型的Pancharatnam相位演化规律差异较大. 这些结论表明，系统的Pancharatnam相位演化与系统所蕴涵的量子代数结构密切相关. 研究结果对量子信息的测量、编码、存储以及量子门的设计都具有指导意义.Key words： deformed； Pancharatnam phase； geometric phase【相关文献】[1] BERRY M V. Quantal phase factors accompanying adiabatic changes[J]. Roy Soc A, 1984, 392:45-47.[2] SHAPERE A, WILCZEK F. Geometric phase in physics[M]. Singapore: World Scientific, 1989.[3] 颜玉珍,胡连. 旋转磁场中的自旋演化及几何位相[J].华南师范大学学报:自然科学版,2004(2):82-85.YAN Yuzhen, HU Lian. Evolution and geotric phase of spin in a rotating magnetic field [J].Journal of South China Normal University:Natural Science Edition, 2004(2):82-85. [4] 高玉梅, 胡连, 张晓燕. 旋转中子及螺旋光纤的几何相[J].华南师范大学学报:自然科学版,2005(1):60-65.GAO Yumei,HU Lian,ZHANG Xiaoyan. Geotric phase for helical neutron and opticalfiber[J].Journal of South China Normal University:Natural Science Edition, 2005(1):60-65.[5] KWIAT P G, CHIAO R. Observation of a nonclassical Berry’s phase for the photon[J]. Phys Rev Lett,1991,66(5): 588-591.[6] WEBB C L, GODUN R M, SUMMY G S, et al. Measurement of Berry’s phase using an atom interferometer[J]. Phys Rev A,1999,60(3):R1783-R1786.[7] PANCHARATNAM S. Generalized theory of interferencia and its applications[J]. Proc Indian Acad Sci A, 1956,44: 247-262.[8] WANG Z S, WU C F, FENG X L, et al. Nonadiabatic geometric quantum computation[J]. Phys Rev A,2007,76: 044303.1-044303.4[9] MARRUCCI L, MANZO C, PAPARO D. Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: Switchable helical mode generation[J]. ApplPhys Lett,2006,88(22): 221102.1-221102.3[10] BONATSOS D, DASKALOYANNIS C, LALAZISSIS G A. Unification of Jaynes-Cummings model[J]. Phys Rev A,1993,47(4):3448-3451.[11] CRNUGELJ J,MARTINIS M,MIKUTA-MARTINIS V. Properties of a deformed Jaynes-Cummings model[J]. Phys Rev A,1994,50(2):1785-1791.[12] GELOUN J B,GOVAERTS J,HOUNKONNOU M N.(p,q) deformations and (p,q)-vector coherent states of the Jaynes-Cummings model in the rotating wave approximation[J]. J Math Phys,2007,48(3): 1-23.[13] WAGH A G,RAKHECHA V C. On measuring the Pancharatnam phase[J].Phys Lett A, 1995, 197:107-111.。