of Abelian Sigma Model 1

有限Abel群的结构定理（Fundamental Theorem ofFinite Abelian Groups）有限Abel群的结构定理(Fundamental Theorem of Finite Abelian Groups) 有限Abel群是群论中已被研究清楚了的重要群类，也是应用比较广泛的群类，本节的主要结论是有限Abel群可以分解成阶为素数的方幂的循环群(循环p-群)的直积，而且表法是唯一的。

我们先看几个具体的例子。

4阶群都是Abel群，它们有两种互不同构的类型，代表分别是。

Z,Z，Z422 ，其中是非Abel群;是Abel群，且6阶群有两种不同的类型，代表分别是ZZ,SS6633。

Z,Z，Z6238阶Abel群有三种不同的类型，代表分别是。

Z,Z，Z,Z，Z，Z8242229阶群都是Abel群，它们有两种互不同构的类型，代表分别是。

Z,Z，Z933 这些有限Abel群都同构于循环群或者循环群的直积，并且每个循环群的阶都是一个素数的方幂，这些循环群的阶组成的有重集合正好是该群阶素数方幂乘积的所有可能组合。

例如8阶32Abel群，有三种情形:，分别对应于8写成素数方幂乘积所有可能的形式{2},{2,2},{2,2,2}32(三种):。

8,2,8,2，2,8,2，2，2下面我们讨论一般有限Abel群的结构。

引理1 设a是群G的一个元素，a的阶等于。

其中与是两个互素的正整数，m,mmmm1212那么a可以唯一的表示成，式中的阶是;;而且都am(i,1,2)a(i,1,2)a,aaaa,aaii12i1221是a的方幂。

证明因为与互素，所以存在整数使得。

于是mmu,uum，um,112121122umumum，umumumumum2211112211222211，令，则，而且a,a,a,aa,aa,aaa,a,aa,aa121221mm12都是的方幂。

因为，所以的阶是的因子。

由于a(i,1,2)adm(i,1,2)ma,e,a,eaiiii112与互素，从而互素，并且，故的阶等于。

a rXiv:h e p-la t/95325v228Mar1995IFUP-TH 11/95The 2-dimensional non-linear σ-model on a random lattice B.All´e s and M.Beccaria Dipartimento di Fisica dell’Universit`a and INFN Piazza Torricelli 2,I-56100Pisa,Italy Abstract The O (n )non-linear σ-model is simulated on 2-dimensional regular and ran-dom lattices.We use two different levels of randomness in the construction of the random lattices and give a detailed explanation of the geometry of such lattices.In the simulations,we calculate the mass gap for n =3,4and 8,analysing the asymptotic scaling of the data and computing the ra-tio of Lambda parameters Λrandom /Λregular .These ratios are in agreement with previous semi-analytical calculations.We also numerically calculate the topological susceptibility by using the cooling method.PACS number(s):05.50.+q 11.15.HaTypeset using REVT E XI.INTRODUCTIONNumerical simulations of an asymptotically freefield theory on a lattice provide informa-tion about continuum physics when they are performed at values of the correlation length which lay within the scaling window.This region is defined by the inequalities1≪ξ/a≪L where a,ξand L are the lattice spacing,the correlation length and the lattice size respec-tively.To this window it corresponds another scaling window in terms of the bare coupling g.To understand the relationship between both regions,it is enough to recall that ξ/a∝Λexp(1/α)whereαis proportional to some power of g.In this expression,Λis the renormalization group independent mass parameter.This parameter depends on the regularization used,thus the scaling window in terms of the bare coupling can be shifted towards the region of lower couplings if we use a lattice regularization in whichΛis small. Gauge theories regularized on a random lattice present aΛparameter some orders of mag-nitude smaller than on a regular square lattice[1].Therefore,the simulations on a random lattice can be performed at lower values of the bare coupling.This fact may make the simulations on random lattices more advantageous than on reg-ular square lattices.The physical signal in the Monte Carlo data can be masked by the presence of pertubative expansions related to mixings,perturbative tails andfinite renor-malizations of composite operators defined on the lattice.If the expansion parameter is smaller(and the perturbative coefficients do not get larger),these perturbative terms may be better controlled and the non-perturbative physical signal be more clearly seen.Also the non-universal terms in the scaling function could be less relevant,thus expediting the asymptotic scaling.The computation of the perturbative coefficients on random lattices might seem rather involved.However,the higher degree of rotational invariance on a random lattice should help in this respect[2].In order to address these prospects on a physically interesting theory,like QCD,in reference[3]one of us started to study some features concerning another asymptotically free field theory,the O(n)non-linearσ-model in two dimensions.The action for this model in the continuum isS=1this model regularized on a random lattice are universal and that theΛparameter dependson the degree of randomnessκ(see below)used in the construction of the random lattice.The present work has several aims.Firstly,we want to verify theκdependence of theΛparameter and support the scenario of a common continuum limit,the same than that ofthe theory defined in Eq.(1.1).For this purpose,the theory with n=3,4and8has beensimulated on both regular and random lattices to extract the topological susceptibility andmass gap.We have also succesfully used the cooling method[7,8]on a random lattice toextract the topological content of the theory.The n=3value was chosen to study boththe topological susceptibility and the mass gap,while the higher values of n were used inorder to have a better asymptotic scaling[9]on the mass gap data.Another reason to studythe topological charge on random lattices is that some sites on these lattices can grouptogether forming clusters with a typical size less than one lattice spacing.This geometrymight allow the existence of very small instantons which could hardly survive on regularlattices.We have always used lattices large compared to the correlation length at ourβ,with L=200,300and400.We have never averaged the results of the simulations amongdifferent random lattices because the previous lattice sizes are large enough to include animplicit average among several subregions of the lattice.The updating has been performedby using a cluster algorithm[10]adapted to the random lattice.Another purpose of the present work is to analyse whether the simulations on a randomlattice improve the asymptotic scaling or not.Actually,this is easy to check with the massgap data because the value of m/Λis known[6].As it was shown in[3],Λrandom≈Λregular for theσ-model.Hence,it is not expected any dramatic shift of the scaling window towardssmall values of the bare coupling.To construct the random lattice,we followed the procedure of T.D.Lee at al.[11].Theonly new ingredient is the introduction of a degree of randomnessκ.The sites of the latticeare the centers of hard spheres,the radius of which is a/2κ.These hard spheres are randomlylocated.At small values ofκ,the lattice is locally less random as we will see in section2.There is aκmin at which all distributions(link lengths,distance between neighbouring sites,plaquette areas,etc.)become Dirac deltas.The plan of the paper is as follows.In section2,we explain how to construct a ran-dom lattice and discuss in detail both theκdependence and some geometrical propertieslike average link lengths and average distance between neighbouring sites.We also give ageometrical argument in favour of a unique random lattice,onceκisfixed,as the numberof sites gets larger.In section3we explain the simulation algorithm used.In section4werecall the scaling andβfunctions for the non-linearσmodel.We also explain the procedurefollowed to extract both the mass gap and the topological susceptibility.A detailed descrip-tion of the cooling method is also given.In section5we show and analyse the Monte Carlodata for the physical observables described in section4.The conclusions are presented insection6.II.CONSTRUCTION AND GEOMETRY OF A RANDOM LATTICEWe define a random lattice as a set of N points located at random on a volume V with the condition that there are no two sites closer than a/κwhere a is the lattice spacing andκis a parameter to befixed.These sites are placed with periodic boundary conditions.Thus, we consider the2-dimensional lattice as a torus.In order to simulate afield theory on this geometry,we must define a net of connections throughout the lattice,linking neighbouring sites.In thefirst part of the section,we will explain how to place the sites on the lattice and in the second part we will review the triangularization process which is used to construct the links and plaquettes.Finally,we will give some properties of the random lattices.If N sites are placed on a2-dimensional volume V,we define the lattice spacing a as a=1Due to the periodic boundary conditions we imposed,the piece of disc which exceeds the allowed volume from below,will reappear on the top of the lattice.κmust be the smallest allowed.The volume occupied by a regular hexagonal lattice with√√N=L2sites is V=L2.(2.1)Vκ2Now,using a2=V/N,we conclude that the total number of proposed sites divided by N is11−νπ.(2.3)1−xππ/κ2This expression,as previously stated,is independent of N.It works well forκ>2.5.Once the lattice volume V has beenfilled with the N sites for some chosen value of κ,we proceed to the triangularization process.We follow the method of reference[12].Itconsists in joining sets of three sites to form a triangle with the only condition that the circle circumscribed by these three points does not contain any other site.The sides of that triangle are the links joining the three sites and the triangle itself is a plaquette.This construction is unique andfills the whole lattice with no overlapping among the triangles[11].We also impose periodic boundary conditions in the triangularization process.In Figure4we show two N=100sites random lattices,withκ=∞andκ=1.3.It is useful to define also the dual lattice.Its dual sites are the centers of the above-mentioned circumscribed circles.It is clear that any link is the common side of two triangles. Thus,every link of the random lattice must be surrounded by two dual sites.The line joining these two dual sites is called the dual link.Therefore,every link is associated with a dual link.Let us call lµij theµ-component of the link vector that points from the site i to the site j.The length of this link is l ij= l ij .The length of the corresponding dual link is s ij. Throughout this work,we will often make use of the matrixλij defined as(2.4)λij≡ s ij/l ij,if i and j are linked;0,otherwise.We introduce another vector,d ij which is twice the vector joining the center of the vector link l ij with the center of the associated dual link s ij.Another useful quantity isσijk defined asσijk≡[(l ij+d ij)×(l ik+d ik)]·ˆz.(2.5) In this equationˆz is the unit vector orthonormal to the plane of the lattice,oriented as ˆz=ˆx׈y.The set of dual links around the site i,{s ij}ifixed,determine a convex region called Voronoi cell.The area of this cell isωi.As soon as the lattice has been constructed,one can devise tests to check the triangular-ization.Thefirst and easiest one is to verify that the number of triangles(links)is equal to twice(three times)the number of sites[11].Other good tests are the integral properties[13] jλij lµij=0, jλij lµij(lνij+dνij)=2ωiδµν.(2.6)A third test consists in demanding that the quadratic part of the action has only one zero mode[3].The action for the2-dimensional O(n)non-linearσ-model on a random lattice can be written as[13]S L=1In this equation,g is the coupling constant.i,j,...denote sites and φi stands for the value of thefield φat the site i.This is the action we will make use in our numerical simulations, both on regular and random lattices.For a regular square lattice,λij is1for linked sites and zero otherwise.Hence,Eq.(2.7)becomes the standard action when it is considered on a regular lattice.One can show that the na¨ıve continuum limit of Eq.(2.7)is the correct action for the model in the continuum,Eq.(1.1).We see from Figure4that random lattices with large(small)values ofκdisplay a higher (lower)degree of randomness.As it was shown in reference[3],random lattices with different κare different regularizations of the same theory.In particular,it was shown that theΛparameter isκ-dependent.In the present paper,we will check this statement by a numerical simulation and,moreover,we will give hints that the non-universal non-leading coefficients of theβfunction of the model are alsoκ-dependent.The level of randomness can be clearly manifested with the distributions of some geo-metrical properties of the lattice.Thefirst property we plot is the distance of one site to its nearest site,r.This distance,referred to the site i,was written before as r i.In Figure 5,we show the probability distribution of this distance,P(r)d r for aκ=∞lattice.The histogram in the Figure is the numerical result,obtained by calculating r i for a single site on30000random lattices of1000sites.The solid line is the theoretical well known Poisson distribution,rP(r)d r=2πdegree of randomness.However,our method to construct the random lattice needs only the knowledge ofκas previously explained.Therefore,we will label our random lattices with thisκparameter.The column for l was obtained averaging the link lengths of random lattices with10000sites forκ=∞,1.3and1000sites forκ=1.2.The column for r was calculated by averaging r on a single site for30000random lattices of1000sites for κ=∞and1.3;forκ=1.2we averaged on14random lattices of100sites.The numbers shown depend on the lattice size.As this size gets larger,the averages tend to stabilize.For instance,the averages for r /a calculated on random lattices of100,1000and10000sites withκ=∞are0.523(2),0.508(2)and0.504(2)respectively.The exact value computed from Eq.(2.8)is r =1√2/distribution of u.To be definite we chose the uniform distribution on the hypersphere by taking at random a point inside the hypercube x∈[0,1]n and rejecting it if x >1.Then, u= x/ x .We now mark with a label the site i and all of its neighbours according to the probability weightsp ij=1−exp min 0,−2λij|C|θC(x)θC(y)(3.3)have already been calculated at each updating step.The above expression is called an improved estimator for the two point function.In Eq.(3.3)N is the total number of sites of the lattice,|C|is the number of sites in the cluster andθC is the characteristic function of the cluster.Another advantage of this estimator is that it can be measured after each updating step without need of decorrelating updatings.IV.THE PHYSICAL OBSER V ABLESThe physical observables that we have measured are the mass gap for several values of n and the topological charge for n=3.For each of these quantities we have studied the asymptotic scaling behaviour.The universal2-loopβfunction of the O(n)non-linear σ-model isβ(g)=−n−24π2g3+O(g4),(4.1)and therefore the lattice spacing obeys the renormalization group law aΛlattice=f(β)≡ 2πβn−2 1+O(1The first non-universal correction for this scaling function on a regular lattice is known [15].Ontheother hand,the value of Λlattice /Λf (β)dim A =Ae ∆1MS ,∆=12 Λ2 (4.6)is enough to reproduce the data [14].To avoid correlation among data at different t ,we used sets of different runs for every t .Hence,the statistical errors obtained are reliable.The above mentioned improved estimator,Eq.(3.3),reduces greatly the statistical noise which affects this measure particularly at large t .We have also measured the topological charge and susceptibility of the O (3)model on a random lattice.The topology of this model is based on the stereographic map from the sphere S 2onto the projective plane [5].The topological charge actually counts the winding or instanton number of this mapping on classical configurations.In the continuum it is defined asQ= d2xQ(x),Q(x)=132π j,kλijλikV i Q i 2 .(4.10)The Monte Carlo data for the topological susceptibilityχL is[17]χL=a2(β)Z(β)2χ+a2(β)A(β) S(x)NP+P(β),(4.11)where a(β)=f(β)/Λlattice,Eq.(4.2), S(x)NPis the non-perturbative vacuum expectationvalue of the density of action and Z(β),A(β)and P(β)are the renormalizations whichcan be calculated perturbatively[17,18].On the regular lattice S(x)NPis negligible[19]and so is the second term in the r.h.s.of Eq.(4.11).We have not calculated all of theserenormalizations on the random lattice but instead we have used the cooling method[7].The cooling procedure is a relaxation process which after a few cooling steps(∼30–40steps)have almost eliminated all short scalefluctuations leaving the long waves still present.If we assume that these short scalefluctuations,of order O(a),are responsible for thequantum noise which shows up as renormalization effects,then after some cooling steps allrenormalizations in Eq.(4.11)will disappear and the physical and Monte Carlo topologicalsusceptibility will be related by the expressionχ=χLHowever the situation for the O(3)σmodel is not so simple.In this model,instantonstend to be small.Indeed the distribution of instantons with radiusρin this theory satisfiesd N/dρ∝1/ρ.As a consequence,the previous cooling process can also eliminate smallO(a)instantons,thus modifying the topological content of the configuration.This unwanted behaviour of the cooling occurs on regular lattices[19].On a random lattice some sites cangroup together forming clusters with a typical size less than one lattice spacing.This facthappens mostly on largeκlattices(see Figure4).This geometry might allow the existenceof very small instantons with a sizeρ≪a.This is the main motivation for studying thetopological properties of the model on random lattices.In Figure9we show the evolution of the topological charge for40uncorrelated config-urations as the cooling process goes on.At zero cooling step the value of the charge onthe lattice is on average Q L=QZ(β).Instead,after30or40steps,this charge reachedan almost integer value which depends only on the underlying instantonic content of theoriginal configuration.This almost integer value remains stable for a long plateau.Thus,we assumed the value of Q L after30or40steps of cooling as the correct topological chargeof the configuration.We checked that the value of the topological susceptibility is also stableon the plateau.Indeed,the value obtained is within errors,the same if30,40or50coolingsteps are performed.It can also be seen from Figure9that the value of Q L after severalcoolings is not exactly an integer.This is a general fact and has to do with the fact thatinstantons are not exact solutions of the theory defined on a lattice.We also checked thatthe susceptibility is insensitive to rounding this number to its nearest integer.We chosefor our analysis the values of Q L after30cooling steps without rounding it to the nearestinteger.We now turn to a detailed description of the cooling step.It consists in locally minimizingthe action with respect to thefield at each site once per step.We used a controlled cooling[8].This means that for a given positive numberδ,the newfield φ′i and the old one φi differ lessthanδ, φi− φ′i ≤δ.First we define the force relative to the site i asFi≡jλij φjφi+ǫ( F i− φi) ,(4.14) whereǫis chosen to beǫ≡δ1−1In a step of cooling we pass through all sites i of the lattice and perform the previous modification on the correspondingfield φi.V.THE MONTE CARLO RESULTSIn this section we will show the set of Monte Carlo data obtained for the mass gap and the topological susceptibility and discuss their consequences.In table2we show the set of data for the mass gap and the O(3)σmodel on lattices of 2002sites.They are also shown in Figure10.The data were obtained from12000measures of the improved correlator,Eq.(3.3)for eachβ.Wefit these data to the scaling function of Eq.(4.2)f(β)=2πβα1exp(−2πβα2),(5.1)whereα1andα2are the parameters of thefit.In particular,α2must be equal to1in order for the data to scale as theβfunction of the theory predicts,Eq.(4.1),andα1must be m/Λlattice.Thefit givesα2=1.02(2),0.97(2),0.95(2)for the regular lattice,κ=1.3andκ=∞random lattices respectively.This is in agreement with the result of[3]concerning the universality of the random lattice:thefirst coefficient of the beta function is the same in any lattice regularization.Now,imposingα2=1and leavingα1free we obtainedα1= 121(2),139(2),91(1)for regular lattice,κ=1.3andκ=∞random lattice respectively, with a value forχ2/n.d.f.equal to6/11,8/11and9/12respectively.Following[6]and[3], the expected results forα1are80.09,87.05and44.49respectively.We excluded anyfinite size explanation to this discrepancy.We arrived at this conclusion because a)the technique of reference[20]did not improve the results and b)thefits with the data of table3for 4002lattices yielded similar results forα1.We think that the disagreement is due to the fact that our range ofβis narrow enough to collect intoα1all the power-law corrections to the asymptotic scaling function.Unfortunately,another single free parameter is not able to account for the whole non-universal terms correcting Eq.(5.1)and it did not improve dramatically the result forα1.From the results of thefits forα1we can get the ratio betweenΛparameters.Let us define R(κ)=Λrandom/Λregular at a givenκ.Then,R(1.3)=0.87(3)and R(∞)=1.33(4). These results are in agreement with the average of the ratios obtained from eachβ:R(1.3)= 0.86(6)and R(∞)=1.29(8).These ratios for eachβare shown in Figure11.The theoretical values[3]are R(1.3)=0.92(2)and R(∞)=1.8(2).Forκ=1.3theoretical and Monte Carlo ratios are in agreement within errors.But this is no so forκ=∞.We again think that this is due to the lack of asymptotic scaling.It is well known that asymptotic scaling is rather elusive in the O(3)σmodel and here we have realised that this problem does not ameliorate if a random lattice is used.For this reason,we also performed runs for the O(4)and O(8)models where it is known that asymptotic scaling is better achieved[9].In table4the mass gap data for O(4)on lattices with4002sites are shown.In table5the same data are shown for O(8)and lattices with3002 sites.In both cases12000measures of the improved estimator,Eq.(3.3),were performed for eachβ.The O(4)data of table4werefitted tof(β)=3 1/6α1exp −πβα2respectively.We repeated the same analysis on larger lattices to check whether this discrep-ancy is due tofinite size effects on the data.In table7we show the data for a regular lattice with3002sites and in table8the data for aκ=∞random lattice with4002sites. They are also obtained by cooling1000uncorrelated configurations.Thefit on the data of table7givesα2=0.76(4)and thefit on table8α2=0.65(4).From thesefigures we could hardly conclude that the lack of correct scaling is due tofinite size effects.We think that this problem must be traced back to the elimination of small instantons during the cooling process.Probably,the use of largeκrandom lattices is not enough to avoid this effect.The lack of asymptotic scaling prevented us from using the topological susceptibility and the mass gap data to check the physical scaling of the O(3)model on random lattices.In Figure13we show the ratio ofΛparameters for each value ofβfrom the data of the topological susceptibility.The average result for this ratio is surprisingly similar to the one obtained from the mass gap data:R(1.3)=0.94(2)and R(∞)=1.37(3).VI.CONCLUSIONSWe have used the random lattice to simulate the2-dimensional O(n)non-linearσ-model. The sites of the random lattice are considered as the centers of hard spheres of radius a/2κwhere a is the lattice spacing.These hard spheres are located at random on the lattice vol-ume.The links between neighbouring sites are established by a well known triangularization process[12].To compare the performance of different lattices,we made the simulations on regular lattices and random lattices with bothκ=1.3andκ=∞.We used the Wolffalgorithm[10]for the simulations as well as an improved estimator[14] for the computation of two-point correlation functions.We measured the mass gap(as the inverse of the correlation length measured from the wall-wall two-point correlation function) and the topological susceptibility.The topological charge was calculated by using a cooling technique[7]and we introduced a regularized operator for the topological charge on the lattice(see Eq.(4.8)).The Monte Carlo results for the mass gap scale as they should,confirming previous claims[3]about the universality of the random lattice regularization.They do not present anyfinite size problems(for2002sites)but for small values of n the asymptotic scaling is not fulfilled.For the O(8)model,where the data display a good asymptotic scaling,we can reproduce the theoretical value for the ratio between Lambda parametersΛrandom/Λregular for κ=∞,thus confirming the semi-analytical prediction of reference[3].Instead,forκ=1.3, the Monte Carlo value for the previous ratio is in good agreement with the theoretical prediction already in the O(3)model.This could mean that both regularizations(regularlattice andκ=1.3random lattice)are quite similar and in particular the non-universalterms in the scaling function,Eq.(4.2),almost coincide.This assumption is also supportedby the fact thatΛrandom/Λregular forκ=1.3is close to1.In any case,these conclusionsare consistent with the scenario where also the non-universal terms in the scaling function,Eq.(4.2)areκ-dependent.The data for the topological susceptibility scale very badly.We think that the coolingprocess removes small instantons thus modifying the topological content of the configura-tion also on random lattices.The cooling smooths outfluctuations with a length of orderO(a).For smallerβthe lattice spacing is longer in physical units,therefore the number of eliminated instantons when smoothing out O(a)fluctuations is also larger.This explainswhy the data in Figure12and Tables6,7and8are shifted downwards for smallβ.In any case,our results prove that the semi-analytical method used in reference[3]isreliable to perform analytical calculations on random lattices.VII.ACKNOWLEDGEMENTSWe thank Federico Farchioni and Andrea Pelissetto for useful discussions.We also ac-knowledgefinancial support from INFN.REFERENCES[1]Z.Qiu,H.C.Ren,X.Q.Wang,Z.X.Yang and E.P.Zhao,Phys.Lett.B198(1987)521;B203(1988)292.[2]Work in progress.[3]B.All´e s,Nucl.Phys.B437(1995)627.[4]A.M.Polyakov,Phys.Lett.B59(1975)79;E.Br´e zin and 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Giacomo,F.Farchioni,A.Papa and E.Vicari,Phys.Lett.B276(1992)148.[19]F.Farchioni and A.Papa,Nucl.Phys.B431(1994)686.[20]M.L¨u scher in“Progress in gaugefield theory”(Carg`e se1983),G.’t Hooft et al.(eds.),Plenum,New York(1984);I.Bender and W.Wetzel,Nucl.Phys.B269(1986)389.Figure captionsFigure1.Location of hard discs for a lattice of32sites withκ=1.The radius of the discs is equal to a/2.In a)it corresponds to a regular square lattice.In b)a column has beenshifted:κstill equals1but the lattice is no longer regular.Figure2.Location of hard discs for a regular hexagonal lattice with9sites.Theyfit tight and no movement is allowed without breaking the hard nature of the discs.Figure3.The ratio between the number of proposed sites N p and total number of sites N in the construction of a random lattice as a function ofκ.The solid and dashed lines are theresults for lattices with N=100and N=1000sites respectively.Figure4.The result of the triangularization process performed on two random lattices of102 sites for a)κ=∞and b)κ=1.3.The different level of randomness is apparent.Figure5.Distribution probability of distances between a site and its closest neighbour on a random lattice withκ=∞.The a in abscisses stands for one lattice spacing.Thehistogram is the numerical result calculated by using a single site on30000randomlattices of1000sites.This histogram should coincide with the Poisson distribution,shown in thefigure with a solid line.Both curves are normalized to1.Figure6.Distribution probability of distances between a site and its closest neighbour on a random lattice withκ=1.3.The a in abscisses stands for one lattice spacing.Thehistogram is the numerical result calculated by using a single site on30000randomlattices of1000sites.The curve is normalized to1.Figure7.Distribution probability of link lengths on a random lattice calculated withκ=∞.The a in abscisses stands for one lattice spacing.The curve is normalized to1.Figure8.Distribution probability of link lengths on a random lattice calculated withκ=1.3.The a in abscisses stands for one lattice spacing.The curve is normalized to1.Figure9.Evolution of the measured topological charge along with50cooling steps for40un-correlated configurations.Notice the clustering towards integer values after a fewcoolings.A random lattice with402sites andκ=∞was used atβ=1.4.Figure10.Monte Carlo data for the mass gap on a lattice with2002sites.The lines are the results of thefits performed on the Monte Carlo data(shown with circles,squaresand diamonds).The solid line(circles),dashed line(squares)and dot dashed line。

六西格玛出自 MBA智库百科()六西格玛(Six Sigma)六西格玛管理法简介六西格玛(6σ)概念于1986年由摩托罗拉公司的比尔·史密斯提出,此概念属于品质管理范畴,西格玛(Σ,σ)是希腊字母,这是统计学里的一个单位,表示与平均值的标准偏差。

旨在生产过程中降低产品及流程的缺陷次数,防止产品变异,提升品质。

六西格玛的由来六西格玛（Six Sigma）是在九十年代中期开始被GE从一种全面质量管理方法演变成为一个高度有效的企业流程设计、改善和优化的技术，并提供了一系列同等地适用于设计、生产和服务的新产品开发工具。

继而与GE的全球化、服务化、电子商务等战略齐头并进，成为全世界上追求管理卓越性的企业最为重要的战略举措。

六西格玛逐步发展成为以顾客为主体来确定企业战略目标和产品开发设计的标尺，追求持续进步的一种管理哲学。

20世纪90年代发展起来的6σ(西格玛)管理是在总结了全面质量管理的成功经验,提炼了其中流程管理技巧的精华和最行之有效的方法,成为一种提高企业业绩与竞争力的管理模式。

该管理法在摩托罗拉、通用、戴尔、惠普、西门子、索尼、东芝行众多跨国企业的实践证明是卓有成效的。

为此,国内一些部门和机构在国内企业大力推6σ管理工作,引导企业开展6σ管理。

源于摩托罗拉的6 sigma系统成为质量管理学发展的里程碑之一。

6 sigma系统由针对制造环节的改进逐步扩大到对几乎所有商业流程的再造，从家电Whirlpool, GE, LG，电脑Dell，物流DHL，化工Dow Chemical, DuPont，制药Agilent, GSK，通信Vodafone, Korea Tel，金融BoA, Merrill Lynch, HSBC，到美国陆海空三军，都引进6 sigma系统。

6σ管理法的概念6σ管理法是一种统计评估法,核心是追求零缺陷生产,防范产品责任风险,降低成本,提高生产率和市场占有率,提高顾客满意度和忠诚度。

Quantization of the O（N） Nonlinear Sigma Model S.I.Muslih【期刊名称】《理论物理通讯：英文版》【年(卷),期】2002(037)005【摘要】The Hamilton-Jacobi method of quantizing singular systems is discussed.The equations of motion are obtained as total differential equations in many variables.It is shown that if the system is integrable,one can obtain the canonical phase space coordinates and set of canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields.As an example we quantize the O(2) nonlinear sigma model using two different approaches:the functional Schrodinger method to obtain the wave functionals for the ground and the exited state and then we quantize the same model using the canonical path integral quantization as an integration over the canonical phase-space coordinates.【总页数】4页(P567-570)【关键词】量子力学;O（N）非线性Sigma模型;量子化【作者】S.I.Muslih【作者单位】DepartmentofPhysics,Al－AzharUniversity,Gaza,Palestine【正文语种】中文【中图分类】O413.1因版权原因，仅展示原文概要，查看原文内容请购买。

数学专业英语词汇(S)s admissible s容许的s matrix s矩阵saddle point 鞍点saddle point method 鞍点法 salient angle 凸角salient point 折点saltusfunction 跳跃函数 sample 样本sample correlation coefficient 样本相关系数sample correlation matrix 样本相关阵sample covariance 样本协方差 sample covariance matrix 样本协方差阵sample dispersion 样本方差 sample function 样本函数 sample mean 样本均值sample median 样本中位数 sample moment 样本矩sample point 样本点sample quartile 样本四分位数 sample size 样本的大小 sample space 样本空间sample survey 样本甸sample unit 抽样单位sample variable 样本变量 sample variance 样本方差 sampling 抽样sampling distribution 样本分布 sampling error 抽样误差 sampling fraction 抽样比 sampling inspection 抽样检查 sampling method 抽样法sampling moment 样本矩 sampling ratio 抽样比sampling survey 样本甸 sampling unit 抽样单位 sarrus rule 萨律法satisfiable 可满足的satisfy 满足saturated set 浸润集saturated subset 浸润子集saturated vertex 饱和顶点 saturation curve 饱和曲线 scalar 纯量scalar curvature 纯量曲率 scalar density 纯量密度scalar field 纯量场scalar flow of vector field 向量场的纯量流scalar matrix 纯量阵scalar multiplication 纯量乘法 scalar product 纯量积scalar quantity 纯量scalar triple product 纯量三重积 scalar valued function 纯量值函数scalar valued map 纯量值映射 scale 图度scale factor 标度因子scale mark 分核度scale parameter 尺度参数scale transformation 标度变换法 scalene triangle 不规则三角形scalenohedron 偏三角面体scalenous triangle 不规则三角形 scatter 散布scatter diagram 散布图scattered set 无核集scattergram 点状图scattering theory 散射理论 schauder theorem 肖德不动点定理scheduling and production planning 生产计划理论scheduling problem 日程计划问题 schematic diagram 简图scheme 模式schlicht 单叶的schlicht domain 单叶域schlicht function 单叶函数 schlicht mapping 单叶映射 schlicht surface 单叶曲面 schmidt orthogonalization 施密特正交化法schwartz space 施瓦尔兹空间 schwarz formula 施瓦尔兹公式 schwarz inequality 施瓦尔兹不等式 scope 辖域screw 螺旋体search process 搜她程secant 正割secant curve 正割曲线second axiom of countability 第二可数公理 second boundary condition 诺伊曼边界条件 second boundary value problem 诺伊曼问题 second comparison test 第二比较检验 second component 第二分量second countability axiom 第二可数公理 second derivative 二次导数second factor 第二因子second fundamental form 第二基本形式 second limit theorem 第二极限定理 second variation 第二变分secondary diagonal 次对角线secondary extremal 配连极值secondary obstruction 第二障碍section 截口section functor 截面函子section graph 部分等距算子sectional area 截面积sectional curvature 截面曲率sector 扇形sectorial harmonic 扇低函数secular equation 长期方程sedenion 十六元数segment 线段segment of a circle 弓形segment of a curve 弧段segmentation 分割selection 选择self adjoint boundary value problem 自伴边值问题self adjoint differential equation 自伴微分方程self adjoint eigenvalue problem 自伴特盏问题 self adjoint element 自伴元self adjoint linear mapping 自伴线性映射 self adjointness 自伴性self checking code 自校验代吗self complementary graph 自补图 self conjugate 自共轭的self conjugate partition 自共轭分拆 self dual 自对偶的self dual category 自对偶范畴self dual group 自对偶群self intersection number 自交数self loop 自身环self orthogonal submodule 自成正交子模 self polar curve 自配极曲线self polar tetrahedron 自配极四面形 self polar triangle 自配极三角形self tangency 自切self weighting sample 自加权样本 selfadjoint linear subspace 自伴线性子空间 selfadjoint operator 自伴算子selfconjugate latin square 自共轭拉丁方 selfosculating point of curve 曲线的自密切点 semantic equivalence 语义等价semantical decision problem 语义判定问题 semantical paradox 语义悖论semantics 语义学semi exact 半正合的semi interquartile 半内四分位数间距 semi invariant 半不变式semi invariant exterior derivative 半不变外微分 semi logarithmic representation 半对数表示 semiadditive category 半加性范畴 semianalytic set 半解析集合semiangle 半角semiaxis 半轴semibounded 半有界的semibounded operator 半有界算子semicircle 半圆semicircular 半圆的semicircular domain 半圆域semicircular protractor 量角器分度规 semicontinuity 半连续性semicontinuous 半连续的semicontinuous function 半连续函数 semicontinuum 半连续统semiconvergent series 半收敛级数 semicubical parabola 半三次抛物线semidefinite eigenvalue problem 半定特盏问题 semidefinite kernel 半定核semidefinite operator 半定算子semidefinite quadratic form 半定二次形式 semidefinite variational problem 半定变分问题 semidiameter 半径semidirect product 半直积semidiscretization 半离散化semifinite 半有限的semifinite trace 半有限迹 semigroup 半群semigroup algebra 半群代数 semigroup of operators 算子半群semihereditary ring 半遗传环 semilinear 半线性的semilinear mapping 半线性映射 semilinear substitution 半线性代换semilinear transformation 半线性变换 semilocal ring 半局部环semilogarithmic diagram 半对数图 semilogarithmic paper 半对数坐标纸semimagic square 半幻方semimajor axis 半长轴semimajorant 半强函数semimartingale 半semimean axis 半中轴semimetric 半度量semiminor axis 半短轴semiminorant 半弱函数semimodular lattice 半模格 semimodularity 半模性seminorm 半范数semiorder 半有序semiordered banach space 半有序巴拿赫空间semiordered set 半有序集 semipath 半通路semiperiod 半周期semipolar set 半极集semiprimary ring 半准素环 semiprime ideal 半素理想 semiprimitive ring 半本原环 semiquaternion 半四元数semireductive 半可简约的 semireflexive 半自反的semireflexive space 半自反空间 semireflexivity 半自反性 semiregular point 半正则点 semiregular space 半正则空间 semiregular topology 半正则拓扑 semiscalar product 半纯量积 semisimple algebra 半单代数 semisimple group 半单群semisimple module 半单模semisimple representation 半单表示 semisimple ring 半单环semisimplicial complex 半单纯复形 semisimplicial map 半单纯映射semisphere 半球semispherical 半球的semitangent 半切线semitransverse axis 半贯轴 semiuniform space 半一致空间 sense of rotation 旋转指向 senseclass 指向类sensepreserving mapping 保向映射 sensitivity analysis 灵敏度分析sentence 命题sentential calculus 命题演算 sentential connective 命题联结词sentential function 谓词separability 可分离性separable closure 可分闭包 separable degree 分离度separable element 可分元separable extension 可分扩张 separable field 可分域separable game 可分对策separable graph 可分图separable polynomial 可分多项式 separable sets 可分集separable space 可分空间separable stochastic process 可分随机过程separable topological space 可分拓扑空间separable transcendental extension 可分超越扩张separate 分离separated equation 分离变数方程 separated sets 隔离集separated space 分离空间separating edge 分离棱separating plane 分离平面separating transcendence basis 可分超越基separation 分离separation axiom 分离公理separation of the zeros 零点分离 separation of variables 分离变量separation principle 分离原理separation relation 分离关系 separation theorem 分离定理separator 分隔符sequence 序列sequence convergent almost everywhere 几乎处处收敛列sequence of arcs 弧序列sequence of complex numbers 复数序列 sequence of differences 差分序列 sequence of distinct points 一一序列 sequence of functions 函数序列sequence of iterations 迭代序列 sequence of numbers 数列sequence of partial sums 部分和列 sequence of points 点序列sequence of sets 集序列sequence of signs 符号序列sequence space 序列空间sequencing 排序sequencing problem 日程计划问题 sequent 串联的sequential analysis 序贯分析 sequential compactness 列紧性sequential control 顺序控制sequential estimation 序列估计 sequential likelihood ratio test 序贯似然比值检验sequential programming 顺序规划 sequential sampling 序贯抽样sequential sampling plan 序列抽样法 sequential switching circuit 依次转接电路 sequential test 序贯检定sequential word function 序贯字函数 sequentially compact set 列紧集sequentially complete space 序列完备空间 serial correlation 自相关serial correlation coefficient 序列相关系数series 级数series development 级数展开series expansion 级数展开series of functions 函数级数 series solution 级数解serpentine 蛇形线service model 服务模型sesquilinear form 半双线性形式 set 集set algebra 集代数set function 集函数set of condensation points 凝聚点集 set of continuum power 连续统势的集 set of limit 极限集合set of lower bounds 下界集 set of measure zero 零测度集 set of numbers 数集set of points 点集set of sets 集的集set of strategies 策略集set of ternary numbers 三进制数集 set of upper bounds 上界集 set theoretic addition 集论的加法 set theoretic image 集合论的象 set theoretic intersection 集论的交 set theoretic operation 集论的运算 set theoretic proof 集论的证 set theoretic union 集论的并集 set theoretical 集论的set theory 集论set topology 集论拓扑set valued functor 集值函子 set valued mapping 集值映射 sexadecimal digit 十六进制数字 sexadecimal notation 十六进记法 sexadecimal number system 十六进制数系 sexagesimal arithmetic 六十进算术 sexagesimal system 六十进制 sextant 六分仪sextic 六次曲线sextic equation 六次方程shadow 影shadow price 影子价格shape 形状shape theory 形状理论sheaf 层sheaf homomorphism 层同态sheaf of germs of continuous functions 连续函数的芽层sheaf of planes 平面束sheaf theoretic 层理论的shear 剪切shearing modulus 刚性模量shearing strain 切应变shearing stress 切应力sheet 叶sheets of rieman surface 黎曼曲面的叶 shell 壳层shift 移动shift operator 位移算子shift register 移位寄存器shilov boundary 希洛夫边界short division 短除法shorten 缩短shortening 缩短shortest 最短线shortest confidence interval 最短置信区间 shortest distance 最短距离shortest path 最短道路shortest path problem 最短道路问题 shortest route 最短道路shortest route problem 最短道路问题 side 边side condition 边条件side elevation 侧视图sieve 筛sieve method 筛法sieve of eratosthenes 厄拉多塞筛 sigma additivity 可列可加性 sigma algebra 代数sigma compact space 紧空间sigma compactness 紧性sigma complete filter 完备滤子 sigma complete lattice 完全铬 sigma complete lower semilattice 完备下半格sigma complete semilattice 完备半格 sigma discrete family of subsets 子集的离散族sigma finite measure 有限测度 sigma function 函数sigma locally finite family of subsets 子集的局部有限族sigma monogenic function 单演函数 sigma space 空间sigmacompleteness 完备性sigmafield of sets 集的域sigmalattice 格sigmoid s形曲线sign 符号sign digit 符号数字sign of equality 等号sign of inclusion 包含记号 sign of inequality 不等号 sign of intersection 相交记号 sign of multiplication 乘号 sign of permutation 置换的符号 sign of subtraction 减法 sign of summation 连加号 sign of the membership relation 从属关系记号sign of union 并号signal 信号signature of permutation 置换的符号 signed numbers 带符号数signed rank test 符号秩检验 signed tree 指定符号的树 significance level 显著性水平 significance of a deviation 偏差的显著性significance test 显著性检定 significant 有效的significant digit 有效数字 significant figure 有效数字 signum 正负号函数similar figures 相似形similar function 相似函数 similar matrix 相似矩阵similar ordered set 相似有序集 similar region 相似域similar test 相似检验similar triangles 相似三角形 similarity 相似similarity principle 相似性原理 similarity theorem 相似性定理similarity transformation 相似变换 similitude 相似similitude transformation 相似变换 simple 单的simple abelian variety 单阿贝耳簇 simple algebra 单代数simple arc 简单弧simple branch point 单分枝点 simple broken line 单折曲线 simple chain 简单链simple character 简单特贞 simple circuit 简单围道simple closed curve 简单闭曲线 simple component 单分量simple compression 单压缩 simple connectedness 单连通性 simple connectivity 单连通性 simple continued fraction 正则连分数 simple continued fraction expansion 简单连分数展开simple convergence 点态收敛 simple correlation coefficient 单相关系数simple cycle 简单循环simple domain 单叶域simple eigenvalue 简单特盏 simple elongation 单伸长simple event 简单事件simple extension 单扩张simple extension field 单扩张域 simple fixed point 单纯不动点 simple fraction 普通分数simple function 单叶函数simple graph 简单图simple group 单群simple harmonic motion 简谐运动 simple hypothesis 简单假设 simple integral 单积分simple intersection point 单纯交点 simple iterative method 单迭代法simple lattice 单格simple lie algebra 单李代数 simple module 单模simple object 简单对象simple path 简单道路simple pendulum 单摆simple point 单点simple polygon 简单多边形 simple polyhedron 简单多面体 simple product 简单积simple proper value 简单特盏 simple quadrilateral 简单四边形 simple random sampling 简单随机样本 simple regression 简单回归 simple regression coefficient 单回归系数simple ring 单环simple root 单根simple sample 简单样本simple sampling 简单抽样simple series 简单级数simple set 单集simple spectrum 单谱simple surface 简单曲面simple tangent 单切线simple theory of types 简单类型论 simple transcendental extension 单超越扩张 simplex 单形simplex method 单形法simplex multiplier 单形乘数simplex tableau 单形表simplex theorem 单形定理simplicial approximation 单纯逼近 simplicial cell 单纯胞腔simplicial chainmapping 单纯链映射 simplicial cochain complex 单纯上链复形 simplicial cohomology group 单纯上同岛 simplicial complex 单纯复形simplicial homology 单纯同调simplicial map 单形映射simplicial mapping cylinder 单形映射柱 simplicial pair 单形对simplicialapproximation theorem 单纯逼近定理 simplification 简化simplified fraction 简化分数simplified newton method 简化牛顿法 simply connected group 单连通群simply connected region 单连通区域 simply connected spatial domain 单连通空间域 simply convergent filter 单收敛滤子 simply ordered group 全有序群simply periodic function 单周期函数 simplyconnected domain 单连通域simpson rule 辛卜生法则simulation 模拟simultaneity 同时性simultaneous confidence intervals 联合置信区间 simultaneous diagonalization 同时对角化 simultaneous differential equation 联立微分方程 simultaneous differential equations 联立微分方程simultaneous equations 方程组simultaneous estimation 联立估计 simultaneous invariant 联立不变式simultaneous substitution 同时代入 sine 正弦函数sine curve 正弦曲线sine function 正弦函数sine integral 正弦积分sine integral function 正弦积分函数 sine law 正弦定律sine spiral 正弦螺线sine theorem 正弦定理sine wave 正弦波single 单的single address 单地址的single address code 一地址代码 single address instruction 单地址指令single address system 单地址系统 single factor method 单因子法 single step method 单步法single step process 单步法single valued 单值的single valued analytic function 单值解析函数single valued correspondence 单值对应 single valued function 单值函数 single valued operation 单值运算 single valued relation 单值关系single valuedness 单值性singlevalued mapping 单值映射 singly periodic function 单周期函数singular 奇异的singular automorphism 奇异自同构 singular bivariate normal distribution 奇异二元正态分布singular boundary 奇异边界singular boundary point 奇异边界点 singular chain 奇异链singular chain complex 奇异链复形 singular cohomology group 奇异上同岛 singular complex 奇异复形singular conic 奇二次曲线singular correspondence 奇对应 singular cycle 连续循环singular distribution 退化分布 singular element 奇元素singular elliptic function 奇异椭圆函数 singular function 奇异函数singular function of bounded variation 有界变差奇异函数singular graph 奇异图singular homology 奇异下同调 singular homology class 奇异同掂singular homology group 连续同岛 singular integral 奇解singular integral element 奇异积分元素 singular integral equation 奇异积分方程singular kernel 奇核singular line element 奇异线素 singular linear operator 奇异线性算子singular linear transformation 奇异线性变换singular locus 奇轨迹singular mapping 奇异映射singular matrix 退化阵singular operator 奇异算子 singular ordinal 特异序数singular part 奇异部分singular plane 奇异平面singular point 奇点singular proposition 特称命题 singular quadric 奇异二次曲面 singular series 奇异级数singular solution 奇解singular space 奇异空间singular submodule 奇子模singular subspace 奇异子空间 singular surface 奇曲面singular transformation 奇异变换 singular value 奇异值singular variational problem 奇异变分问题singular vector 奇异向量singularity 奇点sink 收点sinusoid 正弦摆线sinusoidal 正弦的sinusoidal function 正弦函数 sinusoidal law 正弦定律sinusoidal spiral 正弦螺线 situation 情况size 样本的大小skeleton 骨架skew curve 空间曲线skew derivation 斜微分skew determinant 斜对称行列式skew distribution 偏斜分布 skew field 非交换域skew hermitian form 斜埃尔米德型 skew hermitian matrix 斜埃尔米德矩阵 skew lines 偏斜线skew position 歪扭位置skew quadrilateral 挠四边形 skew surface 非可展直纹曲面 skew symmetric 反对称的skew symmetric determinant 斜对称行列式skew symmetric matrix 斜对称矩阵 skew symmetric tensor 斜对称张量slack variable 松弛变量slide rule 计算尺sliding vector 滑动向量slitregion 裂纹区域slope 斜率slope function 斜率函数slope intercept form 斜截式 slope line 倾斜线slope of a curve 曲线的斜率 slowly increasing sequence 缓增序列small circle 小圆small inductive dimension 小归纳维数 small sample 小样本small set 小集smallest element 最小元smash 收缩smooth curve 平滑曲线smooth map 光滑映射smooth morphism 光滑射smooth projective plane curve 光滑射影平面曲线smoothing 光滑化smoothness 光滑度sobolev embedding theorem 水列夫嵌入定理sobolev space 水列夫空间sojourn time 逗留时间solenoidal group 螺线群solenoidal vector field 螺线向量场 solid 立体;固体solid angle 立体角solid geometry 立体几何solid n sphere n维球体solid of revolution 旋转体soliton 孤立子soluble 可解的solution 解solution curve 积分曲线solution domain 解域solution formula 解公式solution set of equation 方程的解集 solution space 解空间solution surface 积分曲面 solution tree 解树解答树solution vector 解向量solvability 可解性solvable 可解的solvable equation 可解方程 solvable group 可解群solvable ideal 可解理想solve 解sorter 分类器source 发点source free vector field 无源向量场 source function 格林函数source of field 场源source program 源程序space 空间space coordinates 空间坐标 space curve 空间曲线space integral 体积积分space like manifold 类空廖 space of left cosets 左傍系空间 space of matrices 矩阵空间 space of quaternions 四元数空间 space of right cosets 右傍系空间 space quadratic transformation 空间二次变换space region 空间区域span 生成spanning tree 最大树生成树 sparse matrix 稀巯阵spatial 空间的spatial co ordinate 空间坐标 spatial isomorphism 空间同构 spatial point 空间点special divisor class 特殊除子类 special functional equations 特殊函数方程special functions 特殊函数special group 特殊群special homology manifold 特殊同滴 special jordan algebra 特殊约当代数 special linear group 特殊线性群 special linear homogeneous group 特殊线性齐次群special orthogonal group 特殊正交群 special purpose computer 专用计算机 special representation 特殊表示 special unitarian group 特殊酉群special valuation 特殊赋值specialization 特定化specific 特殊的specific address 绝对地址specific heat 比热specificity 特性spectral analysis of operators 算子的谱分析spectral decomposition 谱表示 spectral density 谱线密度spectral distribution 谱分布 spectral distribution curve 光谱分布曲线 spectral function 谱函数spectral functor 谱函子spectral geometry 谱几何spectral integral 谱积分spectral invariant 谱不变量 spectral line 谱线spectral mapping theorem 谱映射定理 spectral measure 谱测度spectral multiplicity 谱重度 spectral norm 谱模spectral point 谱点spectral property 谱性质spectral radius 谱半径spectral representation 谱表示 spectral sequence 谱序列spectral set 谱集spectral space 谱空间spectral subspace 谱子空间spectral synthesis 谱综合spectral theorem 谱定理spectrum 谱spectrum of a matrix 阵的谱 speed 速度sphere 球sphere bundle 球丛sphere of contact 相切球面sphere of the inversion 反演球 spherical 球形的spherical angle 球面角spherical astronomy 球面天文学 spherical asymptote 球面渐近线spherical bessel function 球贝塞耳函数 spherical cap 球冠spherical cohomology class 球上同掂 spherical coordinates 球极坐标spherical curvature 球面曲率spherical curve 球面曲线spherical cyclic curve 球面循环曲线 spherical derivative 球面导数spherical domain 球面域spherical epicycloid 球面外摆线 spherical excess 球面角盈spherical fiber 球面纤维spherical function 球函数spherical geometry 球面几何学 spherical harmonic function 球面低函数spherical harmonics 球面低函数 spherical helix 球面螺旋线spherical homology class 球面同掂 spherical image 球面象spherical indicatrix 球面指标 spherical indicatrix of binormal 副法线球面指标spherical indicatrix of principal normal 吱线球面指标spherical indicatrix of tangents 切线球面指标spherical mean 球中值spherical neighborhood 球形邻域 spherical parallelogram 球面平行四边形 spherical polar coordinates 球极坐标 spherical polygon 球面多边形spherical pyramid 球面棱锥spherical sector 球心角体spherical segment 球截形spherical shell 球壳spherical space 球面空间spherical surface 球面spherical triangle 球面三角形 spherical triangular coordinates 球面三角坐标spherical trigonometry 球面三角学spherical zone 球带spherics 球面几何学sphero quartic 球面四次曲线 spheroid 回转椭圆面spheroidal coordinates 球体坐标 spheroidal function 球体低函数spheroidal harmonic 球体低函数 spheroidal wave function 球体波函数spinode 第一类尖点spinor 旋子spinor field 旋量场spinor genus 旋量狂spinor group 旋子群spinor representation 旋量表示 spiral 螺线spiral point 螺线极点spiral surface 螺面spline 样条spline function 样条函数 spline interpolation 样条内插 splitting 分裂splittingfield 分裂域spur 迹squarable 可平方的square 正方形square bracket 方括弧square contingency 平方列联 square deviation 平方偏差 square matrix 方阵square measure 平方测度square mesh 正方网格square root 平方根square root transformation 平方根变换square summable function 平方可积函数squareintegrable function 平方可积函数stability 稳定性stability conditions 稳定条件 stability criterion 稳定性判据stability group 稳定群stability number 稳定数stability region 稳定区stabilization 稳定stabilization method 稳定法 stabilizer 迷向群stable convergence 稳定收敛 stable equilibrium 稳定平衡 stable homotopy group 稳定同伦群 stable manifold 稳定廖stable orbit 稳定轨道stable point 稳定点stable process 稳定过程 stable set 稳定集合stable solution 稳定解stable state 稳定状态stalk 茎standard complex 标准复形 standard deviation 标准差 standard equation 标准方程 standard form 标准型standard isobaric surfaces 标准等压面standard n simplex 标准n单形 standard regression coefficient 标准回归系数standard simplex 标准单形 standard topology 标准拓扑 standardization 标准化standardize 使标准化standardized normal distribution 标准化正态分布standardized normal variate 标准化正态变量standardized variable 标准化变量 standardized variate 标准化变量star body 星形体star covering 星形覆盖star neighborhood 星形邻域 star of a simplex 单形的星形 star region 拟星形域star shaped domain 拟星形域 starlike domain 拟星形域 starlike mapping 拟星形映射 starlike set 拟星集starrefinement 星型加细 start time 开始时间state 状态state coordinates 状态坐标 state function 状态函数 state of equilibrium 平衡状态 state region 状态区域state space 状态空间state variable 状态变数state vector 状态向量statement 命题static model 静态模型statics 静力学stationary 平稳的stationary curve 平稳曲线stationary distribution 平稳分布 stationary flow 平稳流定常流stationary function 平稳函数stationary osculating plane 平稳密切面 stationary point 平稳点stationary point process 平稳点过程 stationary process 平稳随机过程stationary state 定态stationary time series 平稳时间序列 stationary value 平稳值statistic 统计量statistical 统计的statistical accuracy 统计精确度 statistical analysis 统计分析statistical data analysis 统计数据分析 statistical decision function 统计判决函数 statistical decision problem 统计判决问题 statistical decision procedure 统计判决程序 statistical decision process 统计判决过程 statistical distribution 统计分布 statistical ergodic theorem 平均遍历定理 statistical error 统计误差statistical estimate 估计statistical estimation 统计估计 statistical hypothesis 统计假设statistical hypothesis testing 统计假设检验 statistical inference 统计推断statistical mechanics 统计力学statistical method 平均法statistical model 随机性模型statistical optimization 统计最佳化 statistical quality control 统计质量管理 statistical thermodynamics 统计热力学 statistics 统计学statistics of extreme values 极值统计 statistics of extremes 极值统计 steady 平稳的steady state 定态steenrod algebra 斯丁洛特代数steenrod operation 斯丁洛特运算 steepest descent method 最速下降法steering program 痔序steering routine 痔序steinberg group 斯坦因伯格群step 步长step function 阶梯函数step length 步长stepping stone method 起脚石法 steradian 球面度stereoangle 立体角stereogram 立体频数stereographic projection 球极平面射影 stereography 立体平画法stereometry 立体几何stiefel whitney class 斯蒂费尔惠特尼类 stieltjes integral 斯蒂尔吉斯积分 stirling formula 斯特林公式stochastic 随机的stochastic approximation 随机逼近 stochastic automaton 随机自动机stochastic connection 随机联络 stochastic control 随机控制stochastic dependence 随机相依 stochastic differential equation 随机微分方程stochastic differentiation 随机微分法 stochastic dynamic model 随机动态模型 stochastic dynamic programming 随机动态规划 stochastic filtering 随机滤波stochastic game 随机对策stochastic independence 随机独立 stochastic integral 随机积分stochastic integral equation 随机积分方程 stochastic integration 随机积分 stochastic matrix 随机阵stochastic maximum principle 随机极大原理 stochastic model 随机性模型stochastic optimization 随机规划法 stochastic process 随机过程stochastic programming 随机规划法 stochastic variable 随机变数stochastically dependent event 随机相依事件 stochastically independent event 随机独立事件stokes integral theorem 斯托克斯定理stokes theorem 斯托克斯定理 stop time 停止时间stopped process 停止过程 stopping rule 停止规则 storage 存储store 存储器store capacity 存储容量 store cell 存储单元straight 直的straight angle 平角straighten 弄平straightline 直线strain 应变strain tensor 应变张量 strategic equivalence 策略等价性 strategic model 策略模型 strategy 策略strategy of a game 对策的策略 strategy polygon 策略多角形stratification 层化stratified sample 分层样本 stratified sampling 分层抽样 stratified selection 分层抽样 stratum 层stream function 怜数streamline 吝strength 强度stress 应力stress ellipsoid 应力椭球 stress function 应力函数 stress of a body 体应力 stress tensor 应力张量 stretching transformation 伸缩变换strict convexity 严格凸性 strict decreasing 严格递减 strict epimorphism 严格满射 strict extremum 严格极值 strict implication 严格蕴涵 strict increasing 严格递增 strict inductive limit 严格归纳极限strict inequality 严格不等式 strict isotonicity 严格保序性 strict isotony 严格保序性strict minimum 严格极小strict morphism 严格射strict solution 严格解strict upper bound 严格上界strictly concave function 严格凹函数 strictly convex function 严格凸函数 strictly convex space 严格凸空间 strictly decreasing 严格减少的strictly dominant strategy 严格优策略 strictly finer topology 严格较细拓扑 strictly increasing 严格递增的strictly increasing mapping 严格递增映射 strictly lower triangular matrix 严格下三角矩阵 strictly monotone decreasing 严格单递减的 strictly monotone increasing 严格单递增的 strictly monotonic function 严格单弹数strictly monotonic mapping 严格单党射 strictly monotonic sequence 严格单凋列 strictly normed linear space 严格赋范线性空间 strictly positive measure 严格正测度 strictly upper triangular matrix 严格上三角矩阵 strip 带strip of conditional convergence 条件收敛带 strip region 带形区域strong component 强分支strong convergence 强收敛strong convergent operator 强收敛算子 strong deformation retract 强形变收缩核 strong discontinuity 强间断strong dual 强对偶strong ellipticity 强椭圆型strong epimorphism 强满射strong extremum 强相对极值strong inequality 严格不等式strong law of large numbers 强大数定律 strong markov process 强马尔可夫过程 strong monomorphism 强单射strong operator topology 强算子拓扑 strong solution 强解strong summability 强可和性strong topology 强拓扑strongly coercive 强强制的strongly connected compactum 强连通紧统 strongly connected graph 强连通图strongly continuous map 强连续映射 strongly elliptic operator 强椭圆算子 strongly inaccessible cardinal 强不可达基数 strongly inaccessible ordinal 强不可达序数 strongly mixing transformation 强混合变换 strongly paracompact space 强仿紧空间 strongly plurisubharmonic function 强多重次低函数strongly pseudoconvex domain 强伪凸域 strophoid 环诉structural morphism 结构射structural stability 构造稳定性 structure 结构structure constant 构造常数structure equations 结构方程structure formula 结构公式structure function 结构函数structure group 结构群structure morphism 结构射structure sheaf 结构层structure theorem 结构定理structured complex 结构复形student t distribution 学生t分布 sturm chain 斯图谟链sturm liouville eigenvalue problem 斯图谟刘维尔特盏问题subadditive function 次加性函数 subadditive functional 次加性泛函subadditive interval function 次加性区间函数 subadditive set function 次加性集函数 subadditivity 次可加性subalgebra 子代数subautomaton 子自动机subbase 子基subbundle 子丛subcategory 子范畴subchain 子链subclass 子类subcoalgebra 子上代数subcomplex 子复形subcontinuum 子连续统subcovering 子覆盖subdeterminant 子行列式subdifferential 次微分subdirect sum 次直和subdivide 细分subdivision 重分subdivision chain 剖分链 subdomain 子域subfamily 子族subfield 子域subformula 子公式subgradient 次梯度subgraph 子图subgroup 子群subgroupoid 子群化subharmonic 次低的subharmonic function 次低函数 subideal 子理想subinterval 子区间subject 质subjective probability 诸概率 sublattice 子格sublinear functional 次线性泛函 submanifold 子簇submartingale 半submatrix 子阵submersion 浸没submodel 子模型submodule 子模submultiple 因数subnet 子网subnormal 次法线subobject 子对象subordinate category 从属范畴 subordinate construction 从属构造subordinate partition of unity 从属单位分解subpolyhedron 子多面体subpresheaf 子预层subproduct 子积subprojective manifold 次射影廖 subquasigroup 子拟群subquotient of a module 模的子商 subreflexive 子反射的subregion 子域subrelation 子关系subrepresentation 子表示 subring 子环subroutine 子程序subsample 子样本subsampling 二段抽样subsampling unit 二段抽样单位 subscheme 子概型subscript 下标subsemigroup 子半群subsemiring 子半环subsequence 子序列subseries 子级数subset 子集。

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。

有限Abel群的结构定理（Fundamental Theorem ofFinite Abelian Groups）有限Abel群的结构定理(Fundamental Theorem of Finite Abelian Groups) 有限Abel群是群论中已被研究清楚了的重要群类，也是应用比较广泛的群类，本节的主要结论是有限Abel群可以分解成阶为素数的方幂的循环群(循环p-群)的直积，而且表法是唯一的。

我们先看几个具体的例子。

4阶群都是Abel群，它们有两种互不同构的类型，代表分别是。

Z,Z，Z422 ，其中是非Abel群;是Abel群，且6阶群有两种不同的类型，代表分别是ZZ,SS6633。

Z,Z，Z6238阶Abel群有三种不同的类型，代表分别是。

Z,Z，Z,Z，Z，Z8242229阶群都是Abel群，它们有两种互不同构的类型，代表分别是。

Z,Z，Z933 这些有限Abel群都同构于循环群或者循环群的直积，并且每个循环群的阶都是一个素数的方幂，这些循环群的阶组成的有重集合正好是该群阶素数方幂乘积的所有可能组合。

例如8阶32Abel群，有三种情形:，分别对应于8写成素数方幂乘积所有可能的形式{2},{2,2},{2,2,2}32(三种):。

8,2,8,2，2,8,2，2，2下面我们讨论一般有限Abel群的结构。

引理1 设a是群G的一个元素，a的阶等于。

其中与是两个互素的正整数，m,mmmm1212那么a可以唯一的表示成，式中的阶是;;而且都am(i,1,2)a(i,1,2)a,aaaa,aaii12i1221是a的方幂。

证明因为与互素，所以存在整数使得。

于是mmu,uum，um,112121122umumum，umumumumum2211112211222211，令，则，而且a,a,a,aa,aa,aaa,a,aa,aa121221mm12都是的方幂。

因为，所以的阶是的因子。

由于a(i,1,2)adm(i,1,2)ma,e,a,eaiiii112与互素，从而互素，并且，故的阶等于。