A Complete Analytic Inversion of Supernova Lines in the Sobolev Approximation

英汉语言差异之一综合语vs 分析语Synthetic vs AnalyticWebster’s Ninth New Collegiate Dictionary定义“A synthetic language is characterized by frequent and systematic use of inflected forms to express grammatical relationships.””An analytic language is characterized by a relatively frequent use of function words, auxiliary verbs, and changes in word order to express syntactic relations, rather than of inflected forms.综合语的特征是运用形态变化来表达语法关系。

分析语的特征是不用形态变化而用词序以及虚词来表达语法关系。

现代英语是从古代英语发展出来的，仍然保留着综合语的某些特征，但是也具有分析语的特点：有形态变化，但不象典型的综合语那么复杂；词序比汉语灵活，但相对固定；虚词很多，用的也相当频繁。

现代英语运用遗留下来的形态变化形式（hereditary inflections），相对固定的词序以及丰富的虚词来表达语法关系，因此属综合－分析语（synthetic－analytic language）。

形态变化，词序和虚词是表达语法意义的三大手段。

这些语法手段在英汉语言中都具有不同的特征。

此外，英语采用拼音文字，句有语调intonation；汉字绝大多数为形声字，字有声调tone。

英汉表达方法的差异与这些因素均有密切的关系。

英语（词语）有形态变化，汉语没有严格意义上的形态变化。

英语词序比较灵活，汉语词序相对固定。

英汉都有大量虚词，但是各有特点。

英汉语言对比Synthetic vs. AnalyticⅠ. Answer QuestionsDirections: Answer the Questions in English briefly, with examples if necessary. 1.What are synthetic languages(综合语)? How are they characterized?According to Webster’s Ninth New Collegiate Dictionary, A synthetic language is “Characterized by frequent and systematic use of inflected forms to express grammatical relationships”2.What are analytical languages（分析语）? How are they characterized?According to The Random House College Dictionary, An analytic language is “Characterized by a relatively frequent use of function words, auxiliary verbs and changes in words order to express syntactic relations, rather than of inflected forms”.3.Why English language is more flexible in word order（语序）? Why is it not so inChinese?Because word order is more connected with the formal changes. If the form change is more flexible, the word order is more flexible. Chinese language is an analytic language, the word order cannot be change at random, in which the relationship between the words are closely related with word order and function words.4.What is the usual word order in English sentences? What are the nine types ofinversion in English sentences?The usual word order in English sentence is Subject-Verb-Object. The nine types of inversion in English sentences include: interrogative inversion, imperative inversion, exclamatory inversion, hypothetical inversion, balance inversion, link inversion, signpost inversion, negative inversion, metrical inversion.1. 5.What are the E—C differences in formal words（形式词）?English language is featured in its abundant function words, of which is article. Articles are frequent employed. To use an article or not, and to use a definite one or an indefinite one, makes great difference in expression of meaning. Auxiliary are more frequently used in Chinese. Chinese language contains auxiliary for tense, structural and mood. This last category is especially unique in the language for the richness for delineation of emotions, like the letters 呀、啦、哦、嘛、呢，and others.6.How many kinds of formal words （形式词）are there in English language?2.There are five types of formal words, that is:i.article, including a, an, and the;ii.preposition, like on, in, to, for, as, at, of, with, from, across, etc.;iii.auxiliary verb, like will, do, have;iv.coordinator, including and, but, or;subordinator, like unless, after, for, due to, etc.7.Why Chinese is called Tone Language（声调语言）?Chinese tone can be used for indicating the change of meaning and is not suitable for expressing the certainty, completeness, independence or finality.Ⅱ. Fill in the BlanksDirections: Fill in the blanks with the best answer you have found in the textbook.1.11.综合语的特征是经常性、系统性的运用曲折形式来表现语法关系为特征的语言，分析语的特征是分析语的特征是相对频繁地使用功能词，助词和改变词序来表达句法关系，而不用曲折形式来表达。

a rXiv:h ep-th/067243v311A pr28hep-th/0607243DISTA-UPO-06DFTT-18/2006July,2006Abstract As been recently pointed out,physically relevant models derived from string theory re-quire the presence of non-vanishing form fluxes besides the usual geometrical constraints.In the case of NS-NS fluxes,the Generalized Complex Geometry encodes these informa-tions in a beautiful geometrical structure.On the other hand,the R-R fluxes call for supergeometry as the underlying mathematical framework.In this context,we analyze the possibility of constructing interesting supermanifolds recasting the geometrical data and RR fluxes.To characterize these supermanifolds we have been guided by the fact topological strings on supermanifolds require the super-Ricci flatness of the target space.This can be achieved by adding to a given bosonic manifold enough anticommuting co-ordinates and new constraints on the bosonic sub-manifold.We study these constraints at the linear and non-linear level for a pure geometrical setting and in the presence of p-form field strengths.We find that certain spaces admit several super-extensions and we give a parameterization in a simple case of d bosonic coordinates and two fermionic coordinates.In addition,we comment on the role of the RR field in the construction of the super-metric.We give several examples based on supergroup manifolds and coset supermanifolds.1IntroductionRecently the pure spinor formulation of superstrings[1,2]and the twistor string theory [3]have promoted supermanifolds as a fundamental tool to formulate the string theory and supersymmetric models.The main point is the existence of backgroundfields of extended supergravity theory(and therefore of10dimensional superstrings)which are the p-forms of the fermionic sector of superstrings.Thosefields can be coupled to worldsheet sigma modelsfields by considering the target space as a supermanifold.Indeed,as shown in[4,5]by adding to the bosonic space some anticommuting coordinatesθ’s,one can easily encode the informations regarding the geometry and the p-formfluxes into the supermetric of a supermanifold.In the present paper,we provide a preliminary analysis of the construction of supervarieties on a given bosonic space with or without p-forms (these are usually described by a bispinor Fµνsince they couple to target space spinors). We have to mention the interesting results found in[6,7,8,9]stimulated by the work [3,10].There the case of super-CY is analyzed and it is found that by limiting the number of fermions,the constraints on the bosonic submanifold of the supermanifold become very stringent.The super-CY spaces were introduced in string theory in paper[11].There the sigma models with supertarget spaces,their conformal invariance and the analysis of some topological rings were studied.In addition,we would like also to make a bridge with the recent successes of the Generalized Complex Geometry[12,13]to construct N=1 supersymmetric vacua of type II superstrings[14].In that context,theflux of the NS-NS background contributes to the generalized geometry and space is no longer CY.In the same way for a super-CY its bosonic subspace does not need to be a CY.The differential geometry of supermanifolds is a generalization of the usual geometry bosonic manifolds extended to anticommuting coordinates.There one can define super-vectorfields,superforms and the Cartan calculus.In addition,one can extend the usual Levi-Civita calculus to the fermionic counterparts defining super-Riemann,super-Ricci and super-Weyl tensorfields.A tensor has fermionic components and each component is a superfield.Due to the anticommuting coordinates,one defines also a metric on a supermanifold which can be generically given byds2=G(mn)dx m⊗dx n+G mµdx m⊗dθµ+G[µν]dθµ⊗dθν(1.1) where G(mn)(x,θ),G mµ(x,θ),G[µν](x,θ)are superfields and the tensor product⊗respects the parity of the differentials dx m,dθµ.The superfields are polynomials ofθ’s and the coef-ficients of their expansions are ordinary bosonic and fermionicfields.For example the ex-(x) pansion of G mn(x,θ)=G(0)mn(x)+...has a direct physical interpretation:G(0)mn=g(body)mn where the latter is the metric of the bosonic submanifold and the higher components are fixed by requiring the super-Ricciflatness.The physical interpretation of Gµm and Gµνis more interesting.Indeed,as suggested by the pure spinor string theory[1,15,16]the first components of Gµνcan be identified with a combination of RRfield strengths(in the text,this point will be clarified further).Thefirst component of Gµm arefixed by consistency and,in a suitable gauge,it coincides with a Dirac matrix.This is not the only way in which the RRfields determine the geometrical structure of the supermanifold.We learnt from recent analysis[17,18,19,20]in superstrings and, consequently in supersymmetricfield theory,that the superspace can be deformed by the existence of RRfield strength Fµν(x,θ){θµ,θν}=Fµν.(1.2) This has been only verified for constant RRfield strengths.Notice that a given bispinor Fµνis decomposed into a symmetric F(µν)and antisym-metric part F[µν].Therefore,it is natural to identify the antisymmetric part with the fermion-fermion(FF)components G[µν](x)of the supermetric(1.1)and the symmetric one with the non-vanishing r.h.s.of(1.2).This is very satisfactory since it provides a complete mapping between the superstring backgroundfields and the deformations of the superspace:the non-commutativity and the non-flatness.Indeed,the presence of a superfield Gµν(x,θ)in the supermetric modifies the supercurvature of the superspace. So,finally we can summarize the situation in the following way:given the backgrounds g mn and the NSNS b mn they are responsible for the deformation of the bosonic metric and of the commutation relations between bosonic coordinates[x m,x n]=θ[mn](where θmn=(b−1)mn).On the other side,the RRfields deforms the supermetric and the anti-commutation relations.There is another important aspect to be consider.The anticommuting deformation of superspace is treatable whether thefield strength Fµνof the RRfield is constant and,in four dimensions[19],whether is self-dual.In that case,there is no back-reaction of the metric and it is a solution of the supergravity equations.In the case of antisymmetric RR fields,which deform the supermetric(1.1),we have to impose a new condition in order that the worldsheet sigma model is conformal and therefore treatable.We require that the supermanifold is super-RicciflatR MN=0.(1.3) In this way,we can still use some of the conventional technique of conformalfield theory and of topological strings to study such models.The main point is that even if there is back-reaction,this is under control since the total superspace is super-Ricciflat[4,5,21, 22,23].We would like to remind the reader that the super-Ricciflatness is the condition which guarantees the absence of anomalies in the case of B-model topological strings on supermanifolds[24].This is equivalent to the well-known condition for conventional B-model on bosonic target spaces.The latter have to be Calabi-Yau which are Ricci-flat K¨a hler spaces.A main difference is that the CY must be a3-fold in order to compensate the ghost anomaly.In the case of supermanifold this is given by a virtual dimension which is essentially the difference between the bosonic and fermionic dimensions[25].In the A-model,the CY conditions is required for open string in presence of D-branes(see[24]for a complete discussion).1We start with a metric on a given space,for example the round sphere metric for S n, or Fubini-Study metric for CP n.Then we construct the corresponding supermanifold by adding anticommuting coordinates and additional components of the metric(in some cases,the extension of the metric to a super-metric is not enough and some non-vanishing torsion components are necessary)and we require the super-Ricciflatness.As a conse-quence,we notice that thefirst coefficient of Gµνmust be different from ly,we need a non-vanishing component of the fermion-fermion part of the metric.However,in general for a given space,no invariant spinorial tensor exists to provide such component. Therefore,we argue that this should be identified with p-formfluxes needed to sustain the solution of the supergravity.As a matter of fact,super-Ricciflatness does not seems to relate the value of theflux with the geometry of the bosonic submanifold.In fact,we will see in the forthcoming sections,the value of theflux isfixed by a suitable normalization which is implemented by a gauge condition on the supermetric.So,we conclude that super Ricciflatness turns out to be a necessary ad useful condition,but it does not seem to be sufficient to characterize the vacuum of the theory.We have to divide the analysis in two parts:thefirst one is the analysis of linearized equations and the second one concerns the analysis of the non-linear theory.It turns out that for the linear equation one gets very strong constraints on the bosonic metric such as the scalar curvature must be zero(see also[6,7,9]).This is compatible with theflat or Ricciflat space,but certainly is not true for a sphere or any Einstein manifold.Therefore, it is unavoidable to tackle the non-linear analysis to see whether these constraints can be weakened.Let us explain briefly the lines of analysis.The Ricci-flatness condition(1.3) can be analyzed by expanding the super-metric in components.Since we are interested only in the vacuum solutions,we decompose the superfields of the super-metric into bosonic components by setting to zero all fermionic ones.This implies a set of equations for each single component.Among them,a subset of these equations can be easily solved; indeed,the equations for the components to the order n are solved in terms of some components at the n+1order.However,the equations for highest-components cannot be solved algebraically and in fact,by plugging the solutions of the lower orders in these equations,onefinds high-derivative equations for the lowest components.To show that our analysis is not purely academic,we give some examples of super-manifolds modeled on some bosonic submanifold.Of course,the most famous example is P SU(2,2|4)(from AdS/CFT correspondence)and SL(4|4)(appearing in twistor string theory).Moreover,we give examples based on S n,V(p,q)and CP n.As a by-product we construct a super-Hopffibration connecting the different types of supercoset manifolds. One interesting example is the construction of a supermanifold with T(1,1)as a bosonic submanifold.In addition,we propose a construction which can be applied to any Einstein spaces,and in particular to Sasaki-Einstein space and K¨a hler-Einstein.At the momentwe have not explored the role of Killing spinors,Killing vectors and the role of super-symmetry in the construction.We only argued that if the supermanifold is obtained as a single coset space and not a direct product of coset space,it has more chances to have an enhanced supersymmetry.The relation between supersymmetry and the construction of super-Ricciflat supermanifold will be explored in forthcoming papers.More important,it is not yet established the relation between the solution of supergravity and the super-Ricci flatness.The paper is organized as follows:in sec.2we recall some basic fact about super-geometry..In sec.3,we present a general strategy,we study the linearized version of the constraints and,in the case of d bosonic coordinates and2fermionic directions, we analyze completely the non-linear system.In sec.4,we study the construction of supercosets modeled on bosonic cosets.In sec.5,we add thefluxes(RRfield strengths). Appendices A and B contain some related material.2Elements of supergeometryWe adopt the definition of supermanifold given in[30]based on a superalgebra with the coarse topology.However,this is not the only way and there is an equivalent formulation based on sheaves andflag manifolds[31,32].The equivalence turns out to be useful for studying topological strings on supermanifolds[33].2.1SupermanifoldsA supermanifold M(m|n)is called Riemannian if it is endowed with a metric supertensor which is a real non-singular commuting tensorfield G MN satisfying the following graded-symmetry condition:G MN=(−1)MN G NM.(2.1) The meaning of(2.1)becomes more evident if one distinguishes between bosonic and fermionic components and rewrites the metric supertensor as a block supermatrix GG MN= g mn jµν (2.2)being g an m×m real symmetric commuting matrix,j an n×n imaginary antisymmetric commuting matrix and h an m×n imaginary anticommuting one.Obviously thefirst component of the superfield of the even-even block g is an ordinary metric for the body M(m)of the supermanifold M(m|n).Body means the bosonic submanifold.Note that the supermetric G is preserved by the orthosymplectic supergroup OSp(m|n) whose bosonic submanifold is the direct product SO(m)×Sp(n)of an orthogonal group encompassing the Lorentz transformations of the metric g with a symplectic group mixing the odd directions.Thus,we immediately see that the number n of fermionic dimensions(in the paper this is also denoted by d F)over a Riemannian supermanifold must be even, otherwise Sp(n)would not be defined.The vielbein formalism can be constructed in the same way:supervielbeins will carry a couple of superindices whose different parity combinations induce the matrix block formE A M= E a m Eαµ (2.3)with commuting diagonal blocks and anticommuting off-diagonal blocks.The correct formula to pass from supervielbeins to supermetrics isG MN=(−1)MA E A MηAB E B N(2.4)whereηdenotes theflat superspace metric diag(m1,...,1,−1,...,−1,nσ2,...,σ2).Butwhat isηµν?A question that has been raised several times in the literature(see for example [34]for a review and some comments),and it turns out that theflat super-metric in the fermionic directionsηµνcannot be always defined for any models.For example,in the case of10d supergravity IIB there is no constant tensor which can be used to raise and lower the spinorial indices and to contact to Weyl indices.However,it is also known that a typical bispinor emerges in the quantization of superstrings in the fermionic sector of the theory and these states are known as the RRfields.Indeed,in the case of constant RRfields,one can identify the fermionic components of theflat supermetric with constant RRfield strength.This will be discuss in sec.5.The inverse of a supermetric G MN is computed by inverting the supermatrix G MN. In order that the inverse exists it must be that det(G mn)and det(Gµν)do not vanish.2.2K¨a hler supermetricsA very important subclass of supermetrics consists of so-called K¨a hler supermetrics,which are a straightforward generalization of the usual K¨a hler metrics living on ordinary man-ifolds.As one can expect,the components G M¯N of such a supermetric are obtained applying superderivatives-denoted by“,”-to a superpotential K:G M¯N=K,M¯N.(2.5) As a consequence the body of a K¨a hler supermanifold with a supermetric G M¯N originated by a superpotential K is a K¨a hler manifold endowed with the metric g m¯n obtained by applying two derivatives to the bosonic part of K according to g(body)m¯n=∂m∂¯n K|θ=0. 2.3Superconnections and covariant superderivativeThe idea of connection and covariant derivative have their relative supergeometric coun-terparts.Indicating superconnections withΓ,one writes the defining rules for covariantsuperderivatives-denoted by“;”-as follows:Φ;M=Φ,M,(2.6a)X M;N=X M,N+(−1)P(M+1)X PΓM P N,(2.6b)ΦM;N=ΦM,N−ΦPΓP MN.(2.6c) A superconnection over a supermanifold M(m|n)can be chosen in many ually this freedom is restricted by demanding that the superconnection be metric-compatible, i.e.that the covariant superderivative of the supermetric vanishes,and that the coeffi-cientsΓM NP be graded-symmetric in their lower indices.These two requirements uniquely determine the superconnection to be given by the super Christoffel symbolsM N P =1andR MN=(−1)MN R NM,(2.13) while Bianchi identities readR MNP Q;R+(−1)P(Q+R)R MNQR;P+(−1)R(P+Q)R MNRP;Q=0,(2.14)1R NMN;−2 T M NP−(−1)Q(N+1)G MQ G NR T R QP−(−1)Q(P+1)+NP G MQ G P R T R QN .(2.20) The corresponding Ricci supercurvature can be obtained adding a few terms to the ex-pression valid with vanishing torsion:R MN=(2.10)+(−1)P(M+1) −K P MP;N+(−1)P(M+Q)K P QP K Q MN+(−1)NP K P MN;P−(−1)N(M+P+Q)K P QN K Q MP .(2.21) In terms of the super-vielbeins,the torsion tensor is given byT A BC E C∧E B=T A≡dE A+ΩA B∧E B.(2.22)3Supermanifolds on given bosonic manifoldsIn the present section we are interested in studying examples of supermanifolds which have a given bosonic submanifold.We construct a corresponding supermanifold by adding the anticommuting coordinates and by requiring that the supermanifold is super-Ricciflat.Before proceeding we would like to give an example of a simple model where the procedure for constructing a supermanifold is illustrated.Let us consider the superfield Φ(x,θ)and we set to zero all fermionic components.Then,Φ= nΦn(θ2)n.We can consider the naive generalization of the Klein-Gordon(KG)equation∂m∂mφ+13!Φ30=(∂m∂m+∂µ∂µ)Φ+13!Φ30+2Φ2=0,∂m∂mΦ2+13!∂2Φ30+112Φ50=0.(3.3)This is a weaker condition onΦ0than the usual KG equation and it must be solved in order that the superfieldΦ0can be thefirst component of the superfieldΦwhich solves the new KG equation.The present example has no physical relevance,but it illustrates the problem appearing in the extension of a given bosonic metric to a supermetric and the extension of the bosonic equations to super-equations.Even in this case,we assume that the background has no fermions and we add only superfields with bosonic components.We have to mention that equations similar to(3.1)where already proposed in several papers in thefirst years of supersymmetry(see for example[36,37,38]).However,this construction was abandoned since the resulting theories possess ghosts and higher-spin fields(see for example[34]and the references therein).Here,we do not pretend to interpret the super-Ricciflatness asfield equations with dynamical content,but we consider these equations a way of recasting the informations on the manifold together the RRfluxes in a single mathematical structure.This is very similar to the case of NS-NSfluxes and the generalized geometry of[12,13].3.1General FrameworkIn the present section,we give the general equations of the super-Ricci tensor R MN by expanding the supermetric in components.These are very useful in order to single outthe structure of the equations.Indeed,it can be observed that at each order there is free component of the supermetric that is used to solve the corresponding equation.First, we show that every equation at the level lower than the maximumθ-expansion can be easily algebraically solved,then we show that the remaining equations are high deriva-tives and they provide consistency conditions for the construction of the corresponding supermanifold.As in the above example,using the iterative equations,one derives the highest derivative equation.This is rather cumbersome since the iterative procedure tends to explode soon in long unmanageable expressions.Wefirst tame the linearized system and then we tackle the non-linear one.For the former we are able to provide a complete analysis;for the latter only the case(d|2)will be discussed.In order to study the structure of the equation for super Ricci-flat manifold it is convenient to consider the bosonic and fermionic components separately.For that we display the few terms coming from the superfield expansion of the supermetric:1G mn=g mn+w mνρστθτθσθρ+O(θ5),61Gµν=hµν+hρσ(hσρ;(mn)−t m[σρ];n−t n[σρ];m+f mnσρ)(3.7)2−1Rµν=−12hρσ(lµνσρ+lσρµν+lµρσν+lσνµρ)−14hρσg mn(2t m(µρ)−hµρ,m)(2t n(νσ)−hνσ,n)+O(θ2)=0.(3.8)Eq.(3.7)can be easily solved in terms of f mnρσ.In the same way,(3.8)is solved by a suitable combination of lµνρσ.At the next order,a free bosonicfield will appear from the expansion of the superfields G MN and a new equationfixes it.This iterative procedure allows us to solve the complete set of equations,starting from a given bosonic metric g mn (and with aflat supermetric in the fermionic sector g[µν]),in terms of the components of G MN.At the end of the iterative procedure,wefind the consistency conditions on the bosonic manifold.Wefirst analyze the linearized equations and we expand around the flat space and around a Ricciflat(R(body)mn)bosonic space.It turns out that variation of the bosonic Ricci tensor must satisfy a set of differential equations(with a high-derivative differential operator)and some algebraic conditions(the space must have vanishing scalar curvature).However,some of the constraints are weaker in the non-linear theory.Furthermore,one can also impose additional structure in the superspace such as a su-percomplex structure.Some simplifications are in order in the case of a K¨a hler manifolds.On a K¨a hler supermanifold the hermitian metric supertensor comes from a K¨a hler superpotential,K,by the following formula:G M¯N=K,M¯N.(3.9) As a consequence the Ricci super-tensor components in holomorphic coordinates reduce to the formR M¯N=−(−1)P+Q G¯P Q G Q¯P,M¯N+(−1)P+Q+S+M(P+R)G P¯Q G¯QR,M G R¯S G¯SP,¯N =−(ln sdet G),M¯N(3.10)We can expand again the supermetric in the odd coordinates(θµ,θ¯µ)G m¯n=g m¯n+hρ¯σ,m¯nθ¯σθρ+12f m¯n¯ρ¯σθ¯σθ¯ρ+O(θ4),G m¯ν=h¯νρ,mθρ+t m¯ν¯ρθ¯ρ+O(θ3), Gµ¯n=tµ¯nρθρ+hµ¯ρ,¯nθ¯ρ+O(θ3),Gµ¯ν=hµ¯ν+lµ¯νρ¯σθ¯σθρ+12lµ¯ν¯ρ¯σθ¯σθ¯ρ+O(θ4),f m¯n¯ρ¯σ=f∗n¯mσρ,tρ¯mσ=t∗m¯ρ¯σ,lµ¯ν¯ρ¯σ=l∗ν¯µσρ,(3.11)and obtain the equivalent of the super Ricci-flatness equation,R MN=0,for g,h,l and t.The last line expresses the constraints coming from the hermiticity of the K¨a hlerpotential.At the lowest order in the odd variables one getsR m¯n=R(body)m¯n−h¯ρσhσ¯ρ,m¯n−hµ¯νh¯νρ,m hρ¯σh¯σµ,¯n+O(θ2)==R(body)m¯n+(ln(det h)),m¯n+O(θ2)=0,Rµ¯ν=−g¯p q hµ¯ν,q¯p−h¯ρσlσ¯ρµ¯ν−hρ¯σh¯σµ,p g p¯q hρ¯ν,¯q+hρ¯σt p¯σ¯νg p¯q tρ¯qµ+O(θ2)=0,where R(body)mn is the Ricci tensor on the body of the Ricci-flat K¨a hler supermanifold(SCY)and h=(hµ¯ν),from whichR(body)2 (−1)MN H QN;MP−(−1)Q(M+N)H MN;QP+H QM;NP−(−1)P(M+N)H QP;MN ,(3.14) which,making parity of indices explicit,splits intoδR mn=12(−1)Q G P Q H Qm;µP+H Qµ;mP−(−1)Q H mµ;QP−(−1)P H QP;mµ ,(3.15b)δR µν=12g pqH qm ;np +H qn ;mp −H mn ;qp −H qp ;mn+12g pqH qm ;µp +H qµ;mp −H mµ;qp −H qp ;mµ−12g pqH qµ;νp −H qν;µp −H µν;qp −H qp ;µν+12g pqH (2k )qm [τ1...τ2k ];np +H (2k )qn [τ1...τ2k ];mp −H (2k )mn [τ1...τ2k ];qp−H (2k )qp [τ1...τ2k ];mn+12gpq(2k +2)H (2k +2)qm [µτ1...τ2k +1];p +H (2k +1)qµ[τ1...τ2k +1];mp −H (2k +1)mµ[τ1...τ2k +1];qp−(2k +2)H (2k +2)qp [µτ1...τ2k +1];m−(k +1)jρσ(2k +3)H (2k +3)σm [µρτ1...τ2k +1]+H (2k +2)σµ[ρτ1...τ2k +1];m+(2k +3)H (2k +3)mµ[σρτ1...τ2k +1]+H (2k +2)σρ[µτ1...τ2k +1];m ,(3.18b)δR (2k )µν[τ1...τ2k ]=1Analizing the recursive structure of(3.18)’s,one realizes that the only independentfields which effectively play a role in constraining g mn are obtained by contracting odd indices with j as follows:S(2k) mn =H(2k)mn[τ1...τ2k]jτ1τ2···jτ2k−1τ2k,(3.19a)T(2k+1) m =H(2k+1)mτ1[τ2...τ2k+2]jτ1τ2···jτ2k+1τ2k+2,(3.19b)U(2k)=H(2k)τ1τ2[τ3...τ2k+2]jτ1τ2jτ3τ4···jτ2k+1τ2k+2,(3.19c)V(2k)=H(2k)τ1τ2[τ3...τ2k+2]jτ1τ3jτ2τ4···jτ2k+1τ2k+2.(3.19d) From these definitions one has a few identities:V(0)=0,U(d F)=d F V(d F),S(0)mn=δg mn.(3.20) The calculation is simplified by a gauge-fixing of the supermetric(3.6):a convenient choice isG mµθµ=0,Gµνθνθµ=jµνθνθµ,(3.21) which,in terms of variations,becomesH mµθµ=0,Hµνθνθµ=0.(3.22) We notice that some of thefields defined in(3.19)are vanishing because of the gauge-fixing:T(1)m=0,U(0)=0.(3.23) Introducing S,T,U and V into(3.18)’s,one obtains in principle fourfield equations but one of them follows from the other ones when Ricci curvature of the body vanishes.The remaining threefield equations corresponding to super Ricci-flatness are(2k+2)(2k+1)S(2k+2)mn=(2k+1) T(2k+1)m;n+T(2k+1)n;m −U(2k);mn−g pq S(2k)qm;np+S(2k)qn;mp−S(2k)mn;qp−S(2k)qp;mn ,(3.24a) (2k+2)(2k+1) U(2k+2)+V(2k+2) =2 U(2k)+V(2k) ,(3.24b)2S(2k)mm−S(2k)mnmn;=2 U(2k)+V(2k) .(3.24c) Recalling the identity(3.20)and the gauge-fixing(3.23),from here wefindU(2k)+V(2k)=0,U(d F)=V(d F)=0,(3.25) so that(3.24b)is automatically solved and(3.24c)with k=0shows that,at linear order, the curvature scalar on the body has to be zero:R(body)=0.(3.26)Besides this algebraic constraint on the Ricci curvature,also some differential consis-tency condition has to be taken into account that comes from the remaining constraints. IntroducingˆS(2k+2) mn def=(2k+2)!S(2k+2)mn−(2k+1)! T(2k+1)m;n+T(2k+1)n;m +(2k)!U(2k);mn(3.27)into(3.24a)and(3.24c),one can rewrite them asˆS(2k+2) mn =2ˆS(2k)mn+2ˆS pq(2k)R(body)pmqn(k>0)(3.28)withˆS(2) mn =−2δR(body)mn,ˆS(d F+2)mn=0.(3.29)In other words,at linear order,the only condition to require in addition to(3.26)isL d F/2R R(body)mn=0(3.30) where LRis the Lichnerowicz operator defined on Ricci-flat spaces byL R X mn=2X mn+2X pq R(body)pmqn.(3.31) In conclusion,at linear order,a Ricci-flat manifold can be deformed in the body of a Ricci-flat supermanifold only if conditions(3.26)and(3.30)are satisfied,and,in that case,an overlying supergeometrical structure is constrained by(3.25),(3.28)and(3.29). As a consequence,if a consistent deformation is possible,an infinite number of Ricci-flat supermanifolds with gauge-fixed supermetric can be constructed over the same bosonic body.It would be interesting to compare the present result with the results of paper[9].We plan to explore deeply the relation between the super-Ricciflatness equations and the moduli space of the supervarieties.3.3Non-linear equationsSince a complete general analysis is rather cumbersome,we give the complete expressions only in the case of(d|2)-dimensional supermanifolds.We show that equations at the lowest order can be indeed solved by some free components of the superfields and we derive the equations for the highest components.Following the scheme given in section 3.2,by substituting the lower-order results one gets thefinal constraints on the metric of the bosonic submanifold.As in the present case there are just two fermionic directions,theθ-expansion of the supermetric with the gauge-fixing(3.21)is given byG mn=g mn+12f(2)θ2(3.32)。

解析数论是使用数学分析作为工具来解决数论问题的分支。

微积分和复变函数论发展以后，产生了解析数论。

该学科的第一个主要成就是狄利克雷用解析方法证明了Dirichlet's theorem on arithmetic progressions。

依靠黎曼zeta函数对素数定理的证明是另一个里程碑。

解析数论是解决数论中艰深问题的重要工具，数论中有些问题必须由解析方法才能提出或解决。

中国的华罗庚、王元、陈景润等人在“哥德巴赫猜想”、“华林问题”等解析数论问题上取得世界公认的成就。

黎曼ζ函数Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics. The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.[1]DefinitionThe Riemann zeta-function ζ(s) is the function of a complex variable s initially defined by the following infinite series:As a Dirichlet series with bounded coefficient sequence this series converges absolutely to an analytic function on the open half-plane of s such that Re(s) > 1 and diverges on the open half-plane of s such that Re(s) < 1. The function defined by the series on the half-plane of convergence can however be continued analytically to all complex s≠ 1. For s= 1 the series is formally identical to the harmonic series which diverges to infinity. As a result, the zeta function becomes a meromorphic function of the complex variable s, which is holomorphic in the region {s∈ C : s≠1} of the complex plane and has a simple pole at s= 1 with residue 1.Specific valuesThe values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.this is the harmonic series.this is employed in calculating the critical temperature for a Bose–Einstein condensate in physics, and for spin-wave physics in magnetic systems.the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? [2]this is called Apéry's constant.Stefan–Boltzmann law and Wien approximation in physics.Euler product formulaThe connection between the zeta function and prime numbers was discovered by Leonhard Euler, who proved the identitywhere, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s= 1, diverges, Euler's formula implies that there are infinitely many primes. For s an integer number, the Euler product formula can be used to calculate the probability that s randomly selected integers are relatively prime. It turns out that this probability is indeed 1/ζ(s). The functional equationThe Riemann zeta function satisfies the functional equationvalid for all complex numbers s, which relates its values at points s and 1 −s. Here, Γ denotes the gamma function. This functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place. An equivalent relationship was conjectured by Euler in 1749 for the functionAccording to André Weil, Riemann seems to have been very familiar with Euler's work on the subject.[3]The functional equation given by Riemann has to be interpreted analytically if any factors in the equation have a zero or pole. For instance, when s is 2, the right side has a simple zero in the sine factor and a simple pole in the Gamma factor, which cancel out and leave a nonzero finite value. Similarly, when s is 0, the right side has a simple zero in the sine factor and a simple pole in the zeta factor, which cancel out and leave a finite nonzero value. When s is 1, the right side has a simple pole in the Gamma factor that is not cancelled out by a zero in any other factor, which is consistent with the zeta-function on the left having a simple pole at 1.There is also a symmetric version of the functional equation, given by first definingThe functional equation is then given by(Riemann defined a similar but different function which he called ξ(t).)The functional equation also gives the asymptotic limit(GergőNemes, 2007)Zeros, the critical line, and the Riemann hypothesisThe functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s∈ C: 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set{s∈ C: Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. From the fact that allnon-trivial zeros lie in the critical strip one can deduce the prime number theorem. A better result[4]is that ζ(σ+ i t) ≠ 0 whenever |t| ≥ 3 andThe strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γn) contains the imaginary parts of all zeros in the upper half-plane in ascending order, thenThe critical line theorem asserts that a positive percentage of the nontrivial zeros lies on the critical line.In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + i14.13472514... Directly from the functional equation one sees that the non-trivial zeros are symmetric about the axis Re(s) = 1/2. Furthermore, the fact that ζ(s) = ζ(s*)* for all complex s≠1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The fractal structure of the Riemann zeta zero distribution has been studied using rescaled range analysis.[5] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude, and also for the zeros of other L-functions.The properties of the Riemann zeta function in the complex plane, specifically along parallels to the imaginary axis, has also been studied, by the relation to prime numbers, in recentphysical interference experiments, by decomposing the sum into two parts with opposite phases, ψ and ψ*, which then are brought to interference. [6]For sums involving the zeta-function at integer and half-integer values, see rational zeta series.[O] ReciprocalThe reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius functionμ(n):for every complex number s with real part > 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2. [O] UniversalityThe critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.[O] Representations[O] Mellin transformThe Mellin transform of a function f(x) is defined asin the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we havewhere Γ denotes the Gamma function. By subtracting off the first terms of the power series expansion of 1/(exp(x) −1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we haveand when the real part of s is between −1 and 0,We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, thenfor values with We can relate this to the Mellin transform of π(x) bywhereconverges forA similar Mellin transform involves the Riemann prime-counting function J(x), which counts prime powers p n with a weight of 1/n,so that Now we haveThese expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.Also, from the above (specifically, the second equation in this section), we can write the zeta function in the commonly seen form:[O] Laurent seriesThe Riemann zeta function is meromorphic with a single pole of order one at s= 1. It can therefore be expanded as a Laurent series about s= 1; the series development then isThe constants γn here are called the Stieltjes constants and can be defined by the limitThe constant term γ0 is the Euler-Mascheroni constant.[O] Rising factorialAnother series development valid for the entire complex plane iswhere is the rising factorial This can be used recursively to extend the Dirichlet series definition to all complex numbers.The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing operator acting on x s−1; that context gives rise to a series expansion in terms of the falling factorial.[O] Hadamard productOn the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansionwhere the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler-Mascheroni constant. A simpler infinite product expansion isThis form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ.[O] Globally convergent seriesA globally convergent series for the zeta function, valid for all complex numbers s except s = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930:The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev polynomials, is described in the article on the Dirichlet eta function.[O] ApplicationsAlthough mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning.During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · ·, but we can rewrite it as a sum of reciprocals:The sum S appears to take the form of However, −1 lies outside of the domain for which the Dirichlet series for thezeta-function converges. However, a divergent series of positive terms such as this one can sometimes be represented in a reasonable way by the method of Ramanujan summation (see Hardy, Divergent Series.) Ramanujan summation involves an application of the Euler–Maclaurin summation formula, and when applied to the zeta-function, it extends its definition to the whole complex plane. In particularwhere the notation indicates Ramanujan summation.[7]For even powers we have:and for odd powers we have a relation with the Bernoulli numbers:Zeta function regularization is used as one possible means of regularization of divergent series in quantum field theory. In one notable example, the Riemann zeta-function shows up explicitly in the calculation of the Casimir effect.[O] GeneralizationsThere are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz zeta functionwhich coincides with Riemann's zeta-function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function andL-function.The polylogarithm is given bywhich coincides with Riemann's zeta-function when z = 1.The Lerch transcendent is given bywhich coincides with Riemann's zeta-function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).The Clausen function that can be chosen as the real orimaginary part ofThe multiple zeta functions are defined byOne can analytically continue these functions to then-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.[O] Zeta-functions in fiction。

自动化英语专业英语词汇表文章摘要：本文介绍了自动化英语专业的一些常用的英语词汇，包括自动化技术、控制理论、系统工程、人工智能、模糊逻辑等方面的专业术语。

本文按照字母顺序，将这些词汇分为26个表格，每个表格包含了以相应字母开头的词汇及其中文释义。

本文旨在帮助自动化专业的学习者和从业者掌握和使用这些专业英语词汇，提高他们的英语水平和专业素养。

A英文中文acceleration transducer加速度传感器acceptance testing验收测试accessibility可及性accumulated error累积误差AC-DC-AC frequency converter交-直-交变频器AC (alternating current) electric drive交流电子传动active attitude stabilization主动姿态稳定actuator驱动器，执行机构adaline线性适应元adaptation layer适应层adaptive telemeter system适应遥测系统adjoint operator伴随算子admissible error容许误差aggregation matrix集结矩阵AHP (analytic hierarchy process)层次分析法amplifying element放大环节analog-digital conversion模数转换annunciator信号器antenna pointing control天线指向控制anti-integral windup抗积分饱卷aperiodic decomposition非周期分解a posteriori estimate后验估计approximate reasoning近似推理a priori estimate先验估计articulated robot关节型机器人assignment problem配置问题，分配问题associative memory model联想记忆模型associatron联想机asymptotic stability渐进稳定性attained pose drift实际位姿漂移B英文中文attitude acquisition姿态捕获AOCS (attritude and orbit control system)姿态轨道控制系统attitude angular velocity姿态角速度attitude disturbance姿态扰动attitude maneuver姿态机动attractor吸引子augment ability可扩充性augmented system增广系统automatic manual station自动-手动操作器automaton自动机autonomous system自治系统backlash characteristics间隙特性base coordinate system基座坐标系Bayes classifier贝叶斯分类器bearing alignment方位对准bellows pressure gauge波纹管压力表benefit-cost analysis收益成本分析bilinear system双线性系统biocybernetics生物控制论biological feedback system生物反馈系统C英文中文calibration校准，定标canonical form标准形式canonical realization标准实现capacity coefficient容量系数cascade control级联控制causal system因果系统cell单元，元胞cellular automaton元胞自动机central processing unit (CPU)中央处理器certainty factor确信因子characteristic equation特征方程characteristic function特征函数characteristic polynomial特征多项式characteristic root特征根英文中文charge-coupled device (CCD)电荷耦合器件chaotic system混沌系统check valve单向阀，止回阀chattering phenomenon颤振现象closed-loop control system闭环控制系统closed-loop gain闭环增益cluster analysis聚类分析coefficient of variation变异系数cogging torque齿槽转矩，卡齿转矩cognitive map认知图，认知地图coherency matrix相干矩阵collocation method配点法，配置法combinatorial optimization problem组合优化问题common mode rejection ratio (CMRR)共模抑制比，共模抑制率commutation circuit换相电路，换向电路commutator motor换向电动机D英文中文damping coefficient阻尼系数damping ratio阻尼比data acquisition system (DAS)数据采集系统data fusion数据融合dead zone死区decision analysis决策分析decision feedback equalizer (DFE)决策反馈均衡器decision making决策，决策制定decision support system (DSS)决策支持系统decision table决策表decision tree决策树decentralized control system分散控制系统decoupling control解耦控制defuzzification去模糊化，反模糊化delay element延时环节，滞后环节delta robot德尔塔机器人demodulation解调，检波density function密度函数，概率密度函数derivative action微分作用，微分动作design matrix设计矩阵E英文中文eigenvalue特征值，本征值eigenvector特征向量，本征向量elastic element弹性环节electric drive电子传动electric potential电势electro-hydraulic servo system电液伺服系统electro-mechanical coupling system电机耦合系统electro-pneumatic servo system电气伺服系统electronic governor电子调速器encoder编码器，编码装置end effector末端执行器，末端效应器entropy熵equivalent circuit等效电路error analysis误差分析error bound误差界，误差限error signal误差信号estimation theory估计理论Euclidean distance欧几里得距离，欧氏距离Euler angle欧拉角Euler equation欧拉方程F英文中文factor analysis因子分析factorization method因子法，因式分解法feedback反馈，反馈作用feedback control反馈控制feedback linearization反馈线性化feedforward前馈，前馈作用feedforward control前馈控制field effect transistor (FET)场效应晶体管filter滤波器，滤波环节finite automaton有限自动机finite difference method有限差分法finite element method (FEM)有限元法finite impulse response (FIR) filter有限冲激响应滤波器first-order system一阶系统fixed-point iteration method不动点迭代法flag register标志寄存器flip-flop circuit触发器电路floating-point number浮点数flow chart流程图，流程表fluid power system流体动力系统G英文中文gain增益gain margin增益裕度Galerkin method伽辽金法game theory博弈论Gauss elimination method高斯消元法Gauss-Jordan method高斯-约当法Gauss-Markov process高斯-马尔可夫过程Gauss-Seidel iteration method高斯-赛德尔迭代法genetic algorithm (GA)遗传算法gradient method梯度法，梯度下降法graph theory图论gravity gradient stabilization重力梯度稳定gray code格雷码，反向码gray level灰度，灰阶grid search method网格搜索法ground station地面站，地面控制站guidance system制导系统，导航系统gyroscope陀螺仪，陀螺仪器H英文中文H∞ control H无穷控制Hamiltonian function哈密顿函数harmonic analysis谐波分析harmonic oscillator谐振子，谐振环节Hartley transform哈特利变换Hebb learning rule赫布学习规则Heisenberg uncertainty principle海森堡不确定性原理hidden layer隐层，隐含层hidden Markov model (HMM)隐马尔可夫模型hierarchical control system分层控制系统high-pass filter高通滤波器Hilbert transform希尔伯特变换Hopfield network霍普菲尔德网络hysteresis滞后，迟滞，磁滞I英文中文identification识别，辨识identity matrix单位矩阵，恒等矩阵image processing图像处理impulse response冲激响应impulse response function冲激响应函数inadmissible control不可接受控制incremental encoder增量式编码器indefinite integral不定积分index of controllability可控性指标index of observability可观测性指标induction motor感应电动机inertial navigation system (INS)惯性导航系统inference engine推理引擎，推理机inference rule推理规则infinite impulse response (IIR) filter无限冲激响应滤波器information entropy信息熵information theory信息论input-output linearization输入输出线性化input-output model输入输出模型input-output stability输入输出稳定性J英文中文Jacobian matrix雅可比矩阵jerk加加速度，冲击joint coordinate system关节坐标系joint space关节空间Joule's law焦耳定律jump resonance跳跃共振K英文中文Kalman filter卡尔曼滤波器Karhunen-Loeve transform卡尔胡南-洛维变换kernel function核函数，核心函数kinematic chain运动链，运动链条kinematic equation运动方程，运动学方程kinematic pair运动副，运动对kinematics运动学kinetic energy动能L英文中文Lagrange equation拉格朗日方程Lagrange multiplier拉格朗日乘子Laplace transform拉普拉斯变换Laplacian operator拉普拉斯算子laser激光，激光器latent root潜根，隐根latent vector潜向量，隐向量learning rate学习率，学习速度least squares method最小二乘法Lebesgue integral勒贝格积分Legendre polynomial勒让德多项式Lennard-Jones potential莱纳德-琼斯势level set method水平集方法Liapunov equation李雅普诺夫方程Liapunov function李雅普诺夫函数Liapunov stability李雅普诺夫稳定性limit cycle极限环，极限圈linear programming线性规划linear quadratic regulator (LQR)线性二次型调节器linear system线性系统M英文中文machine learning机器学习machine vision机器视觉magnetic circuit磁路，磁电路英文中文magnetic flux磁通量magnetic levitation磁悬浮magnetization curve磁化曲线magnetoresistance磁阻，磁阻效应manipulability可操作性，可操纵性manipulator操纵器，机械手Markov chain马尔可夫链Markov decision process (MDP)马尔可夫决策过程Markov property马尔可夫性质mass matrix质量矩阵master-slave control system主从控制系统matrix inversion lemma矩阵求逆引理maximum likelihood estimation (MLE)最大似然估计mean square error (MSE)均方误差measurement noise测量噪声，观测噪声mechanical impedance机械阻抗membership function隶属函数N英文中文natural frequency固有频率，自然频率natural language processing (NLP)自然语言处理navigation导航，航行negative feedback负反馈，负反馈作用neural network神经网络neuron神经元，神经细胞Newton method牛顿法，牛顿迭代法Newton-Raphson method牛顿-拉夫逊法noise噪声，噪音nonlinear programming非线性规划nonlinear system非线性系统norm范数，模，标准normal distribution正态分布，高斯分布notch filter凹槽滤波器，陷波滤波器null space零空间，核空间O英文中文observability可观测性英文中文observer观测器，观察器optimal control最优控制optimal estimation最优估计optimal filter最优滤波器optimization优化，最优化orthogonal matrix正交矩阵oscillation振荡，振动output feedback输出反馈output regulation输出调节P英文中文parallel connection并联，并联连接parameter estimation参数估计parity bit奇偶校验位partial differential equation (PDE)偏微分方程passive attitude stabilization被动姿态稳定pattern recognition模式识别PD (proportional-derivative) control比例-微分控制peak value峰值，峰值幅度perceptron感知器，感知机performance index性能指标，性能函数period周期，周期时间periodic signal周期信号phase angle相角，相位角phase margin相位裕度phase plane analysis相平面分析phase portrait相轨迹，相图像PID (proportional-integral-derivative) control比例-积分-微分控制piezoelectric effect压电效应pitch angle俯仰角pixel像素，像元Q英文中文quadratic programming二次规划quantization量化，量子化quantum computer量子计算机quantum control量子控制英文中文queueing theory排队论quiescent point静态工作点，静止点R英文中文radial basis function (RBF) network径向基函数网络radiation pressure辐射压random variable随机变量random walk随机游走range范围，区间，距离rank秩，等级rate of change变化率，变化速率rational function有理函数Rayleigh quotient瑞利商real-time control system实时控制系统recursive algorithm递归算法recursive estimation递归估计reference input参考输入，期望输入reference model参考模型，期望模型reinforcement learning强化学习relay control system继电器控制系统reliability可靠性，可信度remote control system遥控系统，远程控制系统residual error残差误差，残余误差resonance frequency共振频率S英文中文sampling采样，取样sampling frequency采样频率sampling theorem采样定理saturation饱和，饱和度scalar product标量积，点积scaling factor缩放因子，比例系数Schmitt trigger施密特触发器Schur complement舒尔补second-order system二阶系统self-learning自学习，自我学习self-organizing map (SOM)自组织映射sensitivity灵敏度，敏感性sensitivity analysis灵敏度分析，敏感性分析sensor传感器，感应器sensor fusion传感器融合servo amplifier伺服放大器servo motor伺服电机，伺服马达servo valve伺服阀，伺服阀门set point设定值，给定值settling time定常时间，稳定时间T英文中文tabu search禁忌搜索，禁忌表搜索Taylor series泰勒级数，泰勒展开式teleoperation遥操作，远程操作temperature sensor温度传感器terminal终端，端子testability可测试性，可检测性thermal noise热噪声，热噪音thermocouple热电偶，热偶threshold阈值，门槛time constant时间常数time delay时延，延时time domain时域time-invariant system时不变系统time-optimal control时间最优控制time series analysis时间序列分析toggle switch拨动开关，切换开关tolerance analysis公差分析torque sensor扭矩传感器transfer function传递函数，迁移函数transient response瞬态响应U英文中文uncertainty不确定性，不确定度underdamped system欠阻尼系统undershoot低于量，低于值unit impulse function单位冲激函数unit step function单位阶跃函数unstable equilibrium point不稳定平衡点unsupervised learning无监督学习upper bound上界，上限utility function效用函数，效益函数V英文中文variable structure control变结构控制variance方差，变异vector product向量积，叉积velocity sensor速度传感器verification验证，校验virtual reality虚拟现实viscosity粘度，黏度vision sensor视觉传感器voltage电压，电位差voltage-controlled oscillator (VCO)电压控制振荡器W英文中文wavelet transform小波变换weighting function加权函数Wiener filter维纳滤波器Wiener process维纳过程work envelope工作空间，工作范围worst-case analysis最坏情况分析X英文中文XOR (exclusive OR) gate异或门，异或逻辑门Y英文中文yaw angle偏航角Z英文中文Z transform Z变换zero-order hold (ZOH)零阶保持器zero-order system零阶系统zero-pole cancellation零极点抵消。

Chapter 15Audit Sampling for Tests of Controls andSubstantive Tests of TransactionsReview Questions15-1 A representative sample is one in which the characteristics of interest for the sample are approximately the same as for the population (that is, the sample accurately represents the total population). If the population contains significant misstatements, but the sample is practically free of misstatements, the sample is nonrepresentative, which is likely to result in an improper audit decision. The auditor can never know for sure whether he or she has a representative sample because the entire population is ordinarily not tested, but certain things, such as the use of random selection, can increase the likelihood of a representative sample.15-2Statistical sampling is the use of mathematical measurement techniques to calculate formal statistical results. The auditor therefore quantifies sampling risk when statistical sampling is used. In nonstatistical sampling, the auditor does not quantify sampling risk. Instead, conclusions are reached about populations on a more judgmental basis.For both statistical and nonstatistical methods, the three main parts are:1. Plan the sample2. Select the sample and perform the tests3. Evaluate the results15-3In replacement sampling, an element in the population can be included in the sample more than once if the random number corresponding to that element is selected more than once. In nonreplacement sampling, an element can be included only once. If the random number corresponding to an element is selected more than once, it is simply treated as a discard the second time. Although both selection approaches are consistent with sound statistical theory, auditors rarely use replacement sampling; it seems more intuitively satisfying to auditors to include an item only once.15-4 A simple random sample is one in which every possible combination of elements in the population has an equal chance of selection. Two methods of simple random selection are use of a random number table, and use of the computer to generate random numbers. Auditors most often use the computer to generate random numbers because it saves time, reduces the likelihood of error, and provides automatic documentation of the sample selected.15-5In systematic sampling, the auditor calculates an interval and then methodically selects the items for the sample based on the size of the interval. The interval is set by dividing the population size by the number of sample items desired.To select 35 numbers from a population of 1,750, the auditor divides 35 into 1,750 and gets an interval of 50. He or she then selects a random number between 0 and 49. Assume the auditor chooses 17. The first item is the number 17. The next is 67, then 117, 167, and so on.The advantage of systematic sampling is its ease of use. In most populations a systematic sample can be drawn quickly, the approach automatically puts the numbers in sequential order and documentation is easy.A major problem with the use of systematic sampling is the possibility of bias. Because of the way systematic samples are selected, once the first item in the sample is selected, other items are chosen automatically. This causes no problems if the characteristics of interest, such as control deviations, are distributed randomly throughout the population; however, in many cases they are not. If all items of a certain type are processed at certain times of the month or with the use of certain document numbers, a systematically drawn sample has a higher likelihood of failing to obtain a representative sample. This shortcoming is sufficiently serious that some CPA firms prohibit the use of systematic sampling. 15-6The purpose of using nonstatistical sampling for tests of controls and substantive tests of transactions is to estimate the proportion of items in a population containing a characteristic or attribute of interest. The auditor is ordinarily interested in determining internal control deviations or monetary misstatements for tests of controls and substantive tests of transactions.15-7 A block sample is the selection of several items in sequence. Once the first item in the block is selected, the remainder of the block is chosen automatically. Thus, to select 5 blocks of 20 sales invoices, one would select one invoice and the block would be that invoice plus the next 19 entries. This procedure would be repeated 4 other times.15-8 The terms below are defined as follows:15-8 (continued)15-9The sampling unit is the population item from which the auditor selects sample items. The major consideration in defining the sampling unit is making it consistent with the objectives of the audit tests. Thus, the definition of the population and the planned audit procedures usually dictate the appropriate sampling unit.The sampling unit for verifying the occurrence of recorded sales would be the entries in the sales journal since this is the document the auditor wishes to validate. The sampling unit for testing the possibility of omitted sales is the shipping document from which sales are recorded because the failure to bill a shipment is the exception condition of interest to the auditor.15-10 The tolerable exception rate (TER) represents the exception rate that the auditor will permit in the population and still be willing to use the assessed control risk and/or the amount of monetary misstatements in the transactions established during planning. TER is determined by choice of the auditor on the basis of his or her professional judgment.The computed upper exception rate (CUER) is the highest estimated exception rate in the population, at a given ARACR. For nonstatistical sampling, CUER is determined by adding an estimate of sampling error to the SER (sample exception rate). For statistical sampling, CUER is determined by using a statistical sampling table after the auditor has completed the audit testing and therefore knows the number of exceptions in the sample.15-11 Sampling error is an inherent part of sampling that results from testing less than the entire population. Sampling error simply means that the sample is not perfectly representative of the entire population.Nonsampling error occurs when audit tests do not uncover errors that exist in the sample. Nonsampling error can result from:1. The auditor's failure to recognize exceptions, or2. Inappropriate or ineffective audit procedures.There are two ways to reduce sampling risk:1. Increase sample size.2. Use an appropriate method of selecting sample items from thepopulation.Careful design of audit procedures and proper supervision and review are ways to reduce nonsampling risk.15-12 An attribute is the definition of the characteristic being tested and the exception conditions whenever audit sampling is used. The attributes of interest are determined directly from the audit program.15-13 An attribute is the characteristic being tested for in a population. An exception occurs when the attribute being tested for is absent. The exception for the audit procedure, the duplicate sales invoice has been initialed indicating the performance of internal verification, is the lack of initials on duplicate sales invoices.15-14 Tolerable exception rate is the result of an auditor's judgment. The suitable TER is a question of materiality and is therefore affected by both the definition and the importance of the attribute in the audit plan.The sample size for a TER of 6% would be smaller than that for a TER of 3%, all other factors being equal.15-15 The appropriate ARACR is a decision the auditor must make using professional judgment. The degree to which the auditor wishes to reduce assessed control risk below the maximum is the major factor determining the auditor's ARACR.The auditor will choose a smaller sample size for an ARACR of 10% than would be used if the risk were 5%, all other factors being equal.15-16 The relationship between sample size and the four factors determining sample size are as follows:a. As the ARACR increases, the required sample size decreases.b. As the population size increases, the required sample size isnormally unchanged, or may increase slightly.c. As the TER increases, the sample size decreases.d. As the EPER increases, the required sample size increases.15-17 In this situation, the SER is 3%, the sample size is 100 and the ARACR is 5%. From the 5% ARACR table (Table 15-9) then, the CUER is 7.6%. This means that the auditor can state with a 5% risk of being wrong that the true population exception rate does not exceed 7.6%.15-18 Analysis of exceptions is the investigation of individual exceptions to determine the cause of the breakdown in internal control. Such analysis is important because by discovering the nature and causes of individual exceptions, the auditor can more effectively evaluate the effectiveness of internal control. The analysis attempts to tell the "why" and "how" of the exceptions after the auditor already knows how many and what types of exceptions have occurred.15-19 When the CUER exceeds the TER, the auditor may do one or more of the following:1. Revise the TER or the ARACR. This alternative should be followed onlywhen the auditor has concluded that the original specifications weretoo conservative, and when he or she is willing to accept the riskassociated with the higher specifications.2. Expand the sample size. This alternative should be followed whenthe auditor expects the additional benefits to exceed the additionalcosts, that is, the auditor believes that the sample tested was notrepresentative of the population.3. Revise assessed control risk upward. This is likely to increasesubstantive procedures. Revising assessed control risk may bedone if 1 or 2 is not practical and additional substantive proceduresare possible.4. Write a letter to management. This action should be done inconjunction with each of the three alternatives above. Managementshould always be informed when its internal controls are notoperating effectively. If a deficiency in internal control is consideredto be a significant deficiency in the design or operation of internalcontrol, professional standards require the auditor to communicatethe significant deficiency to the audit committee or its equivalent inwriting. If the client is a publicly traded company, the auditor mustevaluate the deficiency to determine the impact on the auditor’sreport on internal control over financial reporting. If the deficiency isdeemed to be a material weakness, the auditor’s report on internalcontrol would contain an adverse opinion.15-20 Random (probabilistic) selection is a part of statistical sampling, but it is not, by itself, statistical measurement. To have statistical measurement, it is necessary to mathematically generalize from the sample to the population.Probabilistic selection must be used if the sample is to be evaluated statistically, although it is also acceptable to use probabilistic selection with a nonstatistical evaluation. If nonprobabilistic selection is used, nonstatistical evaluation must be used.15-21 The decisions the auditor must make in using attributes sampling are: What are the objectives of the audit test?Does audit sampling apply?What attributes are to be tested and what exception conditions are identified?What is the population?What is the sampling unit?What should the TER be?What should the ARACR be?What is the EPER?What generalizations can be made from the sample to thepopulation?What are the causes of the individual exceptions?Is the population acceptable?15-21 (continued)In making the above decisions, the following should be considered: The individual situation.Time and budget constraints.The availability of additional substantive procedures.The professional judgment of the auditor.Multiple Choice Questions From CPA Examinations15-22 a. (1) b. (3) c. (2) d. (4)15-23 a. (1) b. (3) c. (4) d. (4)15-24 a. (4) b. (3) c. (1) d. (2)Discussion Questions and Problems15-25a.An example random sampling plan prepared in Excel (P1525.xls) is available on the Companion Website and on the Instructor’s Resource CD-ROM, which is available upon request. The command for selecting the random number can be entered directly onto the spreadsheet, or can be selected from the function menu (math & trig) functions. It may be necessary to add the analysis tool pack to access the RANDBETWEEN function. Once the formula is entered, it can be copied down to select additional random numbers. When a pair of random numbers is required, the formula for the first random number can be entered in the first column, and the formula for the second random number can be entered in the second column.a. First five numbers using systematic selection:Using systematic selection, the definition of the sampling unit for determining the selection interval for population 3 is the total number of lines in the population. The length of the interval is rounded down to ensure that all line numbers selected are within the defined population.15-26a. To test whether shipments have been billed, a sample of warehouse removal slips should be selected and examined to see ifthey have the proper sales invoice attached. The sampling unit willtherefore be the warehouse removal slip.b. Attributes sampling method: Assuming the auditor is willing toaccept a TER of 3% at a 10% ARACR, expecting no exceptions inthe sample, the appropriate sample size would be 76, determinedfrom Table 15-8.Nonstatistical sampling method: There is no one right answer tothis question because the sample size is determined usingprofessional judgment. Due to the relatively small TER (3%), thesample size should not be small. It will most likely be similar in sizeto the sample chosen by the statistical method.c. Systematic sample selection:22839 = Population size of warehouse removal slips(37521-14682).76 = Sample size using statistical sampling (students’answers will vary if nonstatistical sampling wasused in part b.300 = Interval (22839/76) if statistical sampling is used(students’ answers will vary if nonstatisticalsampling was used in part b).14825 = Random starting point.Select warehouse removal slip 14825 and every 300th warehouseremoval slip after (15125, 15425, etc.)Computer generation of random numbers using Excel (P1526.xls):=RANDBETWEEN(14682,37521)The command for selecting the random number can be entereddirectly onto the spreadsheet, or can be selected from the functionmenu (math & trig) functions. It may be necessary to add theanalysis tool pack to access the RANDBETWEEN function. Oncethe formula is entered, it can be copied down to select additionalrandom numbers.d. Other audit procedures that could be performed are:1. Test extensions on attached sales invoices for clericalaccuracy. (Accuracy)2. Test time delay between warehouse removal slip date andbilling date for timeliness of billing. (Timing)3. Trace entries into perpetual inventory records to determinethat inventory is properly relieved for shipments. (Postingand summarization)15-26 (continued)e. The test performed in part c cannot be used to test for occurrenceof sales because the auditor already knows that inventory wasshipped for these sales. To test for occurrence of sales, the salesinvoice entry in the sales journal is the sampling unit. Since thesales invoice numbers are not identical to the warehouse removalslips it would be improper to use the same sample.15-27a. It would be appropriate to use attributes sampling for all audit procedures except audit procedure 1. Procedure 1 is an analyticalprocedure for which the auditor is doing a 100% review of the entirecash receipts journal.b. The appropriate sampling unit for audit procedures 2-5 is a line item,or the date the prelisting of cash receipts is prepared. The primaryemphasis in the test is the completeness objective and auditprocedure 2 indicates there is a prelisting of cash receipts. All otherprocedures can be performed efficiently and effectively by using theprelisting.c. The attributes for testing are as follows:d. The sample sizes for each attribute are as follows:15-28a. Because the sample sizes under nonstatistical sampling are determined using auditor judgment, students’ answers to thisquestion will vary. They will most likely be similar to the samplesizes chosen using attributes sampling in part b. The importantpoint to remember is that the sample sizes chosen should reflectthe changes in the four factors (ARACR, TER, EPER, andpopulation size). The sample sizes should have fairly predictablerelationships, given the changes in the four factors. The followingreflects some of the relationships that should exist in student’ssample size decisions:SAMPLE SIZE EXPLANATION1. 90 Given2. > Column 1 Decrease in ARACR3. > Column 2 Decrease in TER4. > Column 1 Decrease in ARACR (column 4 is thesame as column 2, with a smallerpopulation size)5. < Column 1 Increase in TER-EPER6. < Column 5 Decrease in EPER7. > Columns 3 & 4 Decrease in TER-EPERb. Using the attributes sampling table in Table 15-8, the sample sizesfor columns 1-7 are:1. 882. 1273. 1814. 1275. 256. 187. 149c.d. The difference in the sample size for columns 3 and 6 result fromthe larger ARACR and larger TER in column 6. The extremely largeTER is the major factor causing the difference.e. The greatest effect on the sample size is the difference betweenTER and EPER. For columns 3 and 7, the differences between theTER and EPER were 3% and 2% respectively. Those two also hadthe highest sample size. Where the difference between TER andEPER was great, such as columns 5 and 6, the required samplesize was extremely small.Population size had a relatively small effect on sample size.The difference in population size in columns 2 and 4 was 99,000items, but the increase in sample size for the larger population wasmarginal (actually the sample sizes were the same using theattributes sampling table).f. The sample size is referred to as the initial sample size because itis based on an estimate of the SER. The actual sample must beevaluated before it is possible to know whether the sample issufficiently large to achieve the objectives of the test.15-29 a.* Students’ answers as to whether the allowance for sampling error risk is sufficient will vary, depending on their judgment. However, they should recognize the effect that lower sample sizes have on the allowance for sampling risk in situations 3, 5 and 8.b. Using the attributes sampling table in Table 15-9, the CUERs forcolumns 1-8 are:1. 4.0%2. 4.6%3. 9.2%4. 4.6%5. 6.2%6. 16.4%7. 3.0%8. 11.3%c.d. The factor that appears to have the greatest effect is the number ofexceptions found in the sample compared to sample size. For example, in columns 5 and 6, the increase from 2% to 10% SER dramatically increased the CUER. Population size appears to have the least effect. For example, in columns 2 and 4, the CUER was the same using the attributes sampling table even though the population in column 4 was 10 times larger.e. The CUER represents the results of the actual sample whereas theTER represents what the auditor will allow. They must be compared to determine whether or not the population is acceptable.15-30a. and b. The sample sizes and CUERs are shown in the following table:a. The auditor selected a sample size smaller than that determinedfrom the tables in populations 1 and 3. The effect of selecting asmaller sample size than the initial sample size required from thetable is the increased likelihood of having the CUER exceed theTER. If a larger sample size is selected, the result may be a samplesize larger than needed to satisfy TER. That results in excess auditcost. Ultimately, however, the comparison of CUER to TERdetermines whether the sample size was too large or too small.b. The SER and CUER are shown in columns 4 and 5 in thepreceding table.c. The population results are unacceptable for populations 1, 4, and 6.In each of those cases, the CUER exceeds TER.The auditor's options are to change TER or ARACR, increase the sample size, or perform other substantive tests todetermine whether there are actually material misstatements in thepopulation. An increase in sample size may be worthwhile inpopulation 1 because the CUER exceeds TER by only a smallamount. Increasing sample size would not likely result in improvedresults for either population 4 or 6 because the CUER exceedsTER by a large amount.d. Analysis of exceptions is necessary even when the population isacceptable because the auditor wants to determine the nature andcause of all exceptions. If, for example, the auditor determines thata misstatement was intentional, additional action would be requiredeven if the CUER were less than TER.15-30 (Continued)e.15-31 a. The actual allowance for sampling risk is shown in the following table:b. The CUER is higher for attribute 1 than attribute 2 because the sample sizeis smaller for attribute 1, resulting in a larger allowance for sampling risk.c. The CUER is higher for attribute 3 than attribute 1 because the auditorselected a lower ARACR. This resulted in a larger allowance for sampling risk to achieve the lower ARACR.d. If the auditor increases the sample size for attribute 4 by 50 items and findsno additional exceptions, the CUER is 5.1% (sample size of 150 and three exceptions). If the auditor finds one exception in the additional items, the CUER is 6.0% (sample size of 150, four exceptions). With a TER of 6%, the sample results will be acceptable if one or no exceptions are found in the additional 50 items. This would require a lower SER in the additional sample than the SER in the original sample of 3.0 percent. Whether a lower rate of exception is likely in the additional sample depends on the rate of exception the auditor expected in designing the sample, and whether the auditor believe the original sample to be representative.15-32a. The following shows which are exceptions and why:b. It is inappropriate to set a single acceptable tolerable exception rateand estimated population exception rate for the combinedexceptions because each attribute has a different significance tothe auditor and should be considered separately in analyzing theresults of the test.c. The CUER assuming a 5% ARACR for each attribute and a samplesize of 150 is as follows:15-32 (continued)d.*Students’ answers will most likely vary for this attribute.e. For each exception, the auditor should check with the controller todetermine an explanation for the cause. In addition, the appropriateanalysis for each type of exception is as follows:15-33a. Attributes sampling approach: The test of control attribute had a 6% SER and a CUER of 12.9%. The substantive test of transactionsattribute has SER of 0% and a CUER of 4.6%.Nonstatistical sampling approach: As in the attributes samplingapproach, the SERs for the test of control and the substantive testof transactions are 6% and 0%, respectively. Students’ estimates ofthe CUERs for the two tests will vary, but will probably be similar tothe CUERs calculated under the attributes sampling approach.b. Attributes sampling approach: TER is 5%. CUERs are 12.9% and4.6%. Therefore, only the substantive test of transactions resultsare satisfactory.Nonstatistical sampling approach: Because the SER for the test ofcontrol is greater than the TER of 5%, the results are clearly notacceptable. Students’ estimates for CUER for the test of controlshould be greater than the SER of 6%. For the substantive test oftransactions, the SER is 0%. It is unlikely that students will estimateCUER for this test greater than 5%, so the results are acceptablefor the substantive test of transactions.c. If the CUER exceeds the TER, the auditor may:1. Revise the TER if he or she thinks the original specificationswere too conservative.2. Expand the sample size if cost permits.3. Alter the substantive procedures if possible.4. Write a letter to management in conjunction with each of theabove to inform management of a deficiency in their internalcontrols. If the client is a publicly traded company, theauditor must evaluate the deficiency to determine the impacton the auditor’s report on internal control over financialreporting. If the deficiency is deemed to be a materialweakness, the auditor’s report on internal control wouldcontain an adverse opinion.In this case, the auditor has evidence that the test of control procedures are not effective, but no exceptions in the sampleresulted because of the breakdown. An expansion of the attributestest does not seem advisable and therefore, the auditor shouldprobably expand confirmation of accounts receivable tests. Inaddition, he or she should write a letter to management to informthem of the control breakdown.d. Although misstatements are more likely when controls are noteffective, control deviations do not necessarily result in actualmisstatements. These control deviations involved a lack ofindication of internal verification of pricing, extensions and footingsof invoices. The deviations will not result in actual errors if pricing,extensions and footings were initially correctly calculated, or if theindividual responsible for internal verification performed theprocedure but did not document that it was performed.e. In this case, we want to find out why some invoices are notinternally verified. Possible reasons are incompetence,carelessness, regular clerk on vacation, etc. It is desirable to isolatethe exceptions to certain clerks, time periods or types of invoices.Case15-34a. Audit sampling could be conveniently used for procedures 3 and 4 since each is to be performed on a sample of the population.b. The most appropriate sampling unit for conducting most of the auditsampling tests is the shipping document because most of the testsare related to procedure 4. Following the instructions of the auditprogram, however, the auditor would use sales journal entries asthe sampling unit for step 3 and shipping document numbers forstep 4. Using shipping document numbers, rather than thedocuments themselves, allows the auditor to test the numericalcontrol over shipping documents, as well as to test for unrecordedsales. The selection of numbers will lead to a sample of actualshipping documents upon which tests will be performed.c. Note: The sampling data sheet that follows assumes an attributessampling approach. The only difference between the sampling datasheet for attributes sampling and for nonstatistical sampling is theactual determination of sample size. For nonstatistical sampling,students’ answers will vary, but will most likely be comparable tothe sample sizes determined under attributes sampling.。