A model for the generic alpha relaxation of viscous liquids

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A remark on the Chisini conjecture

The precise assertion of Theorem 1 is therefore that two generic morphisms (of a priori diﬀerent surfaces!) having the same branch curve are equivalent provided that at least one of them has degree ≥ 12.
which leads to an a priori upper bound for the right hand side of (1). To this
end we shall need the following formulas (cf. Lemmas 6 and 7 in [4]) for the
The conjecture that a generic morphism of degree at least 5 is completely determined by its branch curve was proposed by O. Chisini [2], who also gave an alleged proof of this statement.
2
2. Cusps of branch curves and BMY inequalities. Consider a generic
Hale Waihona Puke morphism f : S → P2 of degree deg f = N with branch curve B ⊂ P2.
Denote by 2d the degree of B (it is always even), by g the genus of the

adopt a different strategy for al model -回复

adopt a different strategy for al model -回复采用不同策略的AL模型引言：自动学习（Autonomous Learning, AL）是一种机器学习方法，它可以让计算机根据经验自动进行学习和优化。

根据不同的应用场景，我们可以选择不同的策略来训练和优化AL模型。

本文将介绍一个实用的AL模型，以及一种采用不同策略的方法来训练和优化该模型。

第一部分：AL模型的基本原理和应用领域在开始讨论策略之前，我们先来了解一下AL模型的基本原理和应用领域。

AL模型是一种无监督学习方法，它可以从无标签的数据中学习知识和模式。

它可以应用于各个领域，如图像识别、自然语言处理和推荐系统等。

第二部分：采用传统机器学习策略的AL模型的训练和优化方法传统的AL模型的训练和优化方法主要包括以下几个步骤：1. 数据准备：选择一个合适的数据集，并对数据进行预处理和特征提取，以使其适合AL模型的训练和优化。

2. 模型选择：选择一个合适的AL模型，如自动编码器或生成对抗网络等。

3. 模型训练：使用数据集对选定的AL模型进行训练。

训练过程可以采用传统的机器学习算法，如梯度下降法或随机森林等。

4. 模型优化：通过调整模型的参数和超参数来优化AL模型的性能。

可以使用交叉验证或网格搜索等方法来确定最佳的参数和超参数组合。

第三部分：采用不同策略的AL模型的训练和优化方法除了传统的机器学习策略外，我们还可以采用其他策略来训练和优化AL模型。

以下是一些常用的策略：1. 强化学习策略：通过与环境的交互来训练和优化AL模型。

可以采用深度强化学习方法，如Q学习或策略梯度等。

2. 迁移学习策略：利用已经学习到的知识和模型来加速AL模型的训练和优化。

可以使用预训练模型或领域适应方法等。

3. 元学习策略：通过学习学习的方式来训练和优化AL模型。

可以使用元优化算法或自适应学习方法等。

采用不同策略的AL模型的训练和优化方法与传统的机器学习策略类似，但在具体实施中有一些差异。

严选题第一章第七题

严选题第一章第七题英文回答：In the realm of artificial intelligence, questions such as "What is it, anyway?" and "What are its capabilities?" arise. To unravel the true nature of AI, we embark on a journey to explore its various dimensions.Cognitive abilities: AI systems exhibit impressive cognitive capabilities that mimic human intelligence. They possess learning algorithms that enable them to process vast amounts of data, detect patterns, and make informed predictions. Moreover, they can engage in natural language processing, understanding and generating human language with remarkable accuracy.Machine learning: At the heart of AI's cognitive prowess lies machine learning. This technology empowers AI systems to learn from data without explicit programming. Supervised learning trains models on labeled data, whileunsupervised learning uncovers hidden patterns in unlabeled data. Reinforcement learning rewards desired behaviors, fostering efficient decision-making.Computer vision: AI's ability to "see" and interpret visual information is known as computer vision. Convolutional neural networks (CNNs) enable AI systems to extract meaningful features from images, empowering themwith object recognition, facial detection, and sceneanalysis capabilities.Natural language processing: Natural languageprocessing (NLP) bridges the gap between humans and machines by allowing AI systems to understand and generate human language. NLP tasks encompass machine translation,text summarization, and sentiment analysis, facilitating seamless communication between humans and AI.Robotics: AI plays a pivotal role in robotics, enabling machines to navigate their surroundings, manipulate objects, and interact with the physical world. AI-powered robots possess autonomous navigation, object manipulation, andperception capabilities, paving the way for advancements in manufacturing, healthcare, and exploration.Applications and impacts: AI's versatility extends across a wide range of disciplines, leaving an indelible mark on fields such as healthcare, finance, and transportation. In healthcare, AI assists in medical diagnosis, drug discovery, and personalized treatment plans. In finance, it automates fraud detection, risk assessment, and portfolio management. In transportation, AI powersself-driving cars, traffic optimization, and logistics planning.Ethical considerations: As AI's capabilities continueto expand, so too must we consider its ethical implications. Concerns regarding privacy, bias, and accountability arise, necessitating the development of ethical guidelines and responsible AI practices.The future of AI: The future of AI holds boundless possibilities, with ongoing research and development promising even more advanced and multifaceted capabilities.AI's integration into various industries will continue to revolutionize our lives, ushering in an era of unprecedented technological progress.中文回答：什么是人工智能？人工智能（Artificial Intelligence，简称AI）是一门计算机科学分支，旨在创建能够执行通常需要人类智能的任务的计算机系统。

Hilbert functions of Gorenstein algebras associated to a pencil of forms

a r X i v :m a t h /0412361v 1 [m a t h .A C ] 18 D e c 2004Hilbert functions of Gorenstein algebras associated to apencil of formsAnthony IarrobinoDepartment of Mathematics,Northeastern University,Boston,MA 02115,USA.December 17,2004Abstract Let R be a polynomial ring in r variables over an inﬁnite ﬁeld K ,and denote by D a corresponding dual ring,upon which R acts as diﬀerential operators.We study type two graded level Artinian algebras A =R/I ,having socle degree j .For each such algebra A ,we consider the family of Artinian Gorenstein [AG]quotients of A having the same socle degree.By Macaulay duality,A corresponds to a unique 2-dimensional vector space W A of forms in D j ,and each such AG quotient of A corresponds to a form in W A -up to non-zero multiple.For W A = F,G ,each such AG quotient A λcorresponds to an element of the pencil of forms (one dimensional subspaces)of W A :given F λ=F +λG,λ∈K ∪∞we have A λ=R/Ann (F λ).Our main result is a lower bound for the Hilbert function H (A λgen )of the generic Gorenstein quotient,in terms of H (A ),and the pair H F =H (R/Ann F )and H G =H (R/Ann G ).This result restricts the possible sequences H that may occur as the Hilbert function H (A )for a type two level algebra A .1Introduction Let R =k [x 1,...,x r ]be the polynomial ring in r variables,over an inﬁnite ﬁeld K .We will assume also for simplicity of exposition that char K =0,but all statements may be extended suitably to characteristic p (see Remark 2.10).We will consider only graded Artinian quotients A =R/I of R ,and we denote by m the irrelevant maximal ideal m =(x 1,...,x r ).We denote by A i the i -th graded component of A .Recall that the socle Soc(A )of A satisﬁes Soc(A )=(0:m )={f ∈A |mf =0},(1.1)and the type of A is t (A )=dim K Soc(A ).We will denote by j =j (A )the socle degree of A ,the largest integer such that A j =0,but A j +1=0.An Artinian algebra A =R/I of socle degree j is level ,if any of the following equivalent conditions holdi.Soc(A )=A j ,ii.The canonical module Hom(A,K )of A is generated in a single degree,iii.Each I i ,0≤i ≤j can be recovered from I j as follows:for 0<i ≤j I i =I j :R j −i ={f ∈R i |R j −i ·f ⊂I j }.(1.2)Recall that A is Artinian Gorenstein if A is level of type one.Our main objects of study here will be type two level algebras A and their Artinian Gorenstein [AG]quotients.11.1Recent results on level algebrasWeﬁrst brieﬂy recall some recent work on graded level algebras.First,the Artinian Gorenstein al-gebras have been the object of much study.For height three,the structure theorem of Buchsbaum-Eisenbud[BuEi]has led not only to a characterization of the Hilbert functions that may occur,but also to the irreducibility[Di]and smoothness[Kle2]of the family PGOR(H)parametrizingArtinian Gorenstein quotients of R,r=3,of a given(symmetric)Hilbert function H.A second proof of smoothness follows from results of M.Boij and A.Conca-G.Valla(see[Bj4,CoVa],and[IK,§4.4]for a survey of related topics).One line of study relates punctual subschemes Z of P r−1 to Gorenstein Artinian quotients of their coordinate rings O Z[Bj2,G,EmI,IK,Kle3].Classicalapolarity,or the inverse systems of Macaulay provide a connection to sums of powers of linearforms,and to a classical Waring problem for forms(see[Te,EmI,G,IK]).When r≥4PGOR(H)often has several irreducible components,a factﬁrst noted by M.Boij[Bj3],and elaborated by others(see[IS,Kle3]).The set of Gorenstein sequences—ones that occuras Hilbert functions of Artinian Gorenstein quotients of R—is not known for r≥4;for r≥5 they include non-unimodal sequences,with several maxima.Level algebras A of types t(A)>1are a natural next topic of study after the Gorenstein algebras,particularly in the low embedding dimensions r≤3or even r=4,where the families ofArtinian Gorenstein quotients are better understood.When r=2,the family LevAlg(H)is wellunderstood(see[I3,ChGe]):these families are smooth,of known dimension,and their closures are the union H′≤H LevAlg(H′)of similar strata for termwise no greater Hilbert functions of the same socle degree.When r=3,and t=2,tables of possible H for small socle degree j,possibleresolutions,and many methods that are more general are given in[GHMS];this case is also studied by F.Zanello[Za1].However the possible sequences H are not known when r≥3even in the case t=2;and although specialists believe there should be cases where LevAlg(H)for r=3,t=2have several irreducible components,this problem is still open.There has been work connecting these results with the simultaneous Waring problem for binary and ternary forms[Car,CarCh,I3].Several authors have studied the extremal Hilbert functions for level algebras of given em-bedding dimension r,type t,and socle degree j[BiGe,ChoI].The minimal resolutions for compressed level algebras(those having maximum Hilbert function given(r,t,j))are studied in [Bj1,MMN,Za1].Also Zanello has obtained results about extremal Hilbert functions for level algebras,given the pair H j−1and H j=2[Za2].For r≥3there is much to be learned about families of level algebras of given Hilbert functions,even when t=2.Certainly,pencils of curves on P2have been long a topic of geometric study;however,a stronger connection(but see[ChGe]) has yet to be made between on the one hand the traditional geometrical approach to pencils of curves and their singularities,and on the other hand the study of the level algebras associated to these pencils.In this article we show some inequalities connecting the Hilbert function of type two levelalgebras,and the Hilbert function of their Artinian Gorenstein quotients.These results had been embedded in the longer preprint[I4],which will now be refocussed on reﬁnements of the numerical results and on parametrization.In section1.2we give notation,and brieﬂy state the main results,and in section1.3we presentfurther context,including the questions that motivated us.In Section2we prove our results andgive examples.1.2Inverse systemsLet D=K[X1,...,X r]denote a second polynomial ring.The ring R acts on D as partial diﬀer-ential operators:for h∈R,F∈Dh◦F=h(∂/∂X1,...,∂/∂X r)◦F.(1.3)2The pairingσj:R j×D j→K,σj(h,F)=h◦F,(1.4) is exact.This is the apolarity or Macaulay duality action of R on D.A type t level algebra A=R/I of socle degree j corresponds via the Macaulay duality to a unique t-dimensional vector space W A,W A={F∈D j|I◦F=0}={F∈D j|I j◦F=0}.(1.5) Thus W A is the perpendicular space to I j in the exact duality between R j and D j,and R◦W A, may be regarded as the dualizing moduleˆA=Hom(A,K)to A.The Hilbert function H(A)is the sequence H(A)i=dim K A i.We have,for0≤u≤j,R u◦W A=I⊥j−u⊂D j−u,and(1.6)H(A)j−u=dim K R u◦W A.(1.7)Remark1.1.A one-dimensional subspace E⊂W A corresponds to an Artinian Gorenstein[AG] quotient R/Ann E of the level algebra A,having the same socle degree j as A.We parametrize these spaces E as points of the projective space P(W A)associated to W A.The Hilbert function H E=H(R/Ann E)is evidently semicontinuous:H E>T termwise for someﬁxed sequence T=(t0,...,t j)deﬁnes an open subset of P(W A)since dim R u◦E>t j−u is an open condition. Thus,among the Hilbert functions of Gorenstein quotients of A having the same socle degree j, there is a termwise maximum H(E gen),that occurs for E belonging to an open dense subset of the projective variety P(W A).When the type of A is two,then W A= F,G is two-dimensional;the one-dimensional sub-spaces constitute a pencil of forms Fλ=F+λG,λ∈K∪∞=P1K=P(W A)(here we set F∞=G).Each AG quotient algebra Aλ=R/Ann(F+λG)has the same socle degree j=j(A) as A,and these comprise all the Gorenstein quotients of A having socle degree j(A).Thus,the family Aλ,λ∈P1is the pencil of Artinian Gorenstein quotients associated to the pencil F+λG. We let A F=R/Ann F and A G=R/Ann G,and set H F=H(A F),H G=H(A G).We focus here on the type two case,and on the pencil of degree-j homogeneous forms or hypersurfaces,Fλ∈D j and their symmetric Hilbert functions Hλ=H(Aλ).By(1.7)(Hλ)j−u is the dimension of the space of order u partial derivatives of Fλ.Evidently,the set of Hilbert functions H(Aλ)that occur is a PGL(r−1)invariant of the level algebra A.Also,Remark1.1),that occurs for an open dense set of implies that there is a termwise maximum value H(Aλgenλ∈P1.We now state the most important part of our main result,Theorem2.2.For i=j−u,we letd i(F,G)=dim K R u◦F ∩ R u◦G =(H F)i+(H G)i−H(A)i,be the overlap dimension,satisfying d i(F,G)=H(R/(Ann F+Ann G))i(see equation(2.4)ﬀ). Theorem.Let A=R/Ann(F,G)be a type two level algebra of socle degree j.For all pairs (u,i=j−u)satisfying0<u<j we have)i.(1.8)H(A)u−d i(F,G)≤H(Aλgen)that depends only on H(A).In Theorem2.4we give a lower bound for H(Aλgen1.3Questions and examples:pencils of formsWe oﬀer some questions about pencils of forms and the Hilbert functions they determine,and state their status.This provides some further context for our work,and as well we pose open problems.3Question1.2.What are natural invariants for pencils of forms?i.What sequences H occur as Hilbert functions H(A)?Status:Open for r≥3,even for t=2,but see[GHMS,Za1].ii.Is there a sequence H=(1,3,...,2,0),such that LevAlg(H)has two irreducible components?Status:Open.The answer to the analogous question is”no”for embedding dimension r=2, and”yes”for r≥4.iii.Can we use our knowledge of the Hilbert functions and parameter spaces for Artinian Goren-stein algebras,to study type two level algebras A of embedding dimensions three and four?Status:This has been the main approach to classifying type two level algebras.See[GHMS] and[Za1,Za2],as well as Lemma2.6,Examples2.7and2.8below.iv.Given a type two Artinian algebra A,consider the pencil of Gorenstein quotients Aλ= R/Ann(F+λG)having the same socle degree as A.What can be said about the Hilbert functions H(Aλ)?Status:We begin a study here.See also[I4,Za1,Za2].The Question1.2about natural invariants of A connects also with classical invariant theory, but we do not pursue this here:see[DK]and[RS]for analogous connections in the Gorenstein case.The following example illustrates Question1.2(iv),and as well the main result.Example1.3.Let r=2.F=X4,G=XY3,then H F=(1,1,1,1,1),H G=(1,2,2,2,1),the ideal I=Ann(F,G)=Ann F∩Ann G=(x2y,y4,x5).The type two level algebra A=R/I has Hilbert function H(A)=(1,2,3,3,2).The dualizing moduleˆA=R◦ F,G ⊂D satisﬁesˆA= 1;X,Y;X2,Y X,Y2;X3,Y3,XY2;X4,XY3 .The Gorenstein quotients AλsatisfyH(Aλ)=(1,2,3,2,1)forλ=∞,0.This is the maximum possible(so compressed)Hilbert function for a Gorenstein Artinian quotient of R having socle degree4.The following speciﬁc question arose from a discussion with A.Geramita about the Hilbert functions possible for type two level algebras.It was the starting point of our work here.Question1.4.Let F,G be two degree-j homogeneous polynomials,elements of D=K[X1,...,X r], such thati.F,G together have at least2r−2linearly independentﬁrst partial derivatives,andii.F,G together involve all r variables:this is equivalent to the(j−1)-order partials of F,G spanning X1,...,X r .Does some linear combination Fλ=F+λG have r linearly independentﬁrst partial derivatives?We answer this question positively in Corollary2.5.Here are two examples to illustrate.Example1.5.Let r=3,j=4,F=X4+Y4,G=(X+Y)4+Z4.Then the pencil V= F,G involves all three variables,and these forms together have four linearly independentﬁrst partials X3,Y3,(X+Y)3,Z3.For allλ=0the form Fλ=F+λG has three linearly independentﬁrst partials.4Example1.6.Let r=3,j=4,F=XZ3,G=Y Z3,Then V= F,G involves all three variables and these forms together have only3=2r−3linearly independentﬁrst partials.Each Fλhas only2linearly independentﬁrst partials.Thus,the hypothesis in Question1.4that V has at least 2r−2linearly independentﬁrst partial derivatives is necessary for the desired conclusion.Our work here is focussed primarly on the following question.Question1.7.Given Hilbert functions H F=H(R/Ann F),and H G=H(R/Ann G)for two degree-j forms F,G∈D,or given H(A),A=R/Ann(F,G),determine the possible Hilbert functions H(Aλ)for Aλ=R/Ann Fλ,Fλ=F+λG?Are there numerical restrictions on the generic value H(Aλgen)?It is easy to give a partial answer.Evidently,by(2.1),we can’t have two values ofλwithH(Aλ)i<H(A)i/2.(1.9) We may conclude that small H(Aλ)are rare,given H(A)!In our main results we show that if H F and H G are small in comparison with H(A),thenH(Aλgen )is large(Theorem2.2).We then show a lower bound for H(Aλgen)in terms of H(A)(Theorem2.4).Several examples illustrate the results(see especially Examples2.7,2.8).Exam-ple2.3gives a pencil Aλof Gorenstein Artinian quotients not having a minimum Hilbert function; and Example2.9gives a compressed type two Artinian level algebra A such that Aλgenis not compressed Gorenstein.2Hilbert Functions for pencils of formsIn this section we state and prove our main results.Weﬁrst give an exact sequence relating A F,A G,and A.We deﬁne R-module homomorphismsι:A→R/Ann F⊕R/Ann Gι(f)=(f mod Ann F,−f mod Ann G)π:R/Ann F⊕R/Ann G→R/(Ann F+Ann G):π(a,b)=(a+b)mod(Ann F+Ann G).Lemma2.1.Let F,G∈D j determine a type two level Artinian quotient A=R/I,I=Ann(F,G) of R.There is an exact sequence of R-modules0→Aι−→R/Ann F⊕R/Ann Gπ−→R/(Ann F+Ann G)→0,(2.1) whose dual exact sequence is0→ R◦F ∩ R◦G π∗−→ R◦F ⊕ R◦G ι∗−→R◦ F,G →0.(2.2) Proof.Since the duality between R j and D j is exact,we have(Ann F+Ann G)⊥= R◦F ∩ R◦G .(2.3) Thus the two sequences are dual.Evidentlyιis an inclusion andπis a surjection.The kernel of ι∗consists of pairs(h1◦F,h2◦G)such that h1◦F−h2◦G=0;this is evidently the image ofπ∗, so the sequences are exact.We let J=Ann F+Ann G:it depends of course upon the choice of the pair(F,G)∈ F,G . We denote by H(R/J)=(1,d1,...,d j)the Hilbert function H(R/J)where d i=d i(F,G).Thus5we have from(2.3),that,letting i=j−u,the integer d i measures the overlap in degree i between the inverse systems determined by F and G:d i=dim K R u◦F ∩ R u◦G (2.4)=dim K R u◦F+dim K R u◦G−H(A)i(2.5)=(H F)i+(H G)i−H(A)i.(2.6) The equalities(2.5),(2.6)are immediate from(1.7)and(2.2).We set,again letting i=j−u,t i=dim K(((Ann F)u◦G)∩((Ann G)u◦F)),(2.7) where t i=t i(F,G)depends on the pair(F,G).Recall from Remark1.1that“genericλ”refers to a suitable open dense set ofλ∈P1,that is,to all but aﬁnite number of values ofλ.We denote by d i,t i the integers d i(F,G)and t i(F,G)deﬁned just above.The following main result shows that a)i.small overlap between R u◦F and R u◦G implies a large value for H(AλgenTheorem2.2.Let A=R/Ann(F,G)be a type two level algebra of socle degree j.Ifλis generic, then for all pairs(u,i=j−u)satisfying0<u<j we haveH(A)u−d i≤H(Aλ)i≤H(A)u−t i.(2.8) The upper bound on H(Aλ)holds for allλ=0,∞.Proof.Fix for now,and through the proof of the Claim below,an integer u satisfying0<u<j. By“dim V”below we mean dim K V.Let C u⊂R u be a vector subspace complement to(Ann F)u, so C u⊕(Ann F)u=R u.Let d=d i and let e=dim((Ann(F,G))u):so dim A u=dim R u−e, and let B⊂(Ann F)u be the vector subspace satisfyingB={h∈(Ann F)u|h◦G∈R u◦F}.(2.9) The homomorphism h→h◦g,h∈B,g∈G induces a short exact sequence0→(Ann(F,G))u→B→ R u◦F ∩ R u◦G ,implyingdim B≤d+e.(2.10) Since(Ann F)u◦(F+λG)=(Ann F)u◦G,we haveR u◦(F+λG)=C u◦(F+λG)+(Ann F)u◦G.(2.11) Claim.For genericλdim R u◦(F+λG)≥dim(C u◦F+(Ann F)u◦G)(2.12)=dim C u+dim(Ann F)u−dim B(2.13)=dim R u−dim B≥dim R u−(d+e)=dim A u−d,(2.14) so dim R u◦(F+λG)≥dim A u−d.Proof of Claim.The key step is(2.12),which results from(2.11)and deformation.For the space C u◦(F+λG)+(Ann F)u◦G in(2.11)is a deformation of the space C u◦F+(Ann F)u◦G on the6right of(2.12),and dimension is a semicontinuous invariant.The other steps are straightforward.The Claim shows the left hand inequality in(2.8)for a speciﬁc u.Since P1K is irreducible,the intersection of the dense open subsets of P1K over which the left side of(2.8)is satisﬁed for each u,0<u<j,is itself a dense open subset,completing the proof that the left side of(2.8)holds simultaneously for genericλand all such u.Supposeλ=0,∞.Let C′u be a complement in R u to J u=(Ann F)u+(Ann G)u.Then we haveR u◦(F+λG)=C′u◦(F+λG)+(Ann F)u◦G+(Ann G)u◦F implying dim R u◦(F+λG)≤dim k R u−dim(Ann F)u∩(Ann G)u)−t i=dim A u−t i.This completes the proof of Theorem2.2. Example2.3(No minimum H(Aλ)).Let r=3,G=X8+Y4Z4,and F=L81+···+L85, where the L i=a i1X+a i2Y+a i3Z are general enough linear forms,elements of D1.Here“general enough”means that their coeﬃcients{a ij∈K}lie in the open dense subset of the aﬃne spaceA15such that the powers L j−u1,...,L j−u5are linearly independent in D j−u and maximally disjointfrom R u◦G,2≤u≤6(see[I2]).Then we have for3≤u≤6R u◦F= L j−u1,...,L j−u5,(2.15)satisfying dim k R u◦F=5.This determines H F,and we haveH F=(1,3,5,5,5,5,5,3,1)H G=(1,3,4,5,6,5,4,3,1)H(A)=(1,3,6,10,11,10,9,6,2)=H F+h H G,where by H F+h H G we mean the sequence satisfying(H F+h H G)i=min{dim R i,(H F)i+(H G)i}.(2.16) Theorem1implies that H(Aλgen)=(1,3,6,10,11,10,6,3,1).It is easy to check that there are no other values ofλother than0,∞(corresponding to F,G)such that H(Aλ)is smaller than H(Aλgen),and hence no minimum sequence H(Aλ),since H F and H G are incomparable.Our second main result gives a lower bound for H(Aλgen)solely in terms of H(A). Theorem2.4.Let A be a type two level Artinian algebra of socle degree j,and let u,i satisfy 0<u≤i=j−u.Assume that H(A)i≥2H(A)u−2−3δu,whereδu≥0andδu is an integer. ThenH(Aλgen)i≥H(A)u−δu.(2.17) Proof.Assume the hypotheses of the Theorem,and suppose by way of contradiction,that for genericλthere is an integer a≥0satisfyingH(Aλ)i=H(A)u−δu−1−a.(2.18) Take two generic forms F′,G′in the pencil.Then the overlap between R u◦F′and R u◦G′(see(2.4),(2.5))satisﬁesd i=2(H(A)u−δu−1−a)−H(A)i≤2H(A)u−2δu−2−2a−(2H(A)u−2−3δu)≤δu−2a.(2.19)7By Theorem2.2,for genericλthe AG quotient A′λ=R/Ann(F′+λG′)satisﬁesH(A′λ)i≥H(A)u−(δu−2a),(2.20) a contradiction with equation(2.18).It follows that the assumed equation(2.18)is false,henceH(Aλ)i≥H(A)u−δu(2.21) which is Theorem2.4.Note.We assumed in the statement and proof of Theorem2.4thatδu is an integer.Alternatively we could deﬁneδ′u=(2H(A)u−2−H(A)i)/3and conclude that H(Aλ)i≥H(A)u−⌈δ′u⌉whengen)i≥H(A)u otherwise.δ′u≥0,and H(AλgenIn the following corollary we give a positive answer to Question1.4.Corollary2.5.Let F,G together have at least2r−2linearly independentﬁrst partial derivatives, and suppose that F,G involve all r variables.Then dim K R1◦(F+λG)=r for genericλ. Proof.Take i=j−1,u=1,δ1=0in Theorem2.4.From the assumptions we haveH(A)j−1≥2r−2=2H(A)1−2,which implies by Theorem2.4that for genericλthe dimension H(Aλ)j−1=r.In order to apply Theorem2.4most eﬀectively,we use the following result from[GHMS].For a sequence H=(1,...,H j),H j>0we denote by Hˆthe reverse sequence Hˆi=H j−i,1≤i≤j. Recall that an O-sequence is one that is the Hilbert function of some Artinian algebra[Mac2,BrH]. Lemma2.6.(A.Geramita et al[GHMS])Let A be a type t level algebra with dualizing module ˆA.Let A=R/Ann W,W=W⊂D j,dim K W=t≥2and let V⊂W be a vector subspace ofAcodimension one.Then there is an exact sequence of R-modules relating the type t−1level algebra B V=B/Ann V to A,0→C→A→B V→0(2.22) whose dual exact sequence of R submodules of D is0→ˆB V→ˆA→ˆC→0.(2.23) HereˆC is a simple R-submodule(single generator).We have for their Hilbert functionsH(A)=H(B V)+H(C),and the reverse sequence H(C)ˆ=(1,...)is an O-sequence.Proof.Let F∈V span a complement to V in W A.Let the homomorphismτ(F,W,V):R→R◦F/ R◦W∩R◦F →0have kernel S.The moduleˆC is isomorphic to R/S.This shows(2.23)and thatˆC is simple.Example2.7.Let r=3and suppose A is a type two level algebra satisfying H(A)=(1,3,...,4,2). Then the pencil W A= F,G ⊂D j deﬁning A may be chosen so that H F=(1,3,...,3,1)andH(A)≤(1,3,6,10,...,6,3,1)+h(1,1,...,1),(2.24)8(see(2.16)for the sum+h used above).In particular(1,3,...,8,4,2)and(1,3,...,12,7,4,2)are not sequences possible for the Hilbert function of a level algebra quotient of R=K[x,y,z].Here in equation(2.24),the sequence(1,3,6,...,3,1)is the compressed Gorenstein sequence of socle degree j(see below).Here the Corollary2.5implies that we may choose G∈W A such that H G=(1,3,...,3,1);it follows that H(ˆC)=(1,1,...),so to be an O-sequence,H(ˆC)≤(1,1,...,1);this and(2.23)show (2.24).F.Zanello has extended this kind of result,and in[Za1]shows sharp upper bounds for the Hilbert function H(A)for type two level algebra quotients of R in r-variables given H(A)j−1. Example2.8.Let r=3and let H=(1,3,6,8,6,4,2).Considerﬁrst W1= F,G where H F= (1,3,5,7,5,3,1),and G=L6,L a general enough linear form(element of D1),so H G=(1,1,...,1). Then A1=R/Ann W1is easily shown to have Hilbert function H,as,choosing Fﬁrst,and G second,R u◦G= L6−u and is linearly disjoint in general from R u◦G,by the spanning property of the rational normal curve:powers of linear forms span D i,i=j−u(see[I2]).Next,let W2= F′,G′ where F′is a general element of K[X,Y]6,and G′is a general enough element of K[X+Y,Z]6.Then H F=H G=(1,2,3,4,3,2,1)and A2=R/Ann W2also has Hilbert function H.In either case,Theorem2.2implies thatH(R/Ann Fλgen )=H(R/Ann F′λgen)=(1,3,6,8,6,3,1).In this example,it is the non-generic elements of each pencil—the“unexpected properties”—that serve to distinguish the pencil.A compressed level algebra of given type t,socle degree j,and embedding dimension r is one having the maximum possible Hilbert function given those integers(see[I1,FL,Bj1,MMN]).The following example responds negatively in embedding dimension three to a question of D.Eisenbud,B.Ulrich,and C.Huneke.They asked if a generic socle degree-j AG quotient Aλgen of a compressed type two level algebra A,must also be compressed.This is true for r=2.Example2.9.Let r=3,j=4,take a,b∈K,(a,b)=(0,0),and setF=X3Y+X2Z2+aXZ3+bY Z3,G=X3Z+X2Y2+X2Y Z+3aXY2Z+bY3Z. Here H(A)=(1,3,6,6,2),and is compressed,but(y2−λz2)◦(F+λG)=0and we have that for genericλ,H(Aλ)=(1,3,5,3,1),which is not the compressed sequence(1,3,6,3,1).Note that in applying Theorem2.4with i=2,j=4here we would have6=H(A)2=2H(A)4−2−2−3(4/3),so takingδ=2in equation(2.17),we would conclude only that H(Aλgen)2≥6−2=4. Remark2.10(Characteristic p).Assume that K is an inﬁniteﬁeld ofﬁnite characteristic p. For p>j,the socle degree,there is no diﬀerence in statements.For p≤j one must use the divided power ring in place of D,and the action of R on D is the contraction action.With that substitution,the lemmas and theorems here extend to characteristic p.However,in examples,one must substitute divided powers for regular powers—see[IK,Appendix A]for further discussion. Acknowledgment.This article began after a conversation with Tony Geramita,in Fall2002 during a visit to the MSRI Commutative Algebra Year.We noted that in tables he and colleagues had calculated for Hilbert functions of type two algebras(see[GHMS]),that Corollary2.5was satisﬁed.I believed there would be a general result,which turned out to be Theorem2.2.I thank Tony for this discussion,and for helpful comments.I am appreciative to the organizers of the Siena conference“Projective Varieties with Unexpected Properties”for hosting a lively and informative meeting,and the impetus to write a concise presentation of the results;and I thank the referee.9References[BiGe]Bigatti A.,Geramita A.:Level algebras,lex segments,and minimal Hilbert functions, Comm.Alg.33(3),2003,1427-1451.[Bj1]Boij M.:Betti numbers of compressed level algebras,J.Pure Appl.Algebra134 (1999),no.2,111–131.[Bj2]:Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function,Paciﬁc J.Math.187(1999),1–11.[Bj4],Chipalkatti J.:On Waring’s problem for several algebraic ment.Math.Helv.78(2003),no.3,494–517.[ChGe]Chipalkatti J.,Geramita A.:On parameter spaces for Artin level algebras,Michigan Math.J.51(2003),no.1,187–207.[ChoI]Cho Y.,Iarrobino,A.:Hilbert functions of level algebras,Journal of Algebra241 (2001),745–758.[CoVa]Conca A.,Valla G.:Hilbert functions of powers of ideals of low codimension,Math.Z.230(1999),no.4,753–784.[Di]Diesel S.J.:Some irreducibility and dimension theorems for families of height3 Gorenstein algebras,Paciﬁc J.Math.172(1996),365–397.[DK]Dolgachev,I.,Kanev V.:Polar covariants of plane cubics and quartics,Advances in Math.98(1993),216–301.[EmI]Emsalem J.,Iarrobino A.:Inverse system of a symbolic power I,J.Algebra174 (1995),1080-1090.[FL]Fr¨o berg R.,Laksov D.:Compressed algebras,Conf.on Complete Intersections in Acireale,(S.Greco and R.Strano,eds),Lecture Notes in Math.#1092,Springer-Verlag,Berlin and New York,1984,pp.121–151.[G]Geramita,A.V.:Inverse systems of fat points:Waring’s problem,secant varieties ofVeronese varieties and parameter spaces for Gorenstein ideals.The Curves Seminarat Queen’s,Vol.X(Kingston,ON,1995),2–114,Queen’s Papers in Pure and Appl.Math.,102,Queen’s Univ.,Kingston,ON,1996.10[GHMS]:Compressed algebras:Artin algebras having given socle degrees and maximal length,Transactions of the A.M.S.,vol.285,no.1pp.337-378,1984.[I3]:Pencils of forms and level algebras,in preparation.[IK],Srinivasan H.:Artinian Gorenstein algebras of embedding dimension four: Components of PGOR(H)for H=(1,4,7,...,1),preprint,2004,to appear,J.Pureand Applied Algebra.[Kle1]Kleppe J.O.:Deformations of graded algebras,Math.Scand.45(1979)205–231. [Kle2]:Maximal Families of Gorenstein algebras,preprint,2004,to appear,Trans-actions A.M.S.[Mac1]Macaulay F.H.S.:The Algebra of Modular Systems,Cambridge Univ.Press,Cam-bridge,U.K.(1916);reprinted with a foreword by P.Roberts,Cambridge Univ.Press)[Mac2]:H-vectors and socle vectors of graded Artinian algebras,Ph.D.thesis,Queen’s University,2004.11。

计算机专业英语考试试题及答案

计算机专业英语考试试题及答案一、选择题1. Which of the following is NOT a programming language?A. JavaB. PythonC. HTMLD. CSS2. What does the acronym "SQL" stand for?A. Structured Query LanguageB. Simple Query LanguageC. Script Query LanguageD. Secure Query Language3. Which protocol is commonly used for sending and receiving emails?A. FTPB. HTTPC. SMTPD. TCP4. What does the term "CPU" refer to?A. Central Processing UnitB. Computer Processing UnitC. Central Program UnitD. Computer Program Unit5. Which of the following is NOT a type of network topology?A. StarB. RingC. MeshD. Scroll二、填空题1. HTML stands for Hypertext Markup Language, which is used for ____________.2. The process of converting source code into machine code is called ____________.3. IP address stands for ____________.4. The act of copying files from a remote server to a local computer is known as ____________.5. The programming language developed by Apple Inc. for iOS and macOS is ____________.三、简答题1. What is the difference between a compiler and an interpreter? Provide examples of programming languages that use each of these methods.2. Explain the concept of object-oriented programming (OOP) and provide an example of a programming language that utilizes this paradigm.3. Describe the client-server model and provide an example of a commonly used protocol within this model.四、论述题Discuss the impact of artificial intelligence (AI) on various industries. Provide examples of how AI is being used in fields such as healthcare, finance, and transportation. Analyze the potential benefits and challenges of implementing AI in these industries.答案：一、选择题1. C. HTML2. A. Structured Query Language3. C. SMTP4. A. Central Processing Unit5. D. Scroll二、填空题1. creating and structuring the content of a webpage2. compilation3. Internet Protocol4. downloading5. Swift三、简答题1. A compiler translates the entire source code into machine code before the program is executed. Examples of languages that use compilers are C, C++, and Java. On the other hand, an interpreter translates and executes the source code line by line. Python and Ruby are examples of languages that use interpreters.2. Object-oriented programming (OOP) is a programming paradigm that organizes data and functions into reusable objects. It focuses on the concept of classes and objects, allowing for code reuse and encapsulation. An example of a programming language that uses OOP is Java, where objects are instances of classes and can interact with each other through methods and attributes.3. The client-server model is a distributed computing architecture wherea server provides services or resources to multiple clients. The clients request and receive these resources through the network. An example of a commonly used protocol within this model is the Hypertext Transfer Protocol (HTTP), which is used for communication between web browsers (clients) and web servers.四、论述题Artificial intelligence (AI) has had a significant impact on various industries. In healthcare, AI is being used for diagnoses and treatments, analyzing medical images, and personalized medicine. For example, AI-powered algorithms can help detect diseases like cancer at an early stage, leading to better treatment outcomes. In finance, AI is utilized for fraud detection, algorithmic trading, and customer service. AI algorithms can analyze large amounts of financial data to identify patterns and make accurate predictions. In transportation, AI is being employed for autonomous vehicles, traffic management, and logistics optimization. Self-driving cars, for instance, use AI algorithms to navigate and make decisions on the road.The implementation of AI in these industries brings about many benefits, such as increased efficiency, improved accuracy, and cost savings. AI systems can process and analyze vast amounts of data much faster than humans, leading to faster and more accurate results. However, there are also challenges to consider. Privacy and security concerns arise as AI systems handle sensitive information. There is also the worry of job displacement, as AI automation may replace certain human tasks. Additionally, ethical considerations need to be addressed, such as bias in algorithms and the potential for AI to be used for malicious purposes.Overall, the impact of AI on various industries is undeniable. It has the potential to revolutionize healthcare, finance, transportation, and many other sectors. However, careful implementation and regulation are necessary to ensure its responsible and beneficial use.。

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On the interplay between measurable and topological dynamics

a rX iv:mat h /48328v1[mat h.DS]24Aug24ON THE INTERPLAY BETWEEN MEASURABLE AND TOPOLOGICAL DYNAMICS E.GLASNER AND B.WEISS Contents Introduction 2Part 1.Analogies 31.Poincar´e recurrence vs.Birkhoﬀ’s recurrence 31.1.Poincar´e recurrence theorem and topological recurrence 31.2.The existence of Borel cross-sections 41.3.Recurrence sequences and Poincar´e sequences 52.The equivalence of weak mixing and continuous spectrum 73.Disjointness:measure vs.topological d mixing:measure vs.topological 125.Distal systems:topological vs.measure 196.Furstenberg-Zimmer structure theorem vs.its topological PI version 217.Entropy:measure and topological 227.1.The classical variational principle 227.2.Entropy pairs and UPE systems 227.3.A measure attaining the topological entropy of an open cover 237.4.The variational principle for open covers 287.5.Further results connecting topological and measure entropy 307.6.Topological determinism and zero entropy 31Part 2.Meeting grounds 338.Unique ergodicity 339.The relative Jewett-Krieger theorem3410.Models for other commutative diagrams3911.The Furstenberg-Weiss almost 1-1extension theorem4012.Cantor minimal representations4013.Other related theorems41References442 E.GLASNER AND B.WEISSIntroductionRecurrent-wandering,conservative-dissipative,contracting-expanding,deter-ministic-chaotic,isometric-mixing,periodic-turbulent,distal-proximal,the list can go on and on.These(pairs of)words—all of which can be found in the dictio-nary—convey dynamical images and were therefore adopted by mathematicians to denote one or another mathematical aspect of a dynamical system.The two sister branches of the theory of dynamical systems called ergodic theory(or measurable dynamics)and topological dynamics use these words to describe diﬀerent but parallel notions in their respective theories and the surprising fact is that many of the corresponding results are rather similar.In the following article we have tried to demonstrate both the parallelism and the discord between ergodic theory and topo-logical dynamics.We hope that the subjects we chose to deal with will successfully demonstrate this duality.The table of contents gives a detailed listing of the topics covered.In theﬁrst part we have detailed the strong analogies between ergodic theory and topological dynamics as shown in the treatment of recurrence phenomena,equicontinuity and weak mixing,distality and entropy.In the case of distality the topological version cameﬁrst and the theory of measurable distality was strongly inﬂuenced by the topo-logical results.For entropy theory the inﬂuence clearly was in the opposite direction. The prototypical result of the second part is the statement that any abstract mea-sure probability preserving system can be represented as a continuous transformation of a compact space,and thus in some sense ergodic theory embeds into topological dynamics.We have not attempted in any way to be either systematic or comprehensive. Rather our choice of subjects was motivated by taste,interest and knowledge and to great extent is random.We did try to make the survey accessible to non-specialists, and for this reason we deal throughout with the simplest case of actions of Z.Most of the discussion carries over to noninvertible mappings and to R actions.Indeed much of what we describe can be carried over to general amenable groups.Similarly, we have for the most part given rather complete deﬁnitions.Nonetheless,we did take advantage of the fact that this article is part of a handbook and for some of the deﬁnitions,basic notions and well known results we refer the reader to the earlier introductory chapters of volume I.Finally,we should acknowledge the fact that we made use of parts of our previous expositions[86]and[35].We made the writing of this survey more pleasurable for us by the introduction of a few original results.In particular the following results are entirely or partially new.Theorem1.2(the equivalence of the existence of a Borel cross-section with the coincidence of recurrence and periodicity),most of the material in Section4 (on topological mild-mixing),all of subsection7.4(the converse side of the local variational principle)and subsection7.6(on topological determinism).MEASURABLE AND TOPOLOGICAL DYNAMICS3 Part1.Analogies1.Poincar´e recurrence vs.Birkhoff’s recurrence1.1.Poincar´e recurrence theorem and topological recurrence.The simplest dynamical systems are the periodic ones.In the absence of periodicity the crudest approximation to this is approximate periodicity where instead of some iterate T n x returning exactly to x it returns to a neighborhood of x.Theﬁrst theorem in abstract measure dynamics is Poincar´e’s recurrence theorem which asserts that for aﬁnite measure preserving system(X,B,µ,T)and any measurable set A,µ-a.e.point of A returns to A(see[46,Theorem4.3.1]).The proof of this basic fact is rather simple and depends on identifying the set of points W⊂A that never return to A.These are called the wandering points and their measurability follows from the formulaW=A∩ ∞ k=1T−k(X\A) .Now for n≥0,the sets T−n W are pairwise disjoint since x∈T−n W means that the forward orbit of x visits A for the last time at moment n.Sinceµ(T−n W)=µ(W) it follows thatµ(W)=0which is the assertion of Poincar´e’s theorem.Noting that A∩T−n W describes the points of A which visit A for the last time at moment n, and thatµ(∪∞n=0T−n W)=0we have established the following stronger formulation of Poincar´e’s theorem.1.1.Theorem.For aﬁnite measure preserving system(X,B,µ,T)and any measur-able set A,µ-a.e.point of A returns to A inﬁnitely often.Note that only sets of the form T−n B appeared in the above discussion so that the invertibility of T is not needed for this result.In the situation of classical dynam-ics,which was Poincar´e’s main interest,X is also equipped with a separable metric topology.In such a situation we can apply the theorem to a reﬁning sequence of partitions P m,where each P m is a countable partition into sets of diameter at most1m ofitself,and since the intersection of a sequence of sets of full measure has full measure, we deduce the corollary thatµ-a.e.point of X is recurrent.This is the measure theoretical path to the recurrence phenomenon which depends on the presence of aﬁnite invariant measure.The necessity of such measure is clear from considering translation by one on the integers.The system is dissipative,in the sense that no recurrence takes place even though there is an inﬁnite invariant measure.There is also a topological path to recurrence which was developed in an abstract setting by G.D.Birkhoﬀ.Here the above example is eliminated by requiring that the topological space X,on which our continuous transformation T acts,be compact.It is possible to show that in this setting aﬁnite T-invariant measure always exists,and so we can retrieve the measure theoretical picture,but a purely topological discussion will give us better insight.4 E.GLASNER AND B.WEISSA key notion here is that of minimality.A nonempty closed,T-invariant set E⊂X, is said to be minimal if F⊂E,closed and T-invariant implies F=∅or F=E.If X itself is a minimal set we say that the system(X,T)is a minimal system. Fix now a point x0∈X and consider∞ n=1ω(x0)=MEASURABLE AND TOPOLOGICAL DYNAMICS5 that the converse is also valid—namely if there are no conservative quasi-invariant measures then there is a Borel cross-section.Note that the periodic points of(X,T)form a Borel subset for which a cross-section always exists,so that we can conclude from the above discussion the following statement in which no explicit mention is made of measures.1.2.Theorem.For a system(X,T),with X a completely metrizable separable space, there exists a Borel cross-section if and only if the only recurrent points are the peri-odic ones.1.3.Remark.Already in[42]as well as in[21]oneﬁnds many equivalent conditions for the existence of a Borel section for a system(X,T).However one doesn’tﬁnd there explicit mention of conditions in terms of recurrence.Silvestrov and Tomiyama [76]established the theorem in this formulation for X compact(using C∗-algebra methods).We thank zar for drawing our attention to their paper.1.3.Recurrence sequences and Poincar´e sequences.We will conclude this sec-tion with a discussion of recurrence sequences and Poincar´e sequences.First for some deﬁnitions.Let us say that D is a recurrence set if for any dynamical system(Y,T) with compatible metricρand anyǫ>0there is a point y0and a d∈D withρ(T d y0,y0)<ǫ.Since any system contains minimal sets it suﬃces to restrict attention here to minimal systems.For minimal systems the set of such y’s for aﬁxedǫis a dense open set. To see this fact,let U be an open set.By the minimality there is some N such that for any y∈Y,and some0≤n≤N,we have T n y∈ing the uniform continuity of T n,weﬁnd now aδ>0such that ifρ(u,v)<δthen for all0≤n≤Nρ(T n u,T n v)<ǫ.Now let z0be a point in Y and d0∈D such that(1)p(T d0z0,z0)<δ.For some0≤n0≤N we have T n0z0=y0∈U and from(1)we getρ(T d0y0,y0)<ǫ. Thus points thatǫreturn form an open dense set.Intersecting overǫ→0gives a dense Gδin Y of points y for whichρ(T d y,y)=0.infd∈DThus there are points which actually recur along times drawn from the given recur-rence set.A nice example of a recurrence set is the set of squares.To see this it is easier to prove a stronger property which is the analogue in ergodic theory of recurrence sets.1.4.Deﬁnition.A sequence{s j}is said to be a Poincar´e sequence if for anyﬁnite measure preserving system(X,B,µ,T)and any B∈B with positive measure we haveµ(T s j B∩B)>0for some s j in the sequence.6 E.GLASNER AND B.WEISSSince any minimal topological system(Y,T)hasﬁnite invariant measures with global support,µany Poincar´e sequence is recurrence sequence.Indeed for any presumptive constant b>0which would witness the non-recurrence of{s j}for(Y,T), there would have to be an open set B with diameter less than b and having positiveµ-measure such that T s j B∩B is empty for all{s j}.Here is a suﬃcient condition for a sequence to be a Poincar´e sequence:1.5.Lemma.If for everyα∈(0,2π)limn→∞1nn1U s k(1B−f0) L2−→0or1nn1µ(B∩T−s k B)= f0 2>0which clearly implies that{s k}is a Poincar´e sequence. The proof we have just given is in fact von-Neumann’s original proof for the mean ergodic theorem.He used the fact that N satisﬁes the assumptions of the proposition, which is Weyl’s famous theorem on the equidistribution of{nα}.Returning to the squares Weyl also showed that{n2α}is equidistributed for all irrationalα.For rationalαthe exponential sum in the lemma needn’t vanish,however the recurrence along squares for the rational part of the spectrum is easily veriﬁed directly so that we can conclude that indeed the squares are a Poincar´e sequence and hence a recurrence sequence.The converse is not always true,i.e.there are recurrence sequences that are not Poincar´e sequences.This wasﬁrst shown by I.Kriz[60]in a beautiful example(see also[86,Chapter5]).Finally here is a simple problem.MEASURABLE AND TOPOLOGICAL DYNAMICS7 Problem:If D is a recurrence sequence for all circle rotations is it a recurrence set?A little bit of evidence for a positive answer to that problem comes from looking at a slightly diﬀerent characterization of recurrence sets.Let N denote the collection of sets of the formN(U,U)={n:T−n U∩U=∅},(U open and nonempty),where T is a minimal transformation.Denote by N∗the subsets of N that have a non-empty intersection with every element of N.Then N∗is exactly the class of recurrence sets.For minimal transformations,another description of N(U,U)is obtained by ﬁxing some y0and denotingN(y0,U)={n:T n y0∈U}Then N(U,U)=N(y0,U)−N(y0,U).Notice that the minimality of T implies that N(y0,U)is a syndetic set(a set with bounded gaps)and so any N(U,U)is the set of diﬀerences of a syndetic set.Thus N consists essentially of all sets of the form S−S where S is a syndetic set.Given aﬁnite set of real numbers{λ1,λ2,...,λk}andǫ>0setV(λ1,λ2,...,λk;ǫ)={n∈Z:maxj{ nλj <ǫ}},where · denotes the distance to the closest integer.The collection of such sets forms a basis of neighborhoods at zero for a topology on Z which makes it a topological group.This topology is called the Bohr topology.(The corresponding uniform structure is totally bounded and the completion of Z with respect to it is a compact topological group called the Bohr compactiﬁcation of Z.)Veech proved in[78]that any set of the form S−S with S⊂Z syndetic contains a neighborhood of zero in the Bohr topology up to a set of zero density.It is not known if in that statement the zero density set can be omitted.If it could then a positive answer to the above problem would follow(see also[32]).2.The equivalence of weak mixing and continuous spectrumIn order to analyze the structure of a dynamical system X there are,a priori,two possible approaches.In theﬁrst approach one considers the collection of subsystems Y⊂X(i.e.closed T-invariant subsets)and tries to understand how X is built up by these subsystems.In the other approach one is interested in the collection of factors Xπ→Y of the system X.In the measure theoretical case theﬁrst approach leads to the ergodic decomposition and thereby to the study of the“indecomposable”or ergodic components of the system.In the topological setup there is,unfortunately,no such convenient decomposition describing the system in terms of its indecomposable parts and one has to use some less satisfactory substitutes.Natural candidates for in-decomposable components of a topological dynamical system are the“orbit closures”(i.e.the topologically transitive subsystems)or the“prolongation”cells(which often coincide with the orbit closures),see[4].The minimal subsystems are of particular importance here.Although we can not say,in any reasonable sense,that the study of the general system can be reduced to that of its minimal components,the analysis of8 E.GLASNER AND B.WEISSthe minimal systems is nevertheless an important step towards a better understanding of the general system.This reasoning leads us to the study of the collection of indecomposable systems (ergodic systems in the measure category and transitive or minimal systems in thetopological case)and their factors.The simplest and best understood indecomposable dynamical systems are the ergodic translations of a compact monothetic group(a cyclic permutation on Z p for a prime number p,the“adding machine”on ∞n=0Z2, an irrational rotation z→e2πiαz on S1={z∈C:|z|=1}etc.).It is not hard toshow that this class of ergodic actions is characterized as those dynamical systems which admit a model(X,X,µ,T)where X is a compact metric space,T:X→X a surjective isometry andµis T-ergodic.We call these systems Kronecker or isometric systems.Thus ourﬁrst question concerning the existence of factors should be:given an ergodic dynamical system X which are its Kronecker factors?Recall that a measure dynamical system X=(X,X,µ,T)is called weakly mixing if the product system(X×X,X⊗X,µ×µ,T×T)is ergodic.The following classical theorem is due to von Neumann.The short and elegant proof we give was suggested by Y.Katznelson.2.1.Theorem.An ergodic system X is weakly mixing iﬀit admits no nontrivial Kronecker factor.Proof.Suppose X is weakly mixing and admits an isometric factor.Now a factor of a weakly mixing system is also weakly mixing and the only system which is both isometric and weakly mixing is the trivial system(an easy exercise).Thus a weakly mixing system does not admit a nontrivial Kronecker factor.For the other direction,if X is non-weakly mixing then in the product space X×X there exists a T-invariant measurable subset W such that0<(µ×µ)(W)<1.For every x∈X let W(x)={x′∈X:(x,x′)∈W}and let f x=1W(x),a function in L∞(µ).It is easy to check that U T f x=f T−1x so that the mapπ:X→L2(µ)deﬁned byπ(x)=f x,x∈X is a Borel factor map.Denotingπ(X)=Y⊂L2(µ),andν=π∗(µ),we now have a factor mapπ:X→(Y,ν).Now the function π(x) is clearly measurable and invariant and by ergodicity it is a constantµ-a.e.;say π(x) =1. The dynamical system(Y,ν)is thus a subsystem of the compact dynamical system (B,U T),where B is the unit ball of the Hilbert space L2(µ)and U T is the Koopman unitary operator induced by T on L2(µ).Now it is well known(see e.g.[35])that a compact topologically transitive subsystem which carries an invariant probability measure must be a Kronecker system and our proof is complete.Concerning the terminology we used in the proof of Theorem2.1,B.O.Koopman, a student of G.D.Birkhoﬀand a co-author of both Birkhoﬀand von Neumann introduced the crucial idea of associating with a measure dynamical system X= (X,X,µ,T)the unitary operator U T on the Hilbert space L2(µ).It is now an easy matter to see that Theorem2.1can be re-formulated as saying that the system X is weakly mixing iﬀthe point spectrum of the Koopman operator U T comprises the single complex number1with multiplicity1.Or,put otherwise,that the one dimensional space of constant functions is the eigenspace corresponding to the eigenvalue1(thisMEASURABLE AND TOPOLOGICAL DYNAMICS9 fact alone is equivalent to the ergodicity of the dynamical system)and that the restriction of U T to the orthogonal complement of the space of constant functions has a continuous spectrum.We now consider a topological analogue of this theorem.Recall that a topo-logical system(X,T)is topologically weakly mixing when the product system (X×X,T×T)is topologically transitive.It is equicontinuous when the family {T n:n∈Z}is an equicontinuous family of maps.Again an equivalent condition is the existence of a compatible metric with respect to which T is an isometry.And,moreover,a minimal system is equicontinuous iﬀit is a minimal translation on a compact monothetic group.We will need the following lemma.2.2.Lemma.Let(X,T)be a minimal system and f:X→R a T-invariant function with at least one point of continuity(for example this is the case when f is lower or upper semi-continuous or more generally when it is the pointwise limit of a sequence of continuous functions),then f is a constant.Proof.Let x0be a continuity point and x an arbitrary point in X.Since{T n x: n∈Z}is dense and as the value f(T n x)does not depend on n it follows that f(x)=f(x0).2.3.Theorem.Let(X,T)be a minimal system then(X,T)is topologically weakly mixing iﬀit has no non-trivial equicontinuous factor.Proof.Suppose(X,T)is minimal and topologically weakly mixing and letπ:(X,T)→(Y,T)be an equicontinuous factor.If(x,x′)is a point whose T×T orbit is dense in X×X then(y,y′)=(π(x),π(x′))has a dense orbit in Y×Y.However,if(Y,T) is equicontinuous then Y admits a compatible metric with respect to which T is an isometry and the existence of a transitive point in Y×Y implies that Y is a trivial one point space.Conversely,assuming that(X×X,T×T)is not transitive we will construct an equicontinuous factor(Z,T)of(X,T).As(X,T)is a minimal system,there exists a T-invariant probability measureµon X with full support.By assumption there exists an open T-invariant subset U of X×X,such that cls U:=M X×X.By minimality the projections of M to both X coordinates are onto.For every y∈X let M(y)={x∈X:(x,y)∈M},and let f y=1M(y)be the indicator function of the set M(y),considered as an element of L1(X,µ).Denote byπ:X→L1(X,µ)the map y→f y.We will show thatπis a continuous homomorphism,where we consider L1(X,µ)as a dynamical system with the isometric action of the group{U n T:n∈Z}and U T f(x)=f(T x).Fix y0∈X andǫ>0.There exists an open neighborhood V of the closed set M(y0)withµ(V\M(y0))<ǫ.Since M is closed the set map y→M(y),X→2X is upper semi-continuous and we can ﬁnd a neighborhood W of y0such that M(y)⊂V for every y∈W.Thus for every y∈W we haveµ(M(y)\M(y0))<ǫ.In particular,µ(M(y))≤µ(M(y0))+ǫand it follows that the map y→µ(M(y))is upper semi-continuous.A simple computation shows that it is T-invariant,hence,by Lemma2.2,a constant.10 E.GLASNER AND B.WEISSWith y0,ǫand V,W as above,for every y∈W,µ(M(y)\M(y0))<ǫandµ(M(y))=µ(M(y0)),thusµ(M(y)∆M(y0))<2ǫ,i.e., f y−f y0 1<2ǫ.This proves the claim thatπis continuous.Let Z=π(X)be the image of X in L1(µ).Sinceπis continuous,Z is compact. It is easy to see that the T-invariance of M implies that for every n∈Z and y∈X, f T−n y=f y◦T n so that Z is U T-invariant andπ:(Y,T)→(Z,U T)is a homomorphism. Clearly(Z,U T)is minimal and equicontinuous(in fact isometric).Theorem2.3is due to Keynes and Robertson[57]who developed an idea of Fursten-berg,[22];and independently to K.Petersen[70]who utilized a previous work of W.A.Veech,[78].The proof we presented is an elaboration of a work of McMahon[66]due to Blanchard,Host and Maass,[13].We take this opportunity to point outa curious phenomenon which recurs again and again.Some problems in topological dynamics—like the one we just discussed—whose formulation is purely topological, can be solved using the fact that a Z dynamical system always carries an invariant probability measure,and then employing a machinery provided by ergodic theory.In several cases this approach is the only one presently known for solving the problem. In the present case however purely topological proofs exist,e.g.the Petersen-Veech proof is one such.3.Disjointness:measure vs.topologicalIn the ring of integers Z two integers m and n have no common factor if whenever k|m and k|n then k=±1.They are disjoint if m·n is the least common multiple of m and n.Of course in Z these two notions coincide.In his seminal paper of 1967[23],H.Furstenberg introduced the same notions in the context of dynamical systems,both measure-preserving transformations and homeomorphisms of compact spaces,and asked whether in these categories as well the two are equivalent.The notion of a factor in,say the measure category,is the natural one:the dynamical system Y=(Y,Y,ν,T)is a factor of the dynamical system X=(X,X,µ,T)if there exists a measurable mapπ:X→Y withπ(µ)=νthat T◦π=π◦T.A common factor of two systems X and Y is thus a third system Z which is a factor of both.A joining of the two systems X and Y is any system W which admits both as factors and is in turn spanned by them.According to Furstenberg’s deﬁnition the systems X and Y are disjoint if the product system X×Y is the only joining they admit.In the topological category,a joining of(X,T)and(Y,S)is any subsystem W⊂X×Y of the product system(X×Y,T×S)whose projections on both coordinates are full;i.e.πX(W)=X andπY(W)=Y.(X,T)and(Y,S)are disjoint if X×Y is the unique joining of these two systems.It is easy to verify that if(X,T)and(Y,S)are disjoint then at least one of them is minimal.Also,if both systems are minimal then they are disjoint iﬀthe product system(X×Y,T×S)is minimal.In1979,D.Rudolph,using joining techniques,provided theﬁrst example of a pair of ergodic measure preserving transformations with no common factor which are not disjoint[72].In this work Rudolph laid the foundation of joining theory.He introduced the class of dynamical systems having“minimal self-joinings”(MSJ),and constructed a rank one mixing dynamical system having minimal self-joinings of all orders.MEASURABLE AND TOPOLOGICAL DYNAMICS11 Given a dynamical system X=(X,X,µ,T)a probability measureλon the product of k copies of X denoted X1,X2,...,X k,invariant under the product transformation and projecting ontoµin each coordinate is a k-fold self-joining.It is called an oﬀ-diagonal if it is a“graph”measure of the formλ=gr(µ,T n1,...,T n k),i.e.λis the image ofµunder the map x→ T n1x,T n2x,...,T n k x of X into k i=1X i.The joiningλis a product of oﬀ-diagonals if there exists a partition(J1,...,J m)of {1,...,k}such that(i)For each l,the projection ofλon i∈J l X i is an oﬀ-diagonal,(ii) The systems i∈J l X i,1≤l≤m,are independent.An ergodic system X has minimal self-joinings of order k if every k-fold ergodic self-joining of X is a product of oﬀ-diagonals.In[72]Rudolph shows how any dynamical system with MSJ can be used to con-struct a counter example to Furstenberg’s question as well as a wealth of other counter examples to various questions in ergodic theory.In[52]del Junco,Rahe and Swanson were able to show that the classical example of Chac´o n[16]has MSJ,answering a question of Rudolph whether a weakly but not strongly mixing system with MSJ exists.In[38]Glasner and Weiss provide a topological counterexample,which also serves as a natural counterexample in the measure category.The example consists of two horocycleﬂows which have no nontrivial common factor but are nevertheless not disjoint.It is based on deep results of Ratner[71]which provide a complete description of the self joinings of a horocycleﬂow.More recently an even more strik-ing example was given in the topological category by E.Lindenstrauss,where two minimal dynamical systems with no nontrivial factor share a common almost1-1 extension,[63].Beginning with the pioneering works of Furstenberg and Rudolph,the notion of joinings was exploited by many authors;Furstenberg1977[24],Rudolph1979[72], Veech1982[81],Ratner1983[71],del Junco and Rudolph1987[53],Host1991 [47],King1992[58],Glasner,Host and Rudolph1992[36],Thouvenot1993[77], Ryzhikov1994[73],Kammeyer and Rudolph1995(2002)[55],del Junco,Lema´n czyk and Mentzen1995[51],and Lema´n czyk,Parreau and Thouvenot2000[62],to men-tion a few.The negative answer to Furstenberg’s question and the consequent works on joinings and disjointness show that in order to study the relationship between two dynamical systems it is necessary to know all the possible joinings of the two systems and to understand the nature of these joinings.Some of the best known disjointness relations among families of dynamical systems are the following:•id⊥ergodic,•distal⊥weakly mixing([23]),•rigid⊥mild mixing([27]),•zero entropy⊥K-systems([23]),in the measure category and•F-systems⊥minimal([23]),•minimal distal⊥weakly mixing,•minimal zero entropy⊥minimal UPE-systems([9]),12 E.GLASNER AND B.WEISSin the topological category.d mixing:measure vs.topological4.1.Deﬁnition.Let X=(X,X,µ,T)be a measure dynamical system.1.The system X is rigid if there exists a sequence n kր∞such thatlimµ(T n k A∩A)=µ(A)for every measurable subset A of X.We say that X is{n k}-rigid.2.An ergodic system is mildly mixing if it has no non-trivial rigid factor. These notions were introduced in[27].The authors show that the mild mixing property is equivalent to the following multiplier property.4.2.Theorem.An ergodic system X=(X,X,µ,T)is mildly mixing iﬀfor every ergodic(ﬁnite or inﬁnite)measure preserving system(Y,Y,ν,T),the product system(X×Y,µ×ν,T×T),is ergodic.Since every Kronecker system is rigid it follows from Theorem2.1that mild mixing implies weak mixing.Clearly strong mixing implies mild mixing.It is not hard to construct rigid weakly mixing systems,so that the class of mildly mixing systems is properly contained in the class of weakly mixing systems.Finally there are mildly but not strongly mixing systems;e.g.Chac´o n’s system is an example(see Aaronson and Weiss[1]).We also have the following analytic characterization of mild mixing.4.3.Proposition.An ergodic system X is mildly mixing iﬀφf(n)<1,lim supn→∞for every matrix coeﬃcientφf,where for f∈L2(X,µ), f =1,φf(n):= U T n f,f . Proof.If X→Y is a rigid factor,then there exists a sequence n i→∞such that U T n i→id strongly on L2(Y,ν).For any function f∈L20(Y,ν)with f =1, we have lim i→∞φf(n i)=1.Conversely,if lim i→∞φf(n i)=1for some n iր∞and f∈L20(X,µ), f =1,then lim i→∞U T n i f=f.Clearly f can be replaced by a bounded function and we let A be the sub-algebra of L∞(X,µ)generated by {U T n f:n∈Z}.The algebra A deﬁnes a non-trivial factor X→Y such that U T n i→id strongly on L2(Y,ν). We say that a collection F of nonempty subsets of Z is a family if it is hereditary upward and proper(i.e.A⊂B and A∈F implies B∈F,and F is neither empty nor all of2Z).With a family F of nonempty subsets of Z we associate the dual familyF∗={E:E∩F=∅,∀F∈F}.It is easily veriﬁed that F∗is indeed a family.Also,for families,F1⊂F2⇒F∗1⊃F∗2, and F∗∗=F.。

Notes on quasiminimality and excellence

Notes on Quasiminimality and ExcellenceJohn T.BaldwinDepartment of Mathematics,Statistics and Computer ScienceUniversity of Illinois at Chicago∗April7,2004AbstractThis paper ties together much of the model theory of the last50years.Shelah’s attempts to generalize the Morley theorem beyondﬁrst order logic led to the notion of excellence,which is a key to the structure theory of uncountable models.The notion of Abstract Elementary Class arose naturally in attempting toprove the categoricity theorem for Lω1,ω(Q).More recently,Zilber has attempted to identify canonicalmathematical structures as those whose theory(in an appropriate logic)is categorical in all powers.Zilber’strichotomy conjecture forﬁrst order categorical structures was refuted by Hrushovski,by the introducion of aspecial kind of Abstract Elementary Class.Zilber uses a powerful and essentailly inﬁnitary variant on thesetechniques to investigate complex exponentiation.This not only demonstrates the relevance of Shelah’smodel theoretic investigations to mainstream mathematics but produces new results and conjectures inalgebraic geometry.Zilber proposes[63]to prove‘canonicity results for pseudo-analytic’rmally,‘canonical’means ‘the theory of the structure in a suitable possibly inﬁnitary language(see Section2)has one model in each uncountable power’while‘pseudoanalytic’means‘the model of power2ℵ0can be taken as a reduct of an expansion of the complex numbers by analytic functions’.This program interacts with two other lines of research.First is the general study of categoricity theorems in inﬁnitary languages.After initial results by Keisler,reported in[31],this line was taken up in a long series of works by Shelah.We place Zilber’s work in this context.The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is‘trivial’,‘vector space-like’,or‘ﬁeld-like’.This construction turns out to be a very concrete example of an Abstract Elementary Class,a concept that arose in Shelah’s analysis. And the construction is a crucial tool for Zilber’s investigations.This paper examines the intertwining of these three themes.For simplicity,we work in a countable vocabulary.The study of(C,+,·,exp)leads one immediately to some extension ofﬁrst order logic;the integers with all their arithmetic areﬁrst order deﬁnable in(C,+,·,exp).Thus,theﬁrst order theory of complex exponentiation is horribly complicated;it is certainly unstable and so itsﬁrst order theory cannot be categorical in power.That is,theﬁrst order theory of complex exponentiation cannot have exactly one model in each uncountable cardinal. One solution is to use inﬁnitary logic to pin down the pathology.Insist that the kernel of the exponential map isﬁxed as a single copy of the integers while allowing the rest of the structure to grow.We describe in Section5 Zilber’s theorem that,modulo certain(very serious)algebraic hypotheses,(C,+,·,exp)can be axiomatized bya categorical Lω1,ω(Q)-sentence.The notion of amalgamation is fundamental to model theory.Even in theﬁrst order case,the notion is subtle because there are several variants depending on the choice of a class of models K and a notion≺of substructure.∗Partially supported by NSF grant DMS-0100594and CDRF grant KM2-2246.1The pair(K,≺)has the amalgamation property if whenever M∈K is embedded by f0,f1into N0,N1so that the image of the embeddings f0M,f1M≺N0,N1respectively,there is an N∗and embeddings g0,g1of N0,N1 into N∗with g0f0and g1f1agreeing on M.If K is the class of models of a completeﬁrst order theory then the amalgamation property holds with≺as elementary embeddings of models.If K is the class of substructures of models of a complete quantiﬁer eliminableﬁrst order theory then the amalgamation property holds for≺as arbitrary embeddings.Morley[39]observed that by adding names for each deﬁnable relation,we can assume, for studying the spectrum problm,that anyﬁrst order theory has elimination of quantiﬁers.Shelah[47],noted that this amalgamation hypothesis allows us to assume the existence of a‘monster model’which serves as a universal domain.In this domain the notion of type of an element a over a set A can be thought of either semantically as the orbit of a under automorphisms thatﬁx A or syntactially as the collection of formulas with parameters from A that are satisﬁed by a.Of course,the extension fromﬁrst order logic causes the failure of the compactness theorem.For example,it is easy to write a sentence in Lω1,ωwhose only model is the natural numbers with successor.But thereare some more subtle losses.The duality between the syntactic and semantic concept of type depends on the amalgamation property.Here is a simple example showing that amalgamation fails in models of a sentence ofLω1,ω.Consider the theory T of a dense linear order without endpoints,a unary predicate P(x)which is denseand codense,and an inﬁnite set of constants arranged in order typeω+ω∗.Let K be class of all models of T which omit the type of a pair of points,which are both in the cut determined by the constants.Now consider the types p and q which are satisﬁed by a point in the cut,which is in P or in¬P respectively.Now p and q are each satisﬁable in a member of K but they are not simultaneously satisﬁable.So the amalgamation property has failed for K and elementary embeddings.This shows that a more subtle notion than consistency is needed to describe types in this wider context.We took‘canonical’above as meaning‘categorical in uncountable cardinalities’.That is,the class has exactly one model in each uncountable cardinality.The analysis ofﬁrst order theories categorical in power is based on ﬁrst studying strongly minimal sets.A set is strongly minimal if every deﬁnable subset of it isﬁnite or coﬁnite.A natural generalization of this,particularly since it holds of simply deﬁned subsets of(C,+,·,exp),is to consider structures where every deﬁnable set is countable or cocountable.As we will see,the useful formulation of this notion requires some auxiliary homogeneity conditions.The role of homogeneity in studying categoricity in inﬁnitary languages has been known for a long time.There is a rough translation between‘homogeneity’hypotheses on a model and and corresponding‘amalgamation’hypotheses on the class of substructures of the model(Section2).A structure isℵ1-homogeneous if for any two countable sequences a,b,which realize the same type,and any c,there is a d such that a c and b d realize the same type.Thus,ℵ1-homogeneity corresponds to amalgamation over arbitrary countable subsets.Keisler[31]proved the natural generalization of Morley’stheorem for a sentenceψin Lω1,ωmodulo two assumptions:1.Every model ofψhas arbitrarily large elementary extensions.2.Every model ofψisℵ1-homogeneous.Keisler asked whether everyℵ1-categorical sentence in Lω1,ωsatisﬁes assumption2.The answer is no.Marcus[37]gave an example of a minimal prime model with inﬁnitely many indiscernibles and a modiﬁcation by Shelahprovides an example of a totally categorical(categorical in each uncountable cardinality)sentence in Lω1,ωwhich has noℵ1-homogeneous models.Shelah’s notion of an excellent class(extremely roughly:‘amalgamation over(independent)n-dimensional cubes for all n’and‘ℵ0-stability’)provides a middle ground.An excellent class(See paragraph2.0.9.)is a strengthening of Keisler’sﬁrst assumption(provides not only arbitrarily large models but a certain control over their construction)while weakening the second to assert amalgamation only over certain conﬁgurations.Recall that the logic L(Q)adds toﬁrst order logic the expression(Qx)φ(x)which holds if there are uncountably many solutions ofφ.I had asked whether a sentence in L(Q)could have have exactly one model and that model2have cardinalityℵ1.Shelah proved in[45]using that anℵ1-categorical sentence in Lω1,ω(Q)must have a modelof powerℵ2.There is a beautiful proof of this result in ZFC in[53].Shelah has moved this kind of argument from(ℵ1,ℵ2)to(λ,λ+)in a number of contexts.But,getting arbitrarily large models just from categoricity in a single cardinal has remained intractable,although Shelah reported substantial but not yet written progress in the summer of2003.Shelah proved an analogue to Morley’s theorem in[48,49]for‘excellent’classes deﬁned in Lω1,ω.Assuming2ℵn<2ℵn+1,for all n<ω,he also proved the following kind of converse:every sentence in Lω1,ωthat iscategorical inℵn for all n<ωis excellent and categorical in all cardinals.The assumption of categoricityall the way up toℵωis shown to be essential in[18]by constructing for each n a sentenceψn of Lω1,ωwhichis categorical up toℵn but has the maximal number of models in all suﬃciently large cardinalities.He alsoasserted that these results‘should be reproved’for Lω1,ω(Q).This‘reproving’has continued for20years andtheﬁnale is supposed to appear in the forthcoming Shelah[50,51].Zilber’s approach to categoricity theorems is more analogous to the Baldwin-Lachlan approach than to Morley’s. Baldwin-Lachlan[8]provide a structural analysis;they show each model of anℵ1-categorical theory is prime over a strongly minimal set.This allows one to transfer the‘geometric’proof of categoricity in power for a strongly minimal theory to show categoricity inℵ1implies categoricity in all cardinalities.In fact,Zilber considers only the quasiminimal case.But a‘Baldwin-Lachlan’style proof was obtained by Lessmann for homogeneous model theory in[35]and for excellent classes in[34].That is,he proves every model is prime and minimal over a quasiminimal set.We begin in Section1by recalling the basic notions of the Fra¨ıss´e construction and the notion of homogeneity.In Section2,we sketch some results on the general theory of categoricity in non-elementary logics.In particular,we discuss both reductions to the‘ﬁrst order logic with omitting types’and the‘syntax-free’approach of Abstract Elementary Classes.We turn to the development of the special case of quasiminimal theories in Section3. This culminates in Zilber’sﬁrst approximation of a quasiminimal axiomatization of complex exponentiation.In Section4we formulate the generalized Fra¨ıss´e construction and place it in the setting of Abstract Elementary Classes.We analyze this method for constructingﬁrst order categorical theories;we then see a variant to get examples in homogeneous model theory.Then we discuss the results and limitations of the program to obtain analytic representations of models obtained by this construction.Finally in Section5we return to Zilber’s use of these techniques to study complex exponentiation.We describe the major algebraic innovations of his approach and the innovations to the Hrushovski construction which result in structures that are excellent but deﬁnitely notﬁrst order axiomatizable.Many thanks to Rami Grossberg and Olivier Lessmann,who were invaluable in putting together this survey, but are not responsible for any ments by Assaf Hasson,David Kueker,David Marker,Charles Steinhorn,Saharon Shelah,and Boris Zilber improved both the accuracy and the exposition.We particularly thank the referee and editor for further clarifying the expositition.1The Fra¨ıss´e ConstructionIn the early1950’s Fra¨ıss´e[13]generalized Hausdorﬀ’s back and forth argument for the uniqueness of the rationals as a countable dense linear order(without end points).He showed that any countable class K ofﬁnite relational structures closed under substructure and satisfying the joint embedding and amalgamation properties (see Deﬁnition4.1.6)has a unique countable(ultra)-homogeneous member(denoted G):any isomorphism betweenﬁnite subsets of G extends to an automorphism.There are easy variants of this notion for locally ﬁnite classes in a language with function symbols.The existence of such structures is proved by iterating the amalgamation property and taking unions of chains.(See[21]for a full account.)J´o nsson[28]extended the notion to arbitrary cardinals and Morley-Vaught[40]created an analogous notion for the class of models ofﬁrst3order theories with elementary embeddings as the morphisms.They characterized the homogeneous universal models in this situation as the saturated models.In general the existence of saturated models in powerκrequires thatκ=κ<κandκ>2|L|;alternatively,one may assume the theory is stable.In particular,κ-saturated models areκ-homogeneous.Morley proved every uncountable model of a theory categorical in an uncountable power is saturated.Abstract versions of the Fra¨ıss´e construction undergird the next section;concrete versions dominate the last two sections of the paper.2Syntax,Stability,AmalgamationThis section is devoted to investigations of categoricity for non-elementary classes.We barely touch the immense literature in this area;see[15].Rather we just describe some of the basic concepts and show how they arisefrom concrete questions of categoricity in Lω1,ωand Lω1,ω(Q).In particular,we show how diﬀerent frameworksfor studying nonelementary classes arise and some relations among them.Any serious study of this topic begins with[30,31].In its strongest form Morley’s theorem asserts:Let T be aﬁrst order theory having only inﬁnite models.If T is categorical in some uncountable cardinal then T is complete and categorical in every uncountable cardinal.This strong form does not generalize to Lω1,ω;take the disjunction of a sentence which is categorical in allcardinalities with one that has models only up to,say, ing both the upward and downward L¨o wenheim-Skolem theorem, L os[36]proved that aﬁrst order theory that is categorical in some cardinality is complete.Since the upwards L¨o wenheim-Skolem theorem fails for Lω1,ω,the completeness cannot be deduced for this logic.However,if the Lω1,ω-sentenceψis categorical inκ,then,applying the downwards L¨o wenheim-Skolem theorem,for every sentenceφeitherψ→φor all models ofφhave cardinality less thanκ.So ifφandψareκ-categoricalsentences with a common model of powerκthey are equivalent.We say a sentence of Lω1,ωis complete if iteither implies or contradicts every other Lω1,ω-sentence.Such a sentence is necessarilyℵ0-categorical(usingdownward L¨o wenheim-Skolem).Moreover,every countable structure is characterized by a complete sentence, which is called its Scott sentence.So if a model satisﬁes a complete sentence,it is L∞,ω-equivalent to a countablemodel.In particular,any model M ofψ∈Lω1,ωis small.That is,for every n it realizes only countably manyLω1,ω-n-types(over the empty set).Moreover,ifφhas a small model thenφis implied by a complete sentencesatisﬁed in that model.In theﬁrst order case it is trivial to reduce the study of categoricity to complete(for Lω,ω)theories.Moreover,ﬁrst order theories share the fundamental properties of sentences–in particular,L¨o wenheim-Skolem down toℵ0.But an Lω1,ω-theory need not have a countable model.The diﬃculty is that an Lω1,ω-theory need not beequivalent to a countable conjunction of sentences,even in a countable language.So while we want to reducethe categoricity problem to that for complete Lω1,ω-sentences,we cannot make the reduction trivially.Weﬁrstshow that ifψ∈Lω1,ωhas arbitrarily large models and is uncountably categorical thenψextends to a completesentence.A key observation is that ifψhas arbitrarily large models thenψhas models that realize few types.Lemma2.0.1Supposeψ∈Lω1,ωhas arbitrarily large models.1.In every inﬁnite cardinalityψhas a model that realizes only countably many Lω1,ω-types over the emptyset.2.Thus,if N is the unique model ofψin some cardinal,ψis implied by a consistent complete sentenceψwhich holds of N.Proof.Sinceψhas arbitrarily large models we can construct a model with indiscernibles(Chapters13-15of [31]).Now take an Ehrenfeucht-Mostowski model M forψover a set of indiscernibles ordered by a k-transitive4dense linear order.(A ordering is k-transitive if any two properly ordered k-tuples are in the same orbit under the automorphism group.These orders exist in every cardinal;take the order type of an orderedﬁeld.)Then for every n,M has only countably many orbits of n-tuples and so realizes only countably many types in anylogic where truth is preserved by automorphism–in particular in Lω1,ω.Ifψisκ-categorical,letψ be theScott sentence of this Ehrenfeucht-Mostowski model with cardinalityκ. 2.0.1 If we do not assumeψhas arbitrarily large models the reduction to complete sentences,sketched below,is more convoluted and uses hypotheses(slightly)beyond ZFC.In particular,the complete sentenceψ does nothold,a priori of the categoricity model.The natural examples of Lω1,ω-sentences which have models of boundedcardinality(e.g.a linear order with a countable dense subset,or coding up an initial segment of the Vαhierarchy of all sets)have the maximal number of models in the largest cardinality where they have a model.Shelah discovers a dichotomy(Theorem2.0.2)between such sentences and‘excellent’sentences.We expand on the notion of excellence at2.0.9and later in the paper.For the moment just think of the assertion that a completeLω1,ω-sentence(equivalently,its class of models)is excellent as a step into paradise.For any class K of models,I(λ,K)denotes the number of isomorphism types of members of K,with cardinality λ.We may writeψinstead of K if K is the class of models ofψ.We say that a class K has many models of cardinalityℵn if I(ℵn,K)≥µ(n)(and few if not;there may not be any).We use as a black box the functionµ(n)(deﬁned precisely in[49]).Either GCH or¬O#implyµ(n)=2ℵn but it is open whether it might be(consistently)smaller.The diﬃcult heart of the argument is the following theorem of Shelah[48,49];we don’t discuss the proof of this result but just show how this solution for complete sentences gives the result forarbitrary sentences of Lω1,ω.Theorem2.0.2 1.(For n<ω,2ℵn<2ℵn+1)A complete Lω1,ω-sentence which has few models inℵn foreach n<ωis excellent(see2.0.9).2.(ZFC)An excellent class has models in every cardinality.3.(ZFC)Suppose thatφis an excellent Lω1,ω-sentence.Ifφis categorical in one uncountable cardinalκthen it is categorical in all uncountable cardinals.So a nonexcellent class deﬁned by a complete Lω1,ω-sentenceψmay not have arbitrarily large models but,ifnot,it must have many models in some cardinal less thanℵω.Combining several results of Keisler,Shelah[48] shows:Lemma2.0.3Assume2ℵ0<2ℵ1.Letψbe a sentence of Lω1,ωthat has at least one but less than2ℵ1modelsof cardinalityℵ1.Thenψhas a small model of cardinalityℵ1.Proof.By Theorem45of[31],for any countable fragment L∗containingψand any N|=ψof cardinalityℵ1, N realizes only countably many L∗types over the empty set.Theorem2.2of[45]says that ifψhas a model M of cardinalityℵ1which realizes only countably many types in each fragment thenψhas a small model of cardinalityℵ1.We sketch a proof of that theorem.Add to the language a linear order<,interpreted as a linearorder of M with order typeωing that M realizes only countably many types in any fragment,write Lω1,ωas a continuous increasing chain of fragments Lαsuch that each type in Lαrealized in M is a formula in Lα+1. Add new2n+1-ary predicates and n+1-ary functions f n.Let M satisfy E n(α,a,b)if and only if a and b realize the same Lα-type and let f n map M n+1into the initialωelements of the order,so that E n(α,a,b) implies f n(α,a)=f n(α,b).Note:i)E n(β,y,z)reﬁnes E n(α,y,z)ifβ>α;ii)E n(0,a,b)implies a and b satisfy the same quantiﬁer free formulas;iii)ifβ>α,E n(β,a,b)implies(∀x)(∃y)E n+1(α,x a,y b).Thus,iv) for any a∈M each equivalence relation E n(a,y,z)has only countably many classes.All these assertions canbe expressed by an Lω1,ωsentenceφ.Now add a unary predicate symbol P and a sentenceχwhich assertsthat M is an end extension of P(M).For everyα<ω1there is a model Mαofφ∧ψ∧χwith order type of5(P(M),<)greater thanα.(Start with P asαand alternately take an elementary submodel for the smallest fragment L∗containingφ∧ψ∧χand close down under<.Afterωsteps we have the P for Mα.)Now by Theorem12of[31]there is countable structure(N0,P(N0))such that P(N0)contains a copy of(Q,<)and N0 is an end extension of P(N0).By Theorem28of[31],N0has an L∗elementary extension of cardinalityℵ1.Fix an inﬁnite decreasing sequence d0>d1>...in N0.For each n,deﬁne E+n(x,y)if for some i,E n(d i,x,y).Now using i),ii)and iii)prove by induction on the quantiﬁer rank ofφthat N1|=E+n(a,b)implies N1|=φ(a)if andonly if N1|=φ(b)for every Lω1,ω-formulaφ.For each n,E n(d0,x,y)reﬁnes E+n(x,y)and by iv)E n(d0,x,y)has only countably many classes;so N is small. 2.0.3Using these two results,we easily derive a version of Morley’s theorem for an Lω1,ω-sentence.Theorem2.0.4Assume2ℵn<2ℵn+1for n<ω.If an Lω1,ω-sentenceψhas an uncountable model,then either1.ψhas many models inℵn for some n<ωor2.ψhas arbitrarily large models and ifψis categorical in one uncountable cardinalκthen it is categoricalall uncountable cardinals.Proof.Supposeψhas few models inℵn for each n<ω.By Lemma2.0.3,choose a small model ofψ,say with Scott sentenceψ .Assuming2ℵn<2ℵn+1for each n,Theorem2.0.21)impliesψ is excellent.By Theorem2.0.2 2)ψ and thusψhave arbitrarily large models.Now supposeψis categorical inκ>ℵ0.Then so isψ whence, by Theorem2.0.23),ψ is categorical in all uncountable powers.To showψis categorical aboveκnote that by downward L¨o wenheim-Skolem all models ofψwith cardinality at leastκsatisfyψ ;the result follows by the categoricity ofψ .Ifψis not categorical in some cardinalityµ<κ, there must be a sentenceθwhich is inconsistent withψ but consistent withψ.Applying the entire analysis to ψ∧θ,weﬁnd a complete sentenceψ which has arbitrarily large models,is consistent withψand contradicts ψ .But this is forbidden by categoricity inκ. 2.0.4 One corollary of this result isCorollary2.0.5Assume2ℵ0<2ℵ1.If an Lω1,ω-sentence is categorical inℵn for n<ω,then it is categoricalin all cardinalities.Hart and Shelah[18]have shown the necessity of the hypothesis of categoricity up toℵω.A key tool in the study of complete Lω1,ω-sentences is the reduction of the class of models of such sentences toclasses which are‘closer’to beingﬁrst order.We now give a full account of this easy reduction.Chang proved in[12]that the class of models of any sentence in Lκ+,ωcould be viewed as the class of reducts to L of models of aﬁrst order theory in an expansion L of L which omitted a family of types.Chang(Lopez-Escobar[12])used this observation to prove that the Hanf number for Lκ+,ωis same as the Hanf number for omitting a family ofκtypes.Shelah[45]took this reduction a step further and showed that the class of models of a complete sentencein Lω1,ωare in1-1correspondence(mapping L∞,ω-submodel to elementary submodel)with the class of atomicmodels of an appropriateﬁrst order theory in an expanded language.That is,to study the generalization ofMorley’s theorem to complete Lω1,ω-sentences it suﬃces to study classes of structures deﬁned by a special typeofﬁnite diagram.By aﬁnite diagram we mean an EC(T,Γ)class:those models ofﬁrst order theory that omit all types from a speciﬁed collectionΓof types inﬁnitely many variables over the empty set.Abusing the EC(T,Γ)notation,EC(T,Atomic)denotes the class of atomic models of T(i.e.to conform to the notationwe should write nonatomic).Most detailed study of the spectrum of Lω1,ω-sentences[45,48,49,34,16,27]just work withﬁnite diagrams or more restrictively atomic models(and usually under stronger homogeneity conditions).In general,an atomic class might be deﬁned by omitting uncountably many types;in the case of interest only countably many types have to be omitted.6Theorem2.0.6Letψbe a complete sentence in Lω1,ω.Then there is a countable language L extending Land aﬁrst order L -theory T such that the reduct map is1-1from the atomic models of T onto the models ofψ.Proof.Let L∗be a countable fragment of Lω1,ωwhich contains all subformulas ofψand the conjunction ofeach Lω1,ω-type that is realized in a model ofψ.(This set is countable since complete sentences are small.)Expand L to L by inductively adding a predicate Pφ(x)for each L∗-formulaφ.Fix a model ofψand expand it to an L -structure by interpreting the new predicates so that the new predicates represent eachﬁnite Boolean connective and quantiﬁcation faithfully:E.g.P¬φ(x)↔¬Pφ(x),andP(∀x)φ(x)↔(∀x)Pφ(x),and that,as far asﬁrst order logic can,the Pφpreserve the inﬁnitary operations:for each i,P Vi φi(x)→Pφi(x).Let T be theﬁrst order theory of any such model and consider the countable setΓof typesp Vi φi(x)={¬P Viφi(x)}∪{Pφi(x):i<ω}.Note that if q is an Lω1,ω-type realized in a model of T,P V q generates a principal type in T.Now if M is amodel of T which omits all the types inΓ(in particular,if M is an atomic model of T),M|L|=ψand each model ofψhas a unique expansion to a model of T which omits the types inΓ(since this is an expansion bydeﬁnitions in Lω1,ω). 2.0.6So in particular,any complete sentence of Lω1,ωcan be replaced(for spectrum purposes)by considering theatomic models of aﬁrst order theory.Since all the new predicates are Lω1,ω-deﬁnable this is the naturalextension of Morley’s procedure of replacing eachﬁrst order formulaφby a predicate symbol Pφ.Morley’s procedure resulted in a theory with elimination of quantiﬁers thus guaranteeing amalgamation over sets for ﬁrst order categorical T.A similar amalgamation result does not follow in this case.Nor,In general,doﬁnite diagrams satisfy the upwards L¨o wenheim-Skolem theorem.Remark2.0.7(Lω1,ω(Q))The situation for Lω1,ω(Q)is more complicated.The example[18]of a sentenceof Lω1,ωthat isℵ1-categorical and not categorial in all uncountable powers is quite complicated.But theL(Q)theory of two disjoint inﬁnite sets illustrates this phenomena trivially.Some of the analysis of[48,49] goes over directly.But many problems intervene and Shelah has devoted several articles(notably[52,50,51]tocompleting the analysis;a deﬁnitive version has not appeared.The diﬃculty in extending from Lω1,ωto Lω1,ω(Q)is in constructing models with the proper interpretation of the Q-quantiﬁer.Following Keisler’s analysis of this problem in[30]the technique is to consider various notions of strong submodel.Two notions are relevant:in the ﬁrst,the relation of M≺K N holds when deﬁnable sets which are intended to be countable(M|=¬(Qx)φ(x)) do not increase from M to N.The seconds adds that deﬁnable sets intended to be uncountable(M|=(Qx)φ(x)) increase from M to N.Theﬁrst notion gives an AEC(Deﬁnition2.0.8);the second does not.The reduction [53,50]is actually to an AEC along with the second relation as an auxiliary that guarantees the existence of standard models.When J´o nsson generalized the Fra¨ısse construction to uncountable cardinalities[28,29],he did so by describ-ing a collection of axioms,which might be satisﬁed by a class of models,that guaranteed the existence of a7。

核心和非核心结构【外文翻译】

外文翻译原文Core vs.Non-Core FrameworkMaterialSource:/articles/2009/01/core-vs-noncore-fra mework,2004.02 Author: Neil Mac Allister, Richard Evans, and Katherine WallaceAcross the pharmaceutical industry,dramatic and durable changes to the operating environment are calling for modifications to companies’strategies and structures. The industry is facing a period of eroding pricing power, falling growth in the consumption of branded drugs, and tighter regulatory standards.As a consequence, companies are finding that revenue growth is becoming both slower and more volatile, and that returns on R&D spending are pushing below the cost of capital. We recommend changes to the current business model that include smaller, more efficient and more flexible cost structures, as well as the increase of efforts to mitigate revenue volatility.Our aim in this article is to apply an analytical framework for how to think about an evolving business model for pharmaceutical companies.The pharmaceutical industry is operating within an increasingly unfavorable political,economic,and regulatory environment,largely as a result of negative public opinion,rising healthcare costs,and increasing involvement from governments in the purchase,reimbursement and market approval of pharmaceuticals.These pressures are being brought to bear on an industry whose structures reflect past rather than present and future conditions,particularly with costs that are both too high and too inflexible.To estimate profitability over long time cycles,we compared year one R&D spending to year 10 net income, a rate of return that has been falling for as long as we can measure. Apparent returns are no longer higher than the industry’s cost of capital.Profits must exceed costs of capital for a business to remain viable.Inpharma, this gap can be widened by either increasing the revenue return generated by each dollar spent on R&D or by reducing the cost of commercializing the industry’s innovationsWe also see revenue growth slowing and becoming more volatile, consisting of interspersed periods of growth and contraction. As real pricing power and per-capita branded volume effects fade,revenue growth slows.Historically,real pricing power and per-capita volume growth made steady, predictable contributions to total revenue growth and more pricing power could be applied when needed to stabilize growth;as they fade,revenue growth defaults to —or at least toward —the remaining variables: population growth and product mix. Population growth is too small to matter,leaving mix as the dominant variable.Product mix is extremely volatile from period to period, consisting of significant gains (i.e. new products) and significant losses (i.e. patent expiry) interspersed at uneven intervals. Absent the buffers of real pricing and per-capita volume gains,it follows that future revenue patterns contain both ups and downs; unless cost structures become more flexible, periods of revenue contraction will result in outright earnings losses.Competitive differentiation must consider whether ownership or control of the activity is important for competitive and/or strategic reasons,and whether or not the company is able to perform the activities at such a level that it provides them with a point of differentiation against their competitors. The availability of sourcing options needs to examine whether or not there are ample vendors performing the activity that can deliver world-class quality at a cost-effective price.Core activities•Enable the overarching business strategy•Are key components of the company’s value proposition•Are a major source of durable competitive advantage (e.g.intellectual capital)•Protect intellectual property•Have internal capabilities that cannot be matched or exceeded by outside vendors or other partnersNon-core activities•Can be pushed outside of pharma to improve flexibility within cost structure •Are general “supportive” activities to the pharma value proposition•Can be conducted by third parties and match or exceed internal quality/economicsDetermining core and non-core activities for your company will depend upon the benefits of outsourcing or partnerships versus keeping the function in-house and the strategic importance of the function relative to your company. The importance and value of these two dimensions will differ depending upon the product stage. For example, elements of screening in discovery that have low risk of IP exposure may be considered non-core,while aspects of lead optimization in development involving high risks of IP exposure are likely to be considered core activities.We have broken R&D down into four phases:basic research,discovery, preclinical, and development. By examining the key activities within each of these phases, a core or non-core determination can be made based on the strategic context underlying each element.In some cases the activity may be “on the fence”–this simply means that the determination will vary for each company depending on their internal capabilities, capacity, and strategic direction.Basic research functions are core only if they provide a point of differentiation for the company. For example, in therapeutic areas in which very few companies are working on a limited number of mechanisms, target identification and validation activities may be a point of differentiation. Beyond this, the activities are thought to be non-core. In therapeutic areas with multiple mechanisms and multiple companies competing,it is likely in the best interest for the company to change their orientation to search for and evaluate targets rather than to generate IP.Basic research around different mechanisms may be found within academia,but increasingly commercial organizations are developing platforms necessary to conduct these activities – providing a variety of capable vendors and partners.Discovery efforts should be considered for outsourcing because they are easily systematized and — in some instances — are automated. For companies that have already built discovery capabilities in-house — e.g. high through-put screening —the cost effectiveness and quality standards must be evaluated against outsourcing options.Preclinical activities are increasingly being outsourced in order to take advantage of specialist modeling capabilities. Assembly of the information gathered in preclinical development will remain in-house,while the actual generation of perspectives will move to outsourcing. The vendor environment for preclinical work,particularly in specialty areas,is maturing quickly,allowing pharmaceutical companies to tap into new efficiencies by outsourcing in these areas.In clinical development,strategy,development plans, and management functions should always be kept in-house. In all development activities, consideration should be given to two rules: the protection of critical relationships and the active management of the outsourcing.Another factor that influences outsourcing in development is the philosophy of the company.This should be considered when examining “on the fence” activities; some companies may view certain activities as generic skills where others see a core strategic advantage.Once the core vs.non-core determination has been made,companies must assess the optimal outsourcing approach for non-core activities.We have divided outsourcing approaches into two categories, functional and integrated, based on the degree of integration necessary between the sponsor and the vendor. As a general rule, if a core activity must be outsourced for some reason, integration with vendors is essential. For non-core activities, a functional approach is usually best.Integrated outsourcing arrangements would focus on accelerated decision making and minimizing time to proof of concept.Vendors would work with integrated workflows and pre-established standard operating procedures (SOPs), with IT decision support. In many cases,integrated outsourcing calls for a component of risk sharing in the compound’s success,and an economic model should be established in which there is an emphasis on quality, not quantity, and it is in the interest of the vendor to “kill” compounds as soon as they begin showing unfavorable results.Clinical data management has been one area of development activity that is routinely outsourced and has been increasingly off-shored. As a very basic example of the application of the core/non-core decision process, we have used the example of the partnership between Accenture and Wyeth to illustrate the use of our framework.To begin,from the company perspective,data management is not a differentiating activity. It does not provide a point of competitive advantage over a competitor, and it is unlikely that any one company possesses a leadership position in this area.As a result,in outsourcing there is very little executional or IP risk associated with outsourcing data management.In terms of capacity,many companies have been moving away from holding data management capabilities in-house over the last decade,so many may not even have internal resources.Additionally,there are no key relationships associated with the data management function.From a company situation standpoint,all variables point toward data management being a non-core activity.The vendor situation for data management points toward the same conclusion. Sophisticated vendor environments already exist for conducting data management, particularly in off-shore communities. Because of the ample supply of vendors, quality standards are high and generally well trusted, and competition has lowered prices to a very attractive level.With both situations pointing toward outsourcing, there is very little by way of company situation that would compel a pharmaceutical company to keep data management activities in-house. Below we have profiled an example of Wyeth’s shifting of data management to an outsource partner and the impact the relationship has had.Wyeth and Accenture formed a deal in which Wyeth gave its entire data management operation to Accenture over a 10-year contact.In doing so,half of Wyeth’s 300 data management positions were eliminated,and the remaining employees were transferred to Accenture for employment. The more mundane data management tasks, such as data entry, would be sent offshore to India to make use of Accenture’s specialized facilities. In order to get the contract, Accenture had to accept a risk-sharing arrangement and meet highly specific performance criteria all while cutting Wyeth’s data management costs by 50%.The result was a deal that provides significant cost savings for Wyeth, shifting fixed costs to variable costs while also tapping into new capabilities through Accenture’s service delivery center in India. New efficiencies can also be reached; functions that would have taken Wyeth more than 100 days must be reduced to about 20 days under the deal, or Accenture will have to pay Wyeth.In contrast to the data management example, program management activities can be examined as an extreme illustration of an activity that is core and must be kept in-house.As a result,we do not have a case studied to apply to this framework because we are not aware of outsourced program management arrangements.While specific company situations will vary,all companies will find a very high executional and IP risk associated with outsourcing the program management function. Program management is one of the remaining areas where companies can hold a significant leadership advantage over competitors and continue to buildinternal know-how (intellectual capital).Additionally, it is critical that internal program management teams develop and leverage key relationships along the value chain.The vendor situation further reinforces that program management remains in-house.While program management functions may be a portion of outsourcing specific activities, vendors solely dedicated to program management do not readily exist. Even if an extreme company situation dictated that program management activity be outsourced, a fully integrated outsourcing approach would be necessary. This arrangement would require oversight from the pharma company and would therefore result in significant overlap of responsibilities.Across the industry’s R&D functions,significant un-tapped degrees of freedom exist for making the cost base more variable. Opportunities for flexibility gains have to be balanced against IP and organizational know-how risks; in general the economic value of flexibility gains increases toward the sell end of the continuum while IP concerns lessen. Means for reducing revenue volatility clearly exist, including co-development of products and/or sharing of commercial rights/returns with commercial partners, and expansion of effective portfolio size in partnership with passive investors.Whether such volatility gains are worth the associated costs can and should be analyzed.Determining a company's degrees of freedom and how they may be leveraged is a two-step process. First, this general framework of core vs. non-core needs to be fine-tuned by persons having greater proximity to each component of the value chain (in general),and to the company’s business circumstances and organizational status (in particular).Second,available degrees of freedom (i.e. owned non-core functions)should be prioritized according to likely gains, associated risks,and the extent to which well-developed external platforms for performing these functions exist.译文核心和非核心结构资料来源：/articles/2009/01/ core-vs-noncore-framework作者：尼尔·麦克利斯特，理查德·埃文斯，凯瑟琳·沃利斯在整个制药行业，戏剧性和持续性的对于经营环境的改变，正在呼吁对公司的战略和结构的修改。

aot generic method

aot generic method"AOT Generic Method: Enhancing Efficiency in Various Domains"Introduction:In today's fast-paced world, efficiency is the need of the hour. Whether it is in business operations, problem-solving, or decision-making, having a method that can streamline processes and maximize outcomes is invaluable. This article explores the concept of the AOT Generic Method and discusses the step-by-step approach to implementing it in different domains. So, let's delve into the details and discover how this method can revolutionize efficiency.Defining the AOT Generic Method:The AOT Generic Method is a systematic approach that focuses on three key stages: Assess, Optimize, and Transform. This method can be applied to various domains, including business management, project execution, and personal development, among others. Each stage plays a crucial role in enhancing performance and driving positive changes.Step 1: Assessment - Identify the Gaps:The first step in implementing the AOT Generic Method is to assess the current state of affairs. This involves identifying the existing gaps, inefficiencies, and bottlenecks. In business management, this could mean analyzing processes, systems, and resources to determine where improvements can be made. For personal development, it may involve a self-assessment to identify strengths and weaknesses.Step 2: Optimization - Streamline Processes:Once the areas requiring improvement have been identified, the focus shifts to optimizing processes. This stage involves analyzing existing workflows, identifying redundancies, and eliminating wasteful activities. Through careful planning and analysis, efficiency can be enhanced significantly. In business, this could mean restructuring departments or implementing automation technologies. For personal development, it may involve setting clear goals, establishing priority tasks, and developing effective time management strategies.Step 3: Transformation - Implement Systemic Changes:In order to achieve sustainable results, the final stage of the AOT Generic Method is transformation. This involves implementing systemic changes to improve overall performance. In business management, this may include retraining staff or adopting new technologies to enhance productivity and output. For personal development, it could mean establishing new habits or acquiring new skills to maximize potential.Case Study: Applying AOT Generic Method in Project Execution:To illustrate the practical application of the AOT Generic Method, let's consider a case study in project execution. The assessment stage involves analyzing the project plan, identifying potential risks and limitations. The optimization stage involves streamlining communication channels, improving resource allocation, and reevaluating project timelines. Finally, the transformation stage involves implementing new project management tools, enhancing collaboration, and establishing accountability measures. Through these steps, project efficiency is significantly enhanced.Conclusion:The AOT Generic Method offers a structured approach to improving efficiency in various domains. By assessing, optimizing, and transforming processes, organizations and individuals can tap into their true potential. This approach provides a roadmap to identify weaknesses and streamline operations, resulting in improved productivity and outcomes. Whether it is in business management, project execution, or personal development, the AOT Generic Method is a powerful tool for achieving success and driving positive change. Embrace this method, and unlock the doorway to enhanced efficiency and growth.。

FE Simulations of Arc-Height Development in Shot-Peened Almen Strips

Zhuo Chen Mechanics and Aerospace Design Laboratory, Mechanical and Industrial Engineering,University of Toronto,5King’s College Road,Toronto,ON M5S3G8,CanadaFan Yang Mechanics and Aerospace Design Laboratory, Mechanical and Industrial Engineering,University of Toronto,5King’s College Road,Toronto,ON M5S3G8,CanadaS.A.Meguid1Fellow ASME Mechanics and Aerospace Design Laboratory, Mechanical and Industrial Engineering,University of Toronto,5King’s College Road,Toronto,ON M5S3G8,Canadae-mail:meguid@mie.utoronto.ca Realistic Finite Element Simulations of Arc-Height Development in Shot-Peened Almen StripsIt is the objective of this study to conduct realistic simulations of the arc-height develop-ment in shot-peened Almen strips using theﬁnite element(FE)method.Unlike our earlier work which is devoted to relaxation of shot peening induced residual stress,in this paper, the focus is on peen forming as a result of repeated spherical impingement.Speciﬁcally, a3D FE model with1500randomly distributed shots bombarding an Almen strip was developed.Strain rate dependent plasticity was considered and an artiﬁcial material damping was applied to control the undesired high-frequency oscillations.The solution further adopts both explicit dynamic and implicit quasi-static analyses to simulate the entire arc-height development in the Almen strips.Quantitative relationships between the resulting equivalent plastic strain and the associated residual stress distribution for a given shot velocity and shot numbers are established and discussed.The work also con-siders the effect of repeated impacts upon the induced residual stressﬁeld using a large number of random shots.Attention was further devoted to the effect of the strip constraint upon the outcome of the impingement.Our results indicate that the proposed FE model is a powerful tool in investigating the underlying mechanisms of the peening treatment. [DOI:10.1115/1.4028006]Keywords:shot peening,ﬁnite element analysis,Almen strip test,residual stress,peening intensity1IntroductionShot peening is a cold working process developed to improve the fatigue life of many critically loaded engineering components, such as compressor and turbine disks in gas turbine engines,bogie beam in landing gears,leaf and coil springs in cars,just to name a few.Similarly,shot peening can be used to form wing skins and rocket shells.In this case,the treatment is known as peen forming [1,2].The plastic indentations resulting from multiple shot impingements result in increased surface area,leading to the desired curvatures or arc-heights.The advantage of this dieless forming process lies in its ability to generate compressive residual stresses and increasedﬂow stresses in the exposed surface layers, leading to improvement in their fatigue resistance[3].In fact, Almen and Black[4]used peen forming as an indirect measure to ensure the consistency of the shot peening treatment.In that case, the bombardment of a thin strip,known as the Almen strip,leads to a speciﬁc arc-height for given peening parameters.A saturation curve is then developed and the peening intensity is identiﬁed to determine the consistency of the treatment.In this paper,we devote our attention to the determination of the peen forming and the development of arc in the Almen strips.One of the key peening parameters is peening intensity.Peen-ing intensity correlates the amount of energy transferred during the impact of a typical shot with the workpiece and is related to the kinetic energy of the blast stream[5].Indirect measurement of peening intensity was originally proposed by Almen and Black [4],relying on a form of Almen test system.In the Almen test, which complies strictly with peening standards and processing sheets,the strip isﬁxed to a mountingﬁxture by means of four roundhead bolts with nuts.Following the full coverage treatment of the strip,the bolts are removed,and the residual arc-height over aﬁxed length is measured by a depth gauge(known as Almen gauge).The arc-height at saturation is deﬁned as the Almen intensity.Some approximate analytical models have been developed to relate residual stresses to Almen intensity[6–8]. Guagliano et al.[9,10]proposed an approach to correlate Almen intensity to shot peening parameters(shot type,shot velocity,and shot size)based on FE simulations,theory of elasticity,and regression analysis.Careful search of the literature indicates that a few numerical attempts have been made to simulate peen forming.For example, a“squeezed pressure layer”model was proposed by Grasty and Andrew[11],who introduced a small plastic deformation into the upper most surface layers.Appropriate squeeze pressure was determined by trial and error that provided identical curvatures resulting from indentation tests.Levers and Prior[12]proposed an indirect approach which employed an equivalent thermal load-ing applied to the work piece to generate nonuniform thermal strains.A thermally loaded unit scheme evolving from the above models was developed by Wang et al.[13].In that model,numeri-cal Almen intensity curve was developed by applying multiple loading cycles to the composite shell elements.It is worth noting that the equivalent thermal loading used in Levers and Prior as well as Wang does not correspond to the actual peening process and is a mere artiﬁcial means used to aid their simulations.A combined FE/DE explicit/implicit simulation strategy was pro-posed by Han et al.[14]in which a residual stress/strain proﬁle was initially identiﬁed by simulating a shot bombarding a small region of the workpiece using explicit dynamic analysis.The resulting deformation and stress distributions of the entire work-piece were then obtained by employing this residual stress/plastic strain proﬁle as initial conditions for a subsequent implicit static solution.Kang et al.[15]used a3D explicit dynamic FE analysis followed by a static springback analysis to simulate the peen1Corresponding author.Contributed by the Materials Division of ASME for publication in the J OURNAL OFE NGINEERING M ATERIALS AND T ECHNOLOGY.Manuscript received February24,2014;ﬁnal manuscript received June30,2014;published online July29,2014.Assoc.Editor:Hareesh Tippur.Journal of Engineering Materials and Technology OCTOBER2014,Vol.136/041002-1Copyright V C2014by ASMEforming of an aluminum2024-T351target.The sequence of the shots followed a prespeciﬁed ordered pattern.The numerical results of component deﬂection showed a good agreement with experimentalﬁndings.In industry,it is of great importance to quantify the effect of the parameters which govern the peen forming process.These include,but are not limited to,shot number and shot velocity on residual stress and arc-height development.The literature contains basic attempts with gross over-simpliﬁcations in treating this problem.For example,some models follow a regular predeﬁned shot sequence.In others,the Almen arc-height was calculated from grossly over-simpliﬁed elastic beam formula.In this paper, we develop a complete integrated model which avoids making ar-tiﬁcial assumptions and/or ad hoc analytical methods to overcome the complexity of the problem.A random distribution of1500 shot impacts was simulated followed by a springback analysis. Strain rate dependent plasticity was considered and an artiﬁcial material damping was applied to control high frequency oscilla-tions.A parametric study was conducted to investigate the effect of shot velocity and shot number upon the plastic strain develop-ment and the induced residual stressﬁeld.Both the residual stress distributions in the strips at constrained and unconstrained state are considered and analyzed to illustrate the arc-height develop-ment mechanism.In our earlier work,the focus was on relaxation of shot peening induced residual stress due to cyclic thermomechanical loads.Our investigations revealed that thermomechanical overload can almost fully relax the entire shot peening residual stresses within theﬁrst cycle due to the combined effects of decreased material yield strength and the cyclic plastic deformation.In this work,our focus is in the study of the mechanisms associated with peen forming of thin structures.Speciﬁcally,this study is concerned with the development of arc-height,as in the commonly adopted Almen strips,and the residual stress variation due to the springback.The paper is organized as follows:Following the Introduction, Sec.2presents the details of the model and discusses its quality assurance.Section3provides the model predictions obtained using3D random impingements at a strain rate sensitive strip made of spring steel.The model seeks the residual stress proﬁles for the strips at constrained and unconstrained states,respectively. Section4concludes the work.2FE Model2.1Details of FE Model.A typical Almen strip test was simulated using a two-step scheme with ABAQUS Explicit and ABAQUS Standard modules[16].The76Â19Â2:39mm type C Almen strip was modeled.In the Almen test,the strip is held byfour screws at the corner,as shown in Fig.1(a).A quarter of the strip,with symmetric boundary conditions,was included in the FE model to reduce the computational cost.Some1500random rigid shots with1mm radius were generated in MATLAB,which corresponds to6000shots impinging the fully dimensional strip. The coordinates of the center of each shot are deﬁned as follows:x¼À19þ38Ârandð1;1Þy¼À4:75þ9:5Ârandð1;1Þz¼1:01þðnÀ1ÞÂvÂD twhere rand(1,1)is a uniform pseudorandom number generator in the interval[0,1],v is the shot velocity,and D t is the interval between sequential shot impingements,which is set to 3:5Â10À6s in our model.The large shot size was used merely to reduce computational cost.However,the approach adopted here can also apply for all shot sizes.The FE model and the imposed boundary conditions are shown in Figs.1(b)and1(c).Corresponding to the real Almen test condi-tions,a rigid surface was used to support the strip.A general con-tact condition with a commonly adopted friction coefﬁcient l¼0:3was deﬁned between them.In addition,the normal dis-placement of the nodes at the speciﬁed location(see Fig.1(b))is also constrained representing the strip holder position restrictions. In theﬁrst step of simulation,plastic stretching of the top layers is developed as a result of the1500random shots impinging the con-strained strip.Following that,the unloading springback was mod-eled by transferring theﬁrst step results to ABAQUS Standard analysis(implicit solver).The external constraints are then removed and the arc-height of the strip is developed due to the unbalanced induced stress.2.2Material Constitutive Models.Almen strip is made of SAE1070steel with Young’s modulus E¼200GPa,Poisson’s ra-tio v¼0.3,and density q¼7800kg/m3.The initial yield stressis Fig.1Investigated FE model:(a)schematic plot of the test strip,(b)top view of the strip withboundary conditions,and(c)3D view of the full FE model041002-2/Vol.136,OCTOBER2014Transactions of the ASMEr 0¼1120MPa and the strain hardening parameters were extracted from the uniaxial stress–strain curve assuming isotropic hardening [10].The strain-rate sensitivity was accounted for using the data of Ref.[17].These data were incorporated in the FE model by scaling the quasi-static stress–strain curve for different strain rates.The shots were modeled as rigid balls with density q shot ¼7850kg/m 3and diameter d shot ¼2mm.According to Meguid et al.[18],the stiffness proportional damping coefﬁcientb was taken as 2Â10À9s.The mass proportional damping coefﬁ-cient a was taken as ð1=H Þðﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2E =q p Þ,which is dependent on the Young’s modulus,density and thickness of the target.Meguid et al.[18]and Kim et al.[19]concluded that the maximum resid-ual stress converges to a steady value for a friction coefﬁcient larger than 0.2for normal impingement,thus a Coulomb friction coefﬁcient of l ¼0:3was used for the dynamic contact between shots and strip.The shot ﬂow rate cannot be easily modeled using random jet spray and accounting for solid–ﬂuid interaction prior to nozzle exit.It was not considered in our model.Instead,use was made of the accumulative number of shots or mass of shots to replace the time duration of peening treatment.2.3Mesh,Elements,and Computational Cost.The target strip was discretized using eight-noded solid elements with reduced integration scheme (C3D8R).The eight-node hexahedral element provides more accurate results than the four-node tetra-hedral element.The reduced integration scheme was chosen because the obtained results showed no discernable difference from those using the full-integration scheme,and the computation time was largely reduced considerably.A gradient meshing scheme was adopted with ﬁne elements near the top surface to account for the possible strain and stress gradients.The top sur-face element thickness was selected to be 0.1mm,and it was increased gradually to 0.25mm near the bottom surface.The rigid shot was implemented in the FE model using an analytical rigid surface with an equivalent point mass and an equivalent point rotational inertia positioned at its center.The surface to surface contact between shots and strip was deﬁned using a penalty algo-rithm in ABAQUS 6.11.A typical model contained 84,774elements and involved 279,000degrees of freedom.A typical run took about 20h on eight INTEL processors each of 2.6GHz on Sharc-net supercomputers.3Results and Discussion3.1Equivalent Plastic Strain and Induced Residual Stresses.An ABAQUS PYTHON script was developed to calculate the averaged equivalent plastic strain and residual stress proﬁles ineach layer along the strip depth direction.Figure 2shows the vari-ation of the equivalent plastic strain in the strips at constrained and unconstrained states using 1500random shots for various shot velocities.It is found that permanent plastic strain exists in the top layer of the component with a depth of half of the shot radius.The maximum equivalent plastic strain appears to be at the top surface layers and decreases gradually along the depth direction.It also indicates that greater equivalent plastic strain is induced with increased shot velocity.Furthermore,the springback showed in-signiﬁcant inﬂuence on the plastic strain.The results indicate that under the current simulation conditions,the springback process would not generate reversedplasticity.Fig.2Effect of springback and shot velocity on equivalent plasticstrainFig.3Effect of the accumulative shot number on equivalent plastic strain for different shot velocity:(a )v 550m/s,(b )v 565m/s,and (c )v 575m/sJournal of Engineering Materials and Technology OCTOBER 2014,Vol.136/041002-3In Fig.3,we investigate the effect of shot number on equivalent plastic strain in the Almen strip for each examined velocity.It can be seen that the increase of shot bombardment induces a larger equivalent plastic strain.A parametric study was further conducted to investigate the effect of various peening parameters upon the obtained residual stress distribution and arc-height development.The investigated parameters include the number of shots,the impinging velocity, and the strip springback behavior.The residual stresses in the strip at both constrained and unconstrained states after springback were examined,and the mechanism of arc-height development is discussed.3.2Effect of Number of Shots.Figure4shows the effect of the accumulative shot number on the residual stress distribution. For that purpose,both the residual stress in the strip at constrained condition(Fig.4(a))and unconstrained condition after springback (Fig.4(b))are examined.For the strip at constrained state,it can be observed that as the accumulative shot number increases,the stress in the constrained strip became more compressive for both the surface and subsurface layers.In addition,as the number of shots increases,the saturation point is reached sooner and residual stress converges to an asymptotic value.However,the effect of the shot number on the induced residual stress is more complex for the strip at unconstrained state,and manifests itself in three ways.First,the induced residual stress ﬁeld is balanced within the exposed and unexposed surface layers; being in compression at and near the exposed surface layers and changing into tension farther away from the exposed regions.Second,the residual stresses in the exposed layer become less compressive and may even change into tensile with an increase in the shot number(increased coverage).Third,the magnitudes of both the maximum compressive residual stress and the maximum tensile stress increase with increasing coverage and kinetic energy (shot number).3.3Effect of Shot Velocity.The effect of shot velocity on re-sidual stress is examined in Fig.5.Figures5(a)and5(b)show the residual stress distribution along the depth direction after1500 random impingements for strip at both the constrained and uncon-strained states,respectively.For the constrained case,both the surface and subsurface stresses become more compressive with an increase in shot velocity.The size of the compressed layer increases as the shot velocity increases.For the unconstrained case,an increase in the shot velocity leads to an increase of the tensile stress at/near the exposed surface and an increase of com-pressive stress at/near the unexposed surface.The location of the maximum stresses was deeper for higher shot velocity.Both the resultant force and resultant moment acting on the cross section remain balanced.3.4Effect of Strain Rate.To evaluate the effect of the rate dependent behavior of the treated strips,two FE analyses were conducted:one using strain rate insensitive material and the other using rate sensitive material following Premack and Douglas[17]. Figures6(a)and6(b)show the strain rate effect upon the resulting equivalent plastic strain and the induced residual stressﬁeld in the strip,respectively.Some1500random shots were simulatedusing Fig.4Effect of shot number on residual stresses in the stripat(a)constrained state and(b)unconstrained state using shotvelocity of50m/sFig.5Effect of shot velocity on the residual stress in the stripat(a)constrained state and(b)unconstrained state after1500random shot impingements041002-4/Vol.136,OCTOBER2014Transactions of the ASMEan impact speed of 75m/s.It is worth noting that strain rate effects lead to a reduction in the plastic zone (Fig.6(a ))and an increase in the resulting residual stress ﬁeld (Fig.6(b ))for the same input impact energy.Our results conﬁrm the earlier ﬁndings stated in Ref.[18].Furthermore,we compared the Almen strip arc-heightfor strain rate sensitive and insensitive targets.Figure 7shows that the rate sensitive behavior leads to an increase in arc-height development.3.5Mechanism of Arc-Height Development.As depicted in Fig.8(a ),unbalanced stress would be induced in a fully con-strained strip which does not allow stretching and bending [20].Once the external constraints are removed,the strip will undergo stretching and bending deformations.The stress will be redistrib-uted until mechanical balance is reached (Fig.8(b )).Thus,the balanced residual stresses could be decomposed into three terms.r res ¼r ind þr axial þr bend(1)where r ind and r res are the induced stress in the strip at con-strained state and the balanced residual stress fortheFig.6Effect of strain-rate sensitivity upon:(a )resulting equiv-alent plastic strain versus depth and (b )induced residual stress versusdepthFig.7Effect of strain-rate sensitivity upon striparc-heightFig.8Sketch of shot peening on a strip:(a )enlarged view of induced stress proﬁle in the strip at constrained state and (b )arc-height development of the strip at unconstrainedstateFig.9Relationship between constrained induced stress and unconstrained residual stress from (a )Homer and Van Luchene [21]and (b )our model using 1500shots at velocity of 75m/sJournal of Engineering Materials and Technology OCTOBER 2014,Vol.136/041002-5unconstrained case,respectively.r axial and r bend are the respective stressﬁelds developed due to strip stretching and strip bending [21].For the remainder of the paper,r axial will be referred to as “axial stress”and r bend as“bending stress.”The balance of residual stress could be regarded as a superposi-tion of constrained induced stress,axial stress and bending stress.A comparison between the induced stress in the constrained con-dition and the balanced residual stress is shown in Fig.9after 1500random impacts of shots traveling at v¼75m/s.It can be seen that our results depict the similar trend as the one proposed by Homer and Van Luchene[21].Due to strip bending and stretching,the compressive residual stress at the surface decreases,while the residual stress at the bottom becomes com-pressive.To balance these,a tensile residual stressﬁeld develops in the middle region of the section.Figures10(a)and10(b)show the contour plots of the stress distributions in the fully dimen-sional strips at constrained and unconstrained states,respectively. The deformation is magniﬁed three times for better viewing.It is worth noting that the blue color indicates the compressive stress, while the red color indicates the tensile stress.The effect of springback on the residual stress distribution can be clearly observed.The peen-formed strip would contain compressive re-sidual stresses near both top and bottom surfaces,which is highly beneﬁcial in enhancing fatigue resistance.In industry,it is important to determine the proper peening pa-rameters to realize the desired peening outcome.For that purpose,a series of simulations were performed to examine the effects of shot number and shot velocity on the Almen arc-height.Full satu-ration is determined as the point on the curve of shot number ver-sus arc-height beyond which the arc-height increases by less than 10%when the number of shots doubles.The Almen intensity is deﬁned as the arc-height of the Almen strip at full saturation. Figure11shows the saturation curve for each examined veloc-ity.The quantitative relationship between the resulting arc-heights and the number of shots was bestﬁtted using Eq.(2)asy¼a1ÀeÀbxÀÁ(2)where y is the arc-height,x is the number of shots,and a;b areﬁt-ting parameters.For each saturation curve,the saturation time(number of shots) and the arc-height at saturation time(Almen intensity)were calcu-lated and listed in Fig.11.It reveals that the Almen intensity increases with increasing shot velocity and that a higher velocity results in an earlier saturation.Increase of the impinged shot num-ber results in an increase of the strip arc-height.However,the change rate of strip deﬂection decreased as the shot number increased.The maximum Almen intensity reached1.2mm after 1500impingements with a shot velocity of75m/s.Typical residual stress distributions under various peening con-ditions are illustrated in Fig.12.For a strip during the constraint state,the compressive stress state is shown in Fig.12(a).Fig.10Contours of the residual stress in Almen strip(v575m/s)for the strip at(a)con-strained state and(b)unconstrained state041002-6/Vol.136,OCTOBER2014Transactions of the ASMECompressive stress is localized at or near the exposed surface layers.No discernible tensile stress is observed.After the removal of the constraint,the stress will redistribute across the strip due to springback.The ﬁnal distribution of the stress in the strip at unconstrained state is closely dependent on the stage of shot peen-ing.Slight peening treatment would lead to an insigniﬁcant strip bending and compressive stress is generated at/near the unexposed surface,as seen in Fig.12(b ).Therefore,it would not achieve the desired beneﬁt of shot peening and peen forming.Moderate peen-ing treatment (see Fig.12(c ))would lead to an appropriate strip deﬂection.Both exposed and unexposed surfaces are subject to compressive stress,with the tensile stress only observed in the middle region of the strip.However,excessive shot peening treat-ment (see Fig.12(d ))would result in large strip bending and sub-stantial change of stress from the constraint state.In that case,the exposed surface would be subject to tensile stress,which is detri-mental in fatigue prevention.4ConclusionsA complete integrated FE model with a two-step scheme is developed to simulate the arc-height development in shot-peened Almen strips.The plastic strains and the associated residual stress in both the constrained and the unconstrained strips are examined.The results reveal that the plastic strains remain the same after springback,while the residual stress ﬁelds would adapt due to the self-equilibrating requirement.For the constrained strip,both the equivalent plastic strains and the residual stress increase with the increase in shot number and shot velocity.However,for the unconstrained strip,with a greater shot number and velocity,the residual stress in the exposed surface layers becomes less com-pressive or more tensile,and that on the unexposed surface layers becomes more compressive.Finally,the mechanism of arc-height development is discussed.It is found that the Almen arc-height increases as the number of shots increases and gradually con-verges to an asymptotic value as observed in the shot peening practice.Earlier saturation is observed for higher impingement velocities.Our simulation results indicate that the proposed FE model and modeling scheme are capable of realistically simulat-ing the peen forming process,thus implying its viability as a useful tool for the control and optimization of the treatment.AcknowledgmentThe authors wish to acknowledge the ﬁnancial support of QNRF under National Priority Research Program No.NPRP 6-292-2-127.References[1]Meguid,S. A.,1986,Impact Surface Treatment ,Elsevier,Amsterdam,Netherlands.[2]Schulze,V.,2006,Modern Mechanical Surface Treatment ,Wiley-VCH,Wein-heim,Germany.[3]Harrison,J.,1987,“Controlled Shot-Peening:Cold Working to ImproveFatigue Strength,”Heat Treat.,19(8),pp.16–18.[4]Almen,J.,and Black,J.P.H.,1963,Residual Stresses and Fatigue in Metals ,McGraw-Hill,Toronto,ON,Canada,pp.64–69.[5]Kyriacou,S.,1996,“Shot-Peening Mechanics,A Theoretical Study,”6th Inter-national Conference on Shot Peening (ICSP-6),San Francisco,CA,September 2–5,pp.505–516.[6]Hills,D.A.,Waterhouse,R.B.,and Noble,B.,1983,“An Analysis of ShotPeening,”J.Strain Anal.Eng.Des.,18(2),pp.95–100.[7]Al-Obaid,Y.F.,1990,“A Rudimentary Analysis of Improving Fatigue Life ofMetals by Shot-Peening,”ASME J.Appl.Mech.,57(2),pp.307–312.[8]Cao,W.,Fathallah,R.,and Castex,L.,1995,“Correlation of Almen Arc HeightWith Residual Stresses in Shot Peening Process,”Mater.Sci.Technol.,11(9),pp.967–973.[9]Guagliano,M.,Vergani,L.,Bandini,M.,and Gili,F.,1999,“An Approach toRelate the Shot Peening Parameters to the Induced Residual Stresses,”7th International Conference on Shot Peening,Warsaw,Poland,September 28–October 1,pp.274–282.[10]Guagliano,M.,2001,“Relating Almen Intensity to Residual Stresses Inducedby Shot Peening:A Numerical Approach,”J.Mater.Process.Technol.,110(3),pp.277–286.[11]Grasty,L.V.,and Andrew,C.,1996,“Shot Peen Forming Sheet Metal:FiniteElement Prediction of Deformed Shape,”Proc.Inst.Mech.Eng.,Part B ,210(4),pp.361–366.[12]Levers,A.,and Prior,A.,1998,“Finite Element Analysis of Shot Peening,”J.Mater.Process.Technol.,80,pp.304–308.[13]Wang,T.,Platts,M.J.,and Levers,A.,2006,“A Process Model for Shot PeenForming,”J.Mater.Process.Technol.,172(2),pp.159–162.[14]Han,K.,Owen,D.R.J.,and Peric,D.,2002,“Combined Finite/Discrete Ele-ment and Explicit/Implicit Simulations of Peen Forming Process,”-put.,19(1),pp.92–118.[15]Kang,X.,Wang,T.,and Platts,J.,2010,“Multiple Impact Modelling for ShotPeening and Peen Forming,”Proc.Inst.Mech.Eng.,Part B ,224(5),pp.689–697.[16]ABAQUS Analysis User’s Manual,2012,version 6.11,Dassault Systemes,Provi-dence,RI.[17]Premack,T.,and Douglas,A.S.,1995,“Three-Dimensional Analysis of theImpact Fracture of 4340Steel,”Int.J.Solids Struct.,32(17),pp.2793–2812.[18]Meguid,S.A.,Shagal,G.,and Stranart,J.C.,2002,“3D FE Analysis of Peen-ing of Strain-Rate Sensitive Materials Using Multiple Impingement Model,”Int.J.Impact Eng.,27(2),pp.119–134.[19]Kim,T.,Lee,H.,Hyun,H.C.,and Jung,S.,2012,“Effects of Rayleigh Damp-ing,Friction and Rate-Dependency on 3D Residual Stress Simulation of Angled Shot Peening,”Mater.Des.,46,pp.26–37.[20]Van Luchene,R.D.,Johnson,J.,and Carpenter,R.G.,1995,“Induced StressRelationships for Wing Skin Forming by Shot Peening,”J.Mater.Eng.Per-form.,4(3),pp.283–290.[21]Homer,S.E.,and Van Luchene,R.D.,1991,“Aircraft Wing Skin Contouringby Shot Peening,”J.Mater.Shaping Technol.,9(2),pp.89–101.Fig.11Shot peening arc-height versus number ofshotsFig.12Enlarged view of residual stress proﬁle under different conditions:(a )peened strip during constrained state,(b )unconstrained state at 20%of total shot number,(c )uncon-strained state at 50%of total shot number,and (d )uncon-strained state at 100%of total shot numberJournal of Engineering Materials and TechnologyOCTOBER 2014,Vol.136/041002-7。

On the strength of Ramsey’s theorem for pairs

ON THE STRENGTH OF RAMSEY’S THEOREM FORPAIRSPETER A.CHOLAK,CARL G.JOCKUSCH,JR.,ANDTHEODORE A.SLAMANAbstract.We study the proof–theoretic strength and eﬀectivecontent of the inﬁnite form of Ramsey’s theorem for pairs.Let RT n kdenote Ramsey’s theorem for k–colorings of n–element sets,and letRT n<∞denote(∀k)RT n k.Our main result on computability is:Forany n≥2and any computable(recursive)k–coloring of the n–element sets of natural numbers,there is an inﬁnite homogeneousset X with X ≤T0(n).Let IΣn and BΣn denote theΣn inductionand bounding schemes,respectively.Adapting the case n=2ofthe above result(where X is low2)to models of arithmetic enablesus to show that RCA0+IΣ2+RT22is conservative over RCA0+IΣ2forΠ11statements and that RCA0+IΣ3+RT2<∞isΠ11-conservativeover RCA0+IΣ3.It follows that RCA0+RT22does not imply BΣ3.In contrast,we reprove old result of Hirst that RCA0+RT2<∞does imply BΣ3,and so RT2<∞is strictly stronger than RT22overRCA0.1.IntroductionRamsey’s theorem was discovered by Ramsey[1930]and used by him to solve a decision problem in logic.Subsequently it has been an important tool in logic and combinatorics.Deﬁnition1.1.i.[X]n={Y⊆X:|Y|=n}.ii.A k–coloring C of[X]n is a function from[X]n into a set of size k.iii.A set H⊆X is homogeneous for a k–coloring C of[X]n if C is constant on[H]n,i.e.all n–element subsets of H are assigned the same color by C.1991Mathematics Subject Classiﬁcation.Primary03F3503C6203D3003D80.Key words and phrases.Ramsey’s Theorem,conservation,reverse mathematics, recursion theory,computability theory.Research partially supported NSF Grants DMS-96-3465(Cholak),DMS-95-03398and DMS-98-03073(Jockusch),and DMS-97-96121(Slaman).12P.CHOLAK,C.JOCKUSCH AND T.SLAMANRamsey’s Theorem.For all k and n,every k-coloring of[N]n hasan inﬁnite homogeneous set.An extensive treatment of Ramsey’s Theorem,emphasizing itsﬁ-nite version,may be found in Graham,Rothschild and Spencer[1990],where many related results and applications are also discussed.There are(at least)two ways to use the tools of mathematical logicto analyze Ramsey’s theorem.One is via computability theory(orequivalently recursion theory):Study the complexity(in terms of thearithmetical hierarchy or degrees)of inﬁnite homogeneous sets for acoloring C relative to that of C.(For simplicity,we can assume thatC is computable(recursive)and relativize.)The other is via reverse mathematics:Study the proof–theoretic strength of Ramsey’s theorem(and its natural special cases)as a formal statement in second orderarithmetic.There has been much work done along these lines.For example,con-sider the independent work by Jockusch[1972],Seetapun,and Slaman(see Seetapun and Slaman[1995]).Our task in this paper is to reviewbrieﬂy the work that has been done and further this analysis.Before getting into details we mention two themes in this work thatwe would like to make explicit.Theﬁrst is that results in computabilitytheory are sometimes the forerunners of results in reverse mathemat-ics.This is certainly the case for Weak K¨o nig’s Lemma and almostall versions of Ramsey’s Theorem.The second theme is the use ofpaths through trees,more speciﬁcally Weak K¨o nig’s Lemma,the LowBasis Theorem,and Scott sets.Almost all of our results use one ormore of these three items in its statement or proof.Whether this useis necessary is unknown.In Section2,there is a brief summary ofprevious work on the analysis of K¨o nig’s Lemma and the inﬁnite formof Ramsey’s Theorem in terms of computability theory and of reversemathematics.Our starting point is the following result,which refutes an old con-jecture of Jockusch(see Jockusch[1972,Corollary4.7]or the secondparagraph after Theorem2.5).Theorem3.1.For any computable coloring of the unordered pairs ofnatural numbers withﬁnitely many colors,there is an inﬁnite low2homogeneous set X,i.e.,X ≤T0 .The proof is not simply an eﬀectivization of the standard proof ofRamsey’s Theorem.Instead,theﬁrst step is to restrict the given com-putable coloring to a low2r–cohesive set A,which exists by Jockuschand Stephan[1993],Theorem2.5.Since for any a the color of theRAMSEY’S THEOREM3 pair{a,b}is independent of b for suﬃciently large b∈A,the color-ing induces a coloring of[A]1which is∆0,A2.Then the relativizationto A of the following new result easily yields the desired inﬁnite low2 homogeneous set.Theorem3.7.If A1,A2,...,A n are∆02sets and∪ni=1A i=N,thensome A i has an inﬁnite low2subset.We give two proofs of the above result,with the common elements of the two proofs presented in Section3.Theﬁrst proof,which was our original proof,is technically easier since it uses the Low Basis theorem to reduce the problem of control-ling the second jump of the constructed set to the easier problem of controlling theﬁrst jump of the constructed set.(A similar approach was used in Jockusch and Stephan[1993]to construct a low2r–cohesive set.)In Section4we will present thisﬁrst proof and also,for the con-venience of the reader,a construction of a low2r–cohesive set based on control of theﬁrst jump.This“ﬁrst jump”method also yields inter-esting additional information on the jumps of degrees of homogeneous sets(see Section12).Our second proof,which will be presented in Section5,is more direct and also somewhat more complicated.It proceeds by direct control of the second jump of the constructed set.It gives no additional information on degrees of homogeneous sets,and the reader interested only in the computability aspect of this paper could well omit reading it.We also give a construction of a low2r-cohesive set using direct control of the second jump.As above,this is more complicated than the construction used in Jockusch and Stephan[1993].The reason for giving these more involved constructions is that they seem to be more suitable to adapting to models of arithmetic to obtain results in reverse mathematics as described below.In Section6,we quickly introduce the reader to second order arith-metic.(The reader unfamiliar with second order arithmetic may want to start there.)A listing of the needed statements of second order arith-metic and the relationships among them can be found in Section7.We will assume that the reader is somewhat familiar with computability theory;a good introduction is Soare[1987].In Section8,we discuss some results concerning Weak K¨o nig’s Lemma;a reasonable portion of this section was known previously but much of it is new.Sections9–11present our conservation theorems for Ramsey’s The-orem for pairs and related principles.Let X→[X]nk be the statement“every k-coloring of[X]n has an inﬁnite homogeneous set.”Thus,Ram-sey’s Theorem states for all k and n,N→[N]nk .RT nkis the statement4P.CHOLAK,C.JOCKUSCH AND T.SLAMANin the language of second order arithmetic“for all k-colorings of[N]n there is an inﬁnite homogeneous set H.”We adapt the forcing used in the“second jump”constructions to models of arithmetic to produce a notion of forcing for adding inﬁnite homogeneous sets to models of second order arithmetic while preserving the appropriate level of induction.We were led to this notion of forcing by a conjecture of Slaman(see Conjecture2.12or Seetapun and Slaman [1995]).(We do not know how to do this for the“ﬁrst jump”proofs.) Using this notion of forcing we obtain the following result.Theorem10.1.RCA0+IΣ2+RT22+WKL isΠ11-conservative overRCA0+IΣ2.This means that anyΠ11statement provable from RCA0+IΣ2+RT22is provable from just RCA0+IΣ2.The following corollary answers the second part of Seetapun and Slaman[1995,Question4.3].Corollary1.2.RT22does not imply PA over RCA0.This improves Seetapun’s result(see Seetapun and Slaman[1995])that RT22does not imply ACA0over RCA0.In the same paper,Slamanshowed in Theorem 3.6that RCA0+RT22is notΠ04–conservativeover RCA0.We should also mention that Hirst[1987]proved thatRCA0+RT22is notΣ03–conservative over RCA0(see Theorem2.11formore details).It turned out that our proof–theoretic results(but not the corre-sponding results in computability theory)are sensitive to whether our colorings use two colors or an arbitraryﬁnite number of colors.X→[ω]n<∞is the statement that“for all k,for all k-colorings of[X]n there is an inﬁnite homogeneous set.”Ramsey’s Theorem impliesfor all k and n,N→[ω]n<∞.RT n<∞is the statement in the languageof second order arithmetic“for all k,for all k-colorings of[N]n there is an inﬁnite homogeneous set H.”Using a modiﬁcation of the above mentioned notion of forcing(work-ing over a Scott set),we proved the following result.Theorem11.1.RCA0+IΣ3+RT2<∞+WKL isΠ11-conservative overRCA0+IΣ3.Thus anyΠ11statement provable from RCA0+IΣ3+RT2<∞+WKLis provable from just RCA0+IΣ3.So RT2<∞does not imply PA overRCA0.In addition,we reprove the following result of Hirst[1987, Theorem6.11].Our proof is a corollary of a slightly stronger theorem. Corollary11.5(Hirst[1987,Theorem6.11]).RCA0+RT2<∞BΣ3.RAMSEY’S THEOREM5 Since IΣ2is stricter weaker than BΣ3(over RCA0)(see Kaye[1991]or H´a jek and Pudl´a k[1993]),it follows that RT22does not imply RT2<∞over RCA0.Theorem3.1also leads to further results on computability and Ram-sey’s theorem which are covered in Section12.For example,the fol-lowing result is obtained for colorings of n–tuples:Theorem12.1.For each n≥2and each computable2-coloring of [N]n,there is an inﬁnite homogeneous set A with A ≤T0(n).Other results on computability include a characterization of the de-grees d such that every computable2–coloring of[N]2has an inﬁnite homogeneous set with jump of degree d(Corollary12.6)and a result combining cone avoidance with some control of theﬁrst jump of an inﬁnite homogeneous set(Theorem12.2).We do not succeed in obtaining a complete understanding of the proof–theoretic strength of Ramsey’s theorem for pairs or of the degrees of inﬁnite homogeneous sets for computable2–colorings of pairs.A number of open questions are listed in theﬁnal section.One theme of this paper is the close relationship between results in computability theory and results in reverse mathematics.Of course, this relationship has turned up in many other contexts,too.We hope that readers will be interested in both aspects of this paper.However, the reader interested only in the computability aspect need read only Sections1–4and12–13.The reader interested only in the reverse mathematics aspect need read only Sections1–3,5–11,and13.2.HistoryThis paper continues a stream of work on analysis of the eﬀective content of mathematical statements and corresponding work on the strength of these statements within second order arithmetic.Here we give a brief summary of some closely related previous work in this area. For further information,see Simpson[1998].See Section6for a summary of the subsystems of second–order arith-metic we shall consider.More extensive treatments may be found in Friedman[1975]and Simpson[1998].Here we brieﬂy remind the reader that our base theory is RCA0,which is based on algebraic axioms andthe schemes of∆01–comprehension andΣ1induction.Theω–models ofRCA0are those nonempty subsets of P(N)closed under⊕and closed downwards under≤T.The stronger system ACA0includes the arith-metic comprehension scheme ACA.Theω–models of ACA0are theωmodels of RCA0which are closed under the jump operation.6P.CHOLAK,C.JOCKUSCH AND T.SLAMANBefore getting to the analysis of Ramsey’s theorem,we consider K¨o nig’s lemma,which in fact will play an important role in this pa-per.Of course,K¨o nig’s lemma is the assertion that any inﬁnite,ﬁnite branching tree has an inﬁnite path.We shall actually be concerned with the case where there is an eﬀective bound on the branching.Let Weak K¨o nig’s Lemma be the assertion that every inﬁnite tree in2<ωhas an inﬁnite path,and let WKL0be RCA0+Weak K¨o nig’s Lemma. It is easy to construct inﬁnite computable trees in2<ωwith no inﬁnite computable paths,using,for example,the existence of disjoint com-putably enumerable sets which are not separable by any computable set.From this it follows that Weak K¨o nig’s Lemma cannot be proved in RCA0.In the other direction G.Kreisel proved the Kreisel basis theorem: Any inﬁnite computable tree in2<ωhas an inﬁnite path computable from the halting problem0 .The corresponding result in reverse math-ematics,due to Steve Simpson,is that Weak K¨o nig’s Lemma can be proved in the system ACA0.Theorem2.1(Jockusch and Soare[1972]).For any noncomputable sets C0,C1,...and any inﬁnite computable tree T⊆2<ωthere is an inﬁnite path f through T such that(∀i)[C i≤T f].The corresponding result in reverse mathematics is the following. Corollary2.2(Simpson[1998]).Arithmetic Comprehension is not provable in WKL0.The following result,due to Jockusch and Soare,is known as the Low Basis Theorem.Theorem8.1(Jockusch and Soare[1972,Theorem2.1]).Any inﬁ-nite computable tree in2<ωhas an inﬁnite low path f,i.e.,f ≤T0 . The forcing conditions used to prove the above result are trees,and this forcing was adapted by Leo Harrington to obtain the following result.Theorem8.4(Harrington,see Simpson[1998]).AnyΠ11statementprovable from WKL0is provable from just RCA0.SinceΣ2induction(without parameters)is not provable in RCA0(see H´a jek and Pudl´a k[1993]),it follows thatΣ2–induction is not provable from WKL0.We now consider the analysis of Ramsey’s Theorem.Theﬁrst re-sult concerning the eﬀective content of the inﬁnite form of Ramsey’s Theorem was obtained in Specker[1971].RAMSEY’S THEOREM7 Theorem2.3(Specker[1971]).There is a computable2–coloring of [N]2with no inﬁnite computable homogeneous set.Since the family of computable sets is anω–model of RCA0,there is an immediate corollary.Corollary2.4(Specker[1971]).RT22is not provable in RCA0.The next work in the area was due to Jockusch.Theorem2.5(Jockusch[1972]).i.For any n and k,any computable k–coloring of[N]n has an inﬁniteΠ0n homogeneous set.ii.For any n≥2,there is a computable2–coloring of[N]n which hasno inﬁniteΣ0n homogeneous set.iii.For any n and k and any computable k–coloring of[N]n,there is an inﬁnite homogeneous set A with A ≤T0(n).iv.For each n≥2,there is a computable2–coloring of[N]n such that 0(n−2)≤T A for each inﬁnite homogeneous set A.Theﬁrst part was proved by induction on n,using aﬁnite injury priority argument for the case n=2and the Low Basis theorem for the induction step.Note that there is a slight gap between the third and fourth items.Fix a2-coloring of[N]2.The third item tells us that there is an inﬁnite homogeneous set A such that A ≤T0 .Jockusch[1972]con-jectured that this cannot be improved to give the existence of an inﬁnite homogeneous set A such that A ≤T0 .By Theorem3.1,we now know that this conjecture was false.Simpson obtained results in reverse mathematics which are related to Theorem2.5.Corollary2.6(Simpson[1998]).i.For each n≥3and k≥2(both n and kﬁxed),the statementsRT nk and RT n<∞are equivalent to ACA0over RCA0.ii.The statement RT is not provable in ACA0.iii.RT does not prove ATR0.iv.ATR0proves RT.(Actually there are stronger results along this line in Simpson[1998].)Sketch of the proof:i.Fix n≥3.A relativized version of Theorem2.5iv“says”that any model of RCA0+RT n2must be closed under the jumpoperator.Hence any such model must contain all sets arithmetically deﬁnable from the reals in it.A relativized version of Theorem2.5i“says”that every coloring of n-tuples has a homogeneous set which is arithmetic in the coloring.Hence8P.CHOLAK,C.JOCKUSCH AND T.SLAMANin any model of ACA0every coloring of n-tuples has a homogeneous set.ii.A relativized version of Theorem2.5iv“says”that any model ofRT n2is closed under the(n−2)-jump.But one canﬁnd non–standardmodels of ACA0which are not closed under the(n)-jump for any non-standard integer n.iii.The family of all arithmetic sets is anω–model of ACA0+RT. Since this is not a model of ATR0,the claim follows.iv.Any model of ATR0is closed under the(n)-jump,for any n in the model.Theorem2.5ii“says”that every k-coloring of[X]n has a homogeneous set which is Turing reducible to X(n).If X is in M then X(n)is in M and therefore a homogeneous set for the above coloring is in M.This is how things stood for twenty years.During that time,thestrength of RT22remained a mystery.Sometimes this was phrasedas the“3-2”question:is RT22equivalent to RT32(over RCA0)?Inground–breaking work,D.Seetapun answered this question negatively by obtaining the following result.Theorem2.7(Seetapun and Slaman[1995]).For any computable2–coloring C of[N]2and any noncomputable sets C0,C1,...,there is an inﬁnite homogeneous set X such that(∀i)[C i≤T X].This allowed Seetapun to construct anω–model of RCA0+RT22 which was not closed under the jump operator and hence deduce the following corollary.Corollary2.8(Seetapun and Slaman[1995]).In RCA0,RT22doesnot imply ACA0.Hence,over RCA0,RT22is strictly weaker than RT32.In the same paper,Slaman obtained the following result going in the opposite direction.Theorem2.9(Seetapun and Slaman[1995]).RT22is notΠ04-conservativeover RCA0.This is good point to mention some forgotten work of Hirst[1987]. First Hirst focuses on Ramsey’s theory for singletons.He shows that RCA0proves RT1n,for n∈ω.Afterward he shows:Theorem2.10(Hirst[1987,Theorem6.4]).Over RCA0,RT1<∞isequivalent to BΣ2.Proof.Throughout this proof we will use the fact that BΣ2is equiva-lent BΠ1over RCA0(see H´a jek and Pudl´a k[1993]).RAMSEY’S THEOREM 9Let M be a model of RT 1<∞,X be the natural numbers of M and θbe a Σ00formula possibly with set parameters.Fix y and assume that (∀x <y )(∃z )(∀w )θ(x,z,w ).Deﬁne F by F (t )is the least n less than t such that (∀x <y )(∃z <n )(∀w <t )θ(x,z,w ),if such a n exists,otherwise let F (t )=t .Suppose H is an inﬁnite set such that F (H )=l ,for some l .Then (∀x <y )(∃z <l )(∀w )θ(x,z,w )as desired.Suppose,on the other hand that no such homogeneous set exists.By RT 1<∞,the range of F is ing ∆01comprehension,wecan construct a sequence t i i ∈X such that for each i ∈X ,t i <t i +1and F (t i )<F (t i +1).Deﬁne G (i )to be the least x less than y such that (∀z <F (t i )−1)(∃w <t i )¬θ(x,z,w ).Let S be an inﬁnite homogeneous for G .Let x 0=G (S ).Choose z 0.Since S is inﬁnite,there is some i ∈S such that F (t i )−1>z 0.So (∃w <t i )¬θ(x 0,z 0,w ).Hence,(∀z )(∃w )¬θ(x 0,z,w ).This is a contradiction.Now let M be a model of B Π1.Let y ∈X and F be a function from X to y .Suppose that (∀x <y )(∃z )(∀w )(w >z →F (w )=x ).Then by B Π1,(∃t )(∀x <y )(∃z <t )(∀w )(w >z →F (w )=x ).In particular,(∃t )(∀x <y )(∀w )(w >t →F (w )=x ).Let t 0be such a t .Then we have (∀x <y )F (t +1)=x ,contradicting the deﬁnition of F .So it must be the case that (∃x <y )(∀z )(∃w )(w >z ∨F (w )=x ).Let x 0be such an x .Then {t :F (t )=x 0}is the desired inﬁnite homogeneous set.As a corollary,he gets RT 1<∞is independent of WKL 0.He then turns to Ramsey’s theory for pairs.Hirst [1987,Theorem 6.8]shows that RT 22implies RT 1<∞.In Lemma 10.6,we will show a slightly stronger theo-rem:SRT 22implies RT 1<∞.As a result,we get the following theorem.Theorem 2.11(Hirst).i .RT 1<∞is not Σ03-conservative over RCA 0.ii .RT 22proves B Σ2.iii .RT 22is not Σ03-conservative over RCA 0.Proof.(ii)By the above results (Theorems 2.10and 10.6)we know RT 22proves B Σ2.(i)and (iii)However since RCA 0proves the same arithmetic statements as I Σ1,RCA 0does not prove B Σ2[see H´a jek and Pudl´a k,1993,Theorem IV.1.29].Hence there is an instance of B Σ2that is not provable in RCA 0.This instance is an Σ03sentence.As we mentioned earlier Hirst also showed:Corollary 11.5(Hirst [1987,Theorem 6.11]).RCA 0+RT 2<∞ B Σ3.10P.CHOLAK,C.JOCKUSCH AND T.SLAMANWe prove a slightly stronger theorem:Theorem11.4.He also showsthat there is aω-model of WKL0which is not a model of RT22.This is what was known up to the time of our work.Note that the series of results on Ramsey’s theorem for pairs is somewhat paral-lel to the results for Weak K¨o nig’s Lemma.In particular,Seetapun’sTheorem2.7and its corollary that RT22does not imply ACA0(Corol-lary2.8)are analogous,to the Jockusch–Soare cone avoidance theoremforΠ01–classes(Theorem2.1)and its corollary that WKL0is strictlyweaker than ACA0(Corollary2.2),respectively.However,in this his-torical survey there is no analogue for Ramsey’s theorem mentioned forthe Low Basis Theorem and Harrington’sΠ11conservation theorem forWKL0,Theorem8.4.It is the analog between Weak K¨o nig’s Lemma and Ramsey’s theorem,which led Slaman to make the following con-jecture.Conjecture2.12(Seetapun and Slaman[1995]).Any proof that ev-ery computable2-coloring of[N]2has an inﬁnite homogeneous low nset should lead to a proof that RCA0+RT22isΠ11-conservative overRCA0+IΣn.It is the main purpose of this paper to conﬁrm Slaman’s conjecture by supplying the analogues of the Low Basis Theorem and Harrington’sΠ11conservation theorem for WKL0for Ramsey’s Theorem,namelyTheorems3.1,10.2,and11.1.3.Low2Homogeneous SetsThe goal of this section is to outline the structure of the proof of the following theorem.For reasons stated in the introduction,we will ac-tually give two proofs of this result.The two proofs,although diﬀering considerably in their details,will both have the structure outlined in this section.Theorem3.1.For any computable k–coloring of[N]2,there is an in-ﬁnite homogeneous set X which is low2(i.e.,X ≤T0 ).Our proof of this theorem is somewhat indirect.The following deﬁ-nition will play a key role.Deﬁnition3.2.An inﬁnite set X is r-cohesive if for each computable set R,X⊆∗R or X⊆∗R.An inﬁnite set is p-cohesive if the above holds for each primitive recursive set R.Theorem3.3(Jockusch and Stephan[1993]).There exists a low2r-cohesive set.RAMSEY’S THEOREM11 A proof of this result can be found in Jockusch and Stephan[1993], Theorem2.5,although the proof presented there has an error which is corrected in Jockusch and Stephan[1997].We will present a“single jump control”proof of this theorem in Section4and a“double jump control”proof of this theorem in Section5.The reason for considering r–cohesive sets is that if X is r–cohesive and C is a2–coloring of[X]2,then the restriction of C to[X]2is stable in the sense of the following deﬁnition.Deﬁnition3.4.A k–coloring of[X]2is called stable if for each a∈X, the pair{a,b}has aﬁxed color c a for all suﬃciently large b∈X(i.e., there is a d a such that for all b greater than d a with b∈X,the color of{a,b}is c a).Stable colorings were considered in Hummel[1994]and play a crucial role in Hummel and Jockusch[n.d.].Now any computable k–coloring of pairs becomes stable when it is restricted to an r-cohesive set X.(Fix i.The sets R c={j:{i,j} has color c}are computable and partition N−{i}as c ranges over the colors.Since X is r–cohesive,there exists a color c such that X⊆∗R c. Thus the color of{i,j}is independent of j for all suﬃciently large j∈X.)Thus,using Theorem3.3,if we can prove that every stable k-coloring of[N]2has an inﬁnite low2homogeneous set,the result for arbitrary computable k-colorings of[N]2follows by relativization.(Any set which is low2relative to a low2set is low2.)The problem ofﬁnding homogeneous sets for computable stable col-orings of pairs is easily reduced,by the Limit Lemma,to the problem ofﬁnding homogeneous sets for∆02colorings of1-tuples.Lemma3.5.For any computable stable k-coloring C of[N]2,there arek disjoint∆02sets A i such that i<k A i=N and any inﬁnite subset ofany A i computes an inﬁnite homogeneous set for C.Proof.Let A i={a:lim b C({a,b})=i}.Suppose that B is an inﬁnite subset of A i.Deﬁne c k by recursion as the least c∈B such that,for all j<k,c>c j and C({c j,c})=i.Then{c i:i∈N}is the desired inﬁnite set C such that C is homogeneous for C and C≤T B.The following results(relativized to a low2r-cohesive set)will com-plete the proof that each computable k-coloring of pairs has an inﬁnite low2homogeneous set.(Theﬁrst is a special case of the second.)Theorem3.6.For each∆02set A there is an inﬁnite low2set G whichis contained in A or A.12P.CHOLAK,C.JOCKUSCH AND T.SLAMANTheorem 3.7.Let {A i }i<k be k disjoint ∆02sets such thati<k A i =N .Then for some k ,there is an inﬁnite low 2set G which is contained in A k .These results will be proved by “single jump control”in Section 4and by “double jump control”in Section 5.Before we proceed,we should note that Theorem 3.7follows by in-duction from Theorem 3.6.(Let {A i }i<k +1be k +1disjoint ∆02sets such that i<k +1A i =N .Let A =A k .Apply Theorem 3.6.If there isa low 2subset of A =A k ,we are done.Otherwise apply the relativized (to the set G )version of the induction hypothesis (i.e.,Theorem 3.7)to {A i ∩G }i<k .)But as we will later see (Theorem 11.4)this does not hold for models of arithmetic;the statement of Theorem 3.6,D 22,in second order arithmetic does not imply the statement of Theorem 3.7,D 2<∞.For this reason we will show,in Section 5.3,how to alter the forcing proof of Theorem 3.6to get a proof of Theorem 3.7.We now complete the proof of Theorem 3.1,assuming Theorems 3.3and 3.7.The idea is that the existence of a low 2r–cohesive set allows us to restrict attention to computable stable partitions of pairs,which are basically the same as ∆02partitions of 1–tuples,and these have inﬁnite low 2homogeneous sets by Theorem 3.7.In more detail,let a computable k –coloring C of [N ]2be given.Let X be a low 2r–cohesive set,and let f be the unique increasing function with range X .Deﬁne an X –computable coloring C 1of [N ]2by C 1({a,b }=C ({f (a ),f (b )}).Then,since the restriction of C to [X ]2is stable,as remarked above,the coloring C 1is a stable k –coloring of [N ]2.By Theorem 3.5relativized to X there are sets A 0,...A k −1with ∪i<k A i =N such that each A i is ∆0,X 2and for any inﬁnite set B contained in any A i ,there is an inﬁnite homogeneous set H for C 1such that H ≤T X ⊕B .By Theorem 3.7relativized to X ,there exists i <k such that A i has an inﬁnite subset B with (X ⊕B ) ≤T X .Let H be a homogeneous set for C 1with H ≤T X ⊕B ,and let H ∗=f (H ).Then H ∗is inﬁnite and homogeneous for C ,and (H ∗) ≤T (X ⊕B ) ≤T X ≤T 0 ,so H ∗is the desired inﬁnite low 2homogeneous set for C .4.Constructing low 2sets by first jump controlIn this section,we prove Theorems 3.3and 3.6by constructing sets A with A of degree at most d where d is an appropriately chosen degree satisfying d ≤0 .Here “appropriately chosen”means that d >>0 ,where the relation >>is deﬁned as follows.RAMSEY’S THEOREM13 Deﬁnition4.1.Let a and b be degrees.Then a>>b means that every b–computable{0,1}–valued partial function has a total a–computable extension.The notation>>was deﬁned and studied in Simpson[1977,pp.652–653].Actually,Simpson deﬁned a>>b to mean that each inﬁnite b–computable tree in2<ωhas an inﬁnite a–computable path.We will see in Section8that this is equivalent to the above deﬁnition.We immediately have the following implications:a≥b ⇒a>>b⇒a>bAlso,for each degree b there is a degree a>>b such that a =b .To prove this,consider the case where b=0and then relativize the result to b.Let P be the class of all{0,1}–valued(total)functions f such that f( e,i )=ϕe(i)wheneverϕe(i)↓≤1.Then P is a nonemptyΠ01subset of2ω,so by the Low Basis Theorem there is a low degree bwhich contains a function f∈P.Clearly b>>0.Of course,it is possible to decide the truth of a givenΠ02sentence inthe integers using a0 –oracle.The following lemma shows that a d–oracle has a somewhat weaker property,which will however be suﬃcient for our construction.It is related to the concept of semirecursiveness studied in Jockusch[1968].Lemma4.2.Suppose that d>>0 and that(γe,0,γe,1)e∈ωis an ef-fective enumeration of all ordered pairs ofΠ02sentences ofﬁrst–orderarithmetic.Then there is a d–computable{0,1}–valued(total)function f such thatγe,f(e)is true wheneverγe,0orγe,1is true.Proof.Let R(e,i,s)be a0 -computable predicate such that,for all e∈ωand i≤1,γe.i is true iﬀ(∀s)R(e,i,s)holds.Letδ(e)be the least s such that either R(e,0,s)or R(e,1,s)is false,if such an s exists,and otherwiseδ(e)is undeﬁned.Letθ(e)=1−i,where i is minimal such that R(e,i,δ(e))is false,providedδ(e)is deﬁned,andθ(e)is undeﬁned otherwise.Thenθis a0 –computable{0,1}–valued partial function and so has a b–computable total extension f.This f satisﬁes the conclusion of the lemma.4.1.Constructing a low2r-cohesive set usingﬁrst jump con-trol.The following theorem easily implies Theorem3.3(see Corol-lary4.5).Theorem4.3(Jockusch and Stephan[1993]).Suppose that the sets R0,R1,...are uniformly computable,and suppose that d>>0 .Then there is an inﬁnite set G such that G has degree at most d,and for all e,either G⊆∗R e or G⊆∗R e.。

AI 艺术时代的原创会呈现怎样的形态中英互译

I want you to envision a single piece of artwork generated by artificial intelligence. When most of us think of AI art, I bet we’re imagining something like this. We’re all probably picturing something totally different.请大家想象一件由人工智能（AI）生成的艺术品。

大多数人想到AI 艺术时，我敢打赌我们想象的是这样的东西。

我们想象的东西可能完全不同。

Today, with machine learning models like DALL-E, Stable Diffusion and Midjourney, we’ve seen AI produce everything from strange life forms to imaginary influencers to entirely foreign, curious kinds of imagery. AI as a technology is fascinating to us because we’re inherently drawn to things we cannot understand.今天，有了像DALL-E、Stable Diffusion 和Midjourney 这样的机器学习模型，我们已经看到AI 可以产生各种各样的东西——从奇怪的生命形式到虚构的网红，再到完全陌生的、奇异的图像。

AI 作为一种技术对我们很有吸引力，因为我们天生就会被自己无法理解的东西所吸引。

And with neural networks processing data from thousands of other images made by people from every possible generation, every art movement, millions of images in one simple scan, they can produce visuals that are so familiar yet strikingly unfamiliar. More poetically, AI mirrors us.而且，通过神经网络处理来自每一代人、每一次艺术运动的成千上万张图像，以及一次简单扫描得到的百万张图像，AI 可以生成既熟悉又惊人陌生的视觉效果。

Survey of Maneuvering Target Tracking—Part II Ballistic models target

A Survey of Maneuvering Target Tracking—Part II:Ballistic Target ModelsX.Rong Li and Vesselin P.JilkovDepartment of Electrical EngineeringUniversity of New OrleansNew Orleans,LA70148,USA504-280-7416,-3950(fax),xli@,vjilkov@AbstractThis paper is the second part in a series that provides a comprehensive survey of the problems and techniques of tracking maneuvering targets in the absence of the so-called measurement-origin uncertainty.It surveys motion models of ballistictargets used for target tracking.Models for all three phases(i.e.,boost,coast,and reentry)of motion are covered.Key Words:Target Tracking,Maneuvering Target,Dynamics Model,Ballistic Target,Survey1IntroductionA survey of dynamics models used in maneuvering target tracking has been reported in[1].It,however,does not cover motion models used for tracking ballistic targets(BT),that is,ballistic missiles,decoys,debris,and satellites.The primary reason for this omission is that these models possess many distinctive features that differ vastly from those covered in[1].To supplement[1],a survey of these models is presented in this paper.To our knowledge,such a survey is not available in the literature.Measurement models are surveyed in Part III[2].The entire trajectory of a BT,from launch to impact,is commonly divided into three basic phases[3,4]:boost1,ballistic (also known as coast),and reentry.The boost phase of motion is the powered,endo-atmosphericﬂight,which lasts from launch to thrust cutoff or burnout.It is followed by the ballistic phase,which is an exo-atmospheric,free-ﬂight motion, continuing until the Earth’s atmosphere is reached again.The atmospheric reentry begins when the atmospheric drag becomes considerable and endures until impact.Target dynamics during the different phases are substantially different.A succinct ouline of the main dynamic phases is given in[5].The boost phase is characterized by a large(primarily axial)thrust acceleration,which in the case of rocket staging is subject to abrupt,jump-wise changes.The effects of the atmospheric drag and the Earth’s gravity are also essential in this phase.After the boost(i.e.,during the post-boost),drag is no longer present and the level of thrust becomes low or vanishes.During the exo-atmospheric ballistic phase the motion is governed essentially by the Earth’s gravity only—the trajectory is more predictable,still,small re-targeting maneuvers are possible.The reentry phase features a rapid drag deceleration with possible lateral accelerations.Overall,a ballistic target has a less uncertain motion than many other types of powered vehicles,such as maneuvering air-craft or agile missiles:Most BTs follow aﬂight path that is to a large extent predetermined by the performance characteristics, speciﬁc for a target type,hence the name ballistic targets.Only some more advanced missiles can undergo small maneuvers, usually for re-targeting.However,this does not mean that the motion of a foreign target can be determined accurately.In fact,as vividly expressed in[4],tracking a foreign BT is likely to be“plagued”with a variety of uncertainties,including those concerning trajectory loft or depression,thrust proﬁle management,target weight,propellant speciﬁc impulse,sensor bias, and atmospheric parameters.Many of these uncertainties stem from the uncertainty in the target/missile type,the principal uncertainty in modeling the motion of a foreign BT for the tracking purpose.Compared with[1],this survey is slightly more tutorial in nature for the beneﬁt of the readers less familiar with the ballistics or aerodynamics.The same disclaimers as made in the Introduction section of[1]apply to this survey,including those on the restriction to point targets for temporal behaviors and the interrelationship between dynamics models and tracking algorithms.The reader should keep in mind that while there have been many extensive studies of BT tracking,not many Research supported by ONR Grant N00014-00-1-0677and NSF Grant ECS-9734285.1And possibly a post-boost phase,which involves small maneuvers.Signal and Data Processing of Small Targets 2001, Oliver E. Drummond, Editor,Proceedings of SPIE Vol. 4473 (2001) © 2001 SPIE · 0277-786X/01/$15.00559results have been published in the open literature.In other words,much of the BT information,particularly target-type speciﬁc information,such as motion proﬁles(or templates),is classiﬁed and not open to the general public.As a result,this survey covers only those dynamics models used for BT tracking in the open literature available to us.While some models covered are applicable to satellite tracking,the emphasis of the survey is on missile tracking.The rest of the paper is organized as follows.Sec.2provides background knowledge in ballistics and aerodynamics that is necessary to understand the BT motion models presented later.Motion models for the simplest phase,the ballisticﬂight, are covered in Sec.3.This is followed in Sec.4by a survey of the models for reentry vehicles.Sec.5then describes models for the boost phase,the most sophisticated phase.2PreliminariesIn this section,we provide necessary,rudimentary background information in aerodynamics and ballistics to help the reader understand the BT motion models presented in the subsequent sections.We hope this will make the text more system-atic and self-contained.2.1Coordinate SystemsThe coordinate systems(CS)commonly used in BT tracking are illustrated in Fig.1.Much more detailed information on coordinate systems can be found in e.g.[4,6,7,8,9,10].Fig.1:Coordinate SystemsThe Earth-centered inertial(ECI)CS isﬁxed in an inertial space(i.e.,ﬁxed relative to the“ﬁxed stars”).It is a right-handed system with the origin at the Earth center,axis pointing in the vernal equinox direction,axis pointing in the direction of the North pole.Its fundamental plane coincides with the Earth’s equatorial plane.The Earth-centered(Earth)ﬁxed(ECF,ECEF,or ECR)CS also has its origin at the Earth center,its axis ,and fundamental plane coincident with the Earth’s equatorial plane.Its axes and,however, rotate with the Earth around the Earth’s spin axis as points to the prime meridian.The East-North-Up(ENU)CS has its origin at some point on the Earth surface or above it(usually at the location of a sensor).Its Up-axis is normal to the Earth’s reference ellipsoid,2usually deﬁned by the geodetic latitude .The axes and are tangential to the Earth reference ellipsoid with pointing North and East.Another CS(not depicted in Fig.1),commonly used in BT tracking,is the radar face(RF)CS3[11,6].It can be deﬁned from the local radar ENU-CS by two angles of rotation[6].For a phased array radar,the and axes of the RF-CS lie on the radar face,with axis along the intersection of the radar face with the local horizontal plane,and is along its normal 2Note that the local vertical direction differs from the radial axis.They coincide if a spherical Earth model is used.3Note that it differs from the so-called radar reference CS,which is actually just an ENU-CS at radar site[11].Proc. SPIE Vol. 4473560(boresight)direction.Such a radar measures the range and the direction cosines and. This nonorthogonal coordinate system is often referred to as the RUV-CS.In Fig.1,points and(i.e.,vectors and)denote target and sensor positions in the ECI-CS or ECF-CS,respectively.Vector deﬁnes the target position with respect to the sensor in the ENU-CS.Note that the velocity in the ECF-CS can be expressed in the ECI-CS as follows(1) where is the Earth rotation rate.The choice of a CS is a complex issue,depending on numerous factors and related with many elements of a tracking system[9,10].For more information,the reader is referred to[2].2.2Total AccelerationFor the tracking purpose,only the most substantial forces that may act on a BT are considered:thrust,aerodynamic forces (most notably,atmospheric drag and possibly lift),the Earth’s gravity,and,depending on the CS used,possibly the Coriolis and centrifugal forces.Not all these forces are present at a level that affects signiﬁcantly the motion of a BT in the different regimes of the trajectory.Speciﬁcally,for most tracking applications the signiﬁcant forces in difference phases are Boost:Thrust,drag,and gravity.Coast:Gravity only.Reentry:Gravity,drag,and lift.The total acceleration of a BT,in the ECI-CS in a fairly general setting,can be decomposed as(2)where(),and denote the acceleration components induced by thrust,aerodynamic forces(drag and lift), and gravity,respectively.Note that the acceleration here is expressed in the ECI-CS(i.e.,)in the absolute sense. If the target motion is considered within a frameﬁxed to the Earth(e.g.,the ENU-CS),then its relative total acceleration (deﬁned as for)should be corrected with the accelerations induced by the Earth’s rotation[3,4]:(3)Coriolis Centrifugalwhere is the Earth’s angular velocity vector.The terms and represent the accelerations due to the Coriolis4and centrifugal forces,respectively.Clearly,the entire end-to-end motion of a BT can be modeled by a“wide-band”dynamic model(e.g.,nearly constant velocity,acceleration,jerk,or Singer model[1])capable of covering the whole range of possible trajectories.Most models developed for the boost phase,the most sophisticated of all three phases,can serve this purpose.This is,however,rather crude and not in common use.What is more natural and rational,as well as common practice,is to develop different models speciﬁc for each trajectory portion that more fully exploit the inherent characteristics of the portion.We survey next dynamics models for the distinct regimes of a BT proposed/used in the available literature.The motion phases are ordered with respect to their sophistication levels,rather than to their chronology in the trajectory.4The Coriolis force is an equivalent force induced by the rotation of the Earth that causes the Coriolis effect—the apparent deﬂection of a body in motion with respect to the Earth,as seen by an observer on the Earth.Proc. SPIE Vol. 44735613Ballistic(Coast)FlightAfter the thrust cut-off or burnout and leaving the atmosphere a BT enters the free-ﬂight portion of its trajectory—no thrust is applied and no drag is experienced.The motion may be considered governed by the gravity alone—other factors (e.g.,perturbations)are neglected.By(2)the total acceleration is and thus obtaining a coast model of the BT amounts to selection of an appropriate gravity model.3.1Gravity ModelsFlat Earth Model.The simplest possible model of gravity assumes aﬂat,nonrotating Earth.In this model,the gravity acting on the target in the ENU-CS is,where is the constant gravitational acceleration of the Earth.The boost and reentry portions of a BT trajectory are relatively short in range compared to the Earth radius and take place in a close vicinity of the Earth.Thus a model ofﬂat,non-rotating Earth may be adequate,particularly in view of the presence of other more dominating uncertainties.On the contrary,the coastﬂight comprises much greater ranges usually and thus accounting for the Earth sphericity(and even ellipticity)and rotation is essential.Spherical Earth Model.Assume that the Earth and the BTs can be represented as point masses at their geocenters5and that the gravitational forces of the moon(and other stars)can be neglected.Since the target has a negligible mass relative to the Earth’s mass,the gravitational acceleration is the solution of a so-called restricted two-body problem,obtained by the Newton’s inverse-square gravity law[4]as(4) where is the vector from the Earth center to the target,is its length,is the unit vector in the direction of ,and is the Earth’s gravitational constant6.This inverse-square gravitational acceleration model(4)is classical and has been commonly used in a variety of BT tracking applications[12,13,14,15,16,17,18,19,9].It is attractive for its simplicity.It has been proven satisfactory for tracking of BTs over a short range and/or period,for example,as a model of the gravitational acceleration as an integral part of the total acceleration within a boost(boost-to-coast)motion model[15,16,19,9]or for track initiation purposes [17].However,its underlying assumptions are rather idealistic,especially the one that the Earth can be treated as a point mass.This may make it inadequate for other BT tracking applications,in particular,precision tracking of coast targets over a long time period or at a low data rate,such as the ballisticﬂight portion of a long-range missile or a satellite.Clearly, these simplifying assumptions becomes less accurate when the targets being tracked travel over a larger geographical region and a longer time period.Also,during the exo-atmospheric ballisticﬂight,gravity is either the only practical or at least the dominating acceleration acting on the targets and thus needs to be modeled more accurately.That is why the employment of more precise gravity models have been proposed.Ellipsoidal Earth Model.More accurate expressions for the gravitational acceleration can be obtained by replacing the above spherical Earth model with an ellipsoidal(or more precisely,spheroidal)Earth model.Such a more precise approx-imation—accounting for the Earth oblateness by including the second-order gravitational harmonic term of the Earth’s gravitationalﬁeld model—is[4,11](5)where is the Earth’s equatorial radius,is a correction constant,and is the unit vector along(see Fig.1).,the best known Jeffery constant,represents the difference between the polar and equatorial moment of inertia.It quantiﬁes the oblateness of the Earth because it is approximately equal to one third of the ellipticity of the Earth,hence also known as the oblateness term.This model is usually considered sufﬁciently accurate for most BT tracking applications[11],at a cost of considerable nonlinearity.5This holds if the Earth and the targets are spherically symmetric with an even distribution of their masses.6For the values of well-known constants used in this survey,the reader is referred to[4].Proc. SPIE Vol. 44735623.2Motion Models in ECI CoordinatesIn the ECI-CS,the target position and velocity vectors are and respectively.Clearly,a state-space model of a coast target with the state vector in the ECI-CS is(6)where is the gravitational acceleration given by,e.g.,(4)or(5).3.2.1Inverse-Square ModelIn this case,the acceleration part of the state-space model(6)is clearly described by the inverse-square model(4)in the ECI-CS as:with(7)A trajectory described by(6)with(7)is conﬁned to a plane in the ECI-CS,called an orbital plane,and is a part of a conic orbit,governed by the Keplerian motion equation[4].For the BTs considered in this survey,this orbit is elliptical.A dynamics model is mainly used in target tracking for propagating the target state(i.e.,state prediction),known as the Kepler problem in astrodynamics,and covariance prediction.While the model of(6)with(7)is highly nonlinear,with it the state propagation can be done in an efﬁcient manner,as outlined below.Given at time the(predicted)target state at time(with)is given by[4](8)(9)where,,and are known functions of.The variable is deﬁned through its time derivative by and can be computed accurately and efﬁciently via the Newton iteration scheme for solving a so-called time-of-ﬂight equation,as given below for the case of an elliptical orbit(i.e.,)[4]:1.Initialize2.RepeatUntilA useful program-like pseudocode7of the above algorithm for all conic(i.e.,elliptical,parabolic,and hyperbolic)trajec-tories can be found in[17],along with the computation of the Jacobian necessary for the extended 7With a few small typographical errors.Proc. SPIE Vol. 4473563Kalmanﬁltering(EKF)[13,20].Explicit evaluation of the Jacobian by direct differentiation of(7)can be found in[14].For a comprehensive treatment of the underlying theoretical background and solutions to the Kepler prediction problem,the reader is referred to[4].3.2.2Model with CorrectionA reﬁned model for the gravitational acceleration is based on the ellipsoidal Earth model(5).Note that andin the ECI-CS.Similar to(7),the acceleration part of the state-space model(6)has the following form[4,16]:with(10)This model is highly accurate.It was chosen as the coast model for a6-state EKF-based ballisticﬁlter in the ECI-CS implemented in a multiple-model tracking system developed in[16,5].However,as pointed out in[16],a model mismatch caused by moderate trajectory perturbations(due to,e.g.,maneuvers from countermeasure thrust)will lead to track diver-gence.This effect can be alleviated by introducing smallﬁctitious zero-mean process noise.This technique was used in[16] to adapt the model for the post-boost phase of the trajectory,where small maneuvers are involved.In the early work on BT(e.g.,satellite)tracking and in particular orbit determination,the target dynamics were usually considered to be deterministic(i.e.,without process noise).This often had led to divergence problems with the EKF[13]. Introducing aﬁctitious process noise input is an effective means to account for such factors as model errors,neglected perturbations,nonlinearities,and computer roundoff errors.With an additional zero-mean process noise input,while state prediction is unaffected,its associated error covariance is ampliﬁed by the process noise covariance,thus reducing the possibility for the EKF to diverge.The price paid is a possible accuracy degradation of theﬁlter when the deterministic model is indeed adequate.This is closely related to the problem of noise identiﬁcation and adaptiveﬁltering.See[13]and the brief surveys included in[21,22].The latest work along this line for tracking an orbital target can be found in[23], where is tuned adaptively using the most recent state estimate and covariance,as well as gravity-gradient model with the inverse-square law.Although a number of choices of have been proposed in the literature,remains a design parameter in practice,adjusted/tuned mostly based on engineering experience and intuition[11,24].3.3Motion Models in ENU CoordinatesIt is often preferable or necessary to formulate the BT motion in the natural sensor ENU coordinate system(Fig.1).For this reason,let,,and denote the target position,velocity,and acceleration in the ENU CS respectively.In this case the total acceleration involves the Coriolis and centrifugal terms as well as the gravitational acceleration:(11) withCoriolis andCentrifugal(12)where the vector and the Earth’s angular velocity vector are expressed in the ENU-CS(see, e.g.,[4]).During the powered-ﬂight and reentry portions of a BT,where the target is in a close vicinity of the Earth and over a short period,the effect of the Coriolis and centrifugal forces may be neglected compared with other factors.For the ballistic portion,however,this effect on the BT motion relative to a non-inertial frame(e.g.,the ENU-CS)is usually signiﬁcant and should be accounted for,in particular,in the case of long-range BTs(relative to the Earth’s radius).The target state-space model in the ENU-CS is clearly given by(13) where is the total acceleration,as a function of the position and velocity,speciﬁed by(11)and(12).Proc. SPIE Vol. 44735643.3.1Inverse-Square ModelWith this spherical Earth model,as shown in Fig.1,the vector from the Earth center to the target in the ENU-CS is,where is the known distance between the sensor and the Earth center,,is the Earth rotation rate,and are the known sensor latitude and altitude, respectively,and is the Earth center.Then it can be easily obtained from(12)that(14)(15)where(16) Thus the motion equation can be given in the following compact form(17)withAs for the models considered before,the standard EKF technique is directly applicable to this model in the ENU-CS.The linearization needed for error covariance propagation is sufﬁciently accurate for relatively short propagation time periods. For cases with high sampling rates,however,a simple yet accurate piecewise-constant acceleration model[18]could be used, discussed later.3.3.2Model with CorrectionWith an ellipsoidal Earth model in the sensor ENU-CS(see,e.g.,[10]),the vector from the Earth center to the target is,where.It is then straightforward to obtain the state-space form of the acceleration model in the sensor ENU-CS from(11)with(5)and(12)as(18)withand(19) The Jacobian of the model needed for the state prediction error covariance can be computed as[11](20)By including the correction term,this model clearly gives a more accurate approximation of the gravitational accelera-tion than the simper spherical Earth based models.A detailed discussion of its further conversion to the radar face CS and the corresponding RUV-CS and of its application in the EKF can be found in[11].A similar model with inverse-square gravity, an ellipsoidal Earth,and Coriolis and centrifugal acceleration terms in the radar face CS,along with the respective EKF with RUV measurements is discussed in[6].Proc. SPIE Vol. 44735653.3.3Piecewise-Constant Acceleration ModelThe idea of this approach,proposed in[18],is simple and natural.At each propagation time step,ﬁrst sample (i.e.,compute)the continuous-time total acceleration process at the estimated point,that is,.For example,if(17)is used to model the total acceleration,we have(21)with.Then,assume that the target motion is uncoupled along,,and directions with a constant acceleration over the time period.This leads to the following discrete-time dynamics model,for example,along the direction(22)Likewise for and directions.This state-dependent piecewise-constant acceleration model is simple and straightforward for application.(22)is used for state prediction to get.For the propagation of the associated error covariance needed in,e.g.,Kalmanﬁltering, it is proposed in[18]to add,in effect,component-uncoupled small noise to in the dynamics equation,where cov diag is a design parameter.As a result,what is proposed is a coordinate-“uncoupled,”state-dependent piecewise-constant nonzero-mean white-noise acceleration model,using the terminology of [20,1].There seems room for improvement here by a better way of adding noise or by adding temporally or spatially correlated noise,but at a price of increased complexity.Note that the motions along,,and directions described by this model are not actually uncoupled over a period of multiple time steps.The coupling arises from the dependence of the acceleration on the common target state in(21).This model has been reported in[18]to provide quite satisfactory performance—competitive with the EKF based on the fully coupled nonlinear inverse-square model—for several BT tracking scenarios with a high sampling rate(of Hz).We emphasize that the underlying idea of this approach is not restricted to coast models or the speciﬁc form of the acceleration model used:The piecewise-constant approximation can be directly applied in any situation where the total acceleration is available as a function of the position and velocity,such as those described in later sections.Its accuracy depends mainly on the sampling rate.Given the attractive simplicity and accuracy of this model,it seems worthwhile to try this model in conjunction with some more precise acceleration models,rather than(17)as originally proposed in[18].3.4Models in Other CoordinatesModels in other CSs(e.g.,radar face CS and spherical CS)can be obtained by conversion from the ENU-CS.For example, an explicit expression of the inverse-square model(neglecting the Coriolis and centrifugal forces)can be found in[25].4ReentryDuring the atmospheric reentry phase,two signiﬁcant forces always act on the target:the Earth’s gravity and atmospheric drag.If maneuvers are possibly present,a third force—aerodynamic lift force—must also be considered.A reentry vehicle (RV)is referred to as a maneuvering RV(MaRV)if it involves maneuvers or ballistic RV(BRV)if it does not.Formally the total acceleration of an RV is given by(2)or(3)with a zero thrust acceleration term().Most RV motion models can be viewed as coast models plus additional but dominating terms to account for drag and possibly lift.In fact,during reentry aerodynamic forces(drag and possibly lift)are usually much greater than gravity:drag in excess ofcan be present over most of the RV trajectory below km,possibly as high as,and lift and other transverse loads can easily exceed.Also,the effect of the Coriolis and centrifugal forces and the Earth oblateness on the target motion is much smaller than during the coastﬂight.Weﬁrst describe the models for drag and lift,and then present models for the motion of a BRV and MaRV respectively. Note that[26]is a useful reference but is largely overlooked by the tracking community so far.Proc. SPIE Vol. 44735664.1Aerodynamic ForcesDrag.The drag force acts opposite to the target velocity vector relative to the atmosphere(e.g.,as seen in the ENU-CS or ECF-CS rather than the ECI-CS),with a magnitude proportional to the air density and the square of the target speed. Speciﬁcally,the drag induced acceleration is given by[3]for(23)where is the air density,denotes the target altitude,and is referred to as drag parameter,given by through the target mass,the reference area(deﬁned as the target body cross-sectional area perpendicular to the velocity),and the so-called drag coefﬁcient.The coefﬁcient generally depends on and,but is sometimes assumed constant[9].The inverse of the drag parameter is known as the ballistic coefﬁcient(BC).Consequently,for the tracking purpose, all uncertainties associated with the drag(other than the air density and velocity)are generally aggregated in(the dynamic models of)the drag parameter(or equivalently the ballistic coefﬁcient)within the simple drag formulation(23).Lift.The lift force on an RV can be any direction in the plane perpendicular to the velocity vector[26].The presence of a lift force causes also a change in the target ballistic proﬁle,which is accounted for by an additional drag,referred to as a residual or induced drag,acting along the negative velocity direction.Like the zero-lift drag,both lift and its induced drag have a magnitude proportional to.This factor quantiﬁes the free-stream dynamic pressure[3]and is indispensable in the RV dynamic models.The complete aerodynamic acceleration is the vector sum of drag and lift:(24)where is the zero-lift drag component,given by(23),is the residual(lift induced)drag,and is the lift acceleration, perpendicular to the velocity vector.Air Density.The air density needed in the above models is usually approximated as an exponential function[27,26,28] or more accurately as a locally exponential function8of the target altitude[6,29,30]:(25) where and are known constants.See,e.g.,[30]for the parameters of the local exponential functions.More precise but nonanalytic models of the air density are available[26].4.2Ballistic(Nonmaneuvering)RV MotionIn simple terms,a nonmaneuvering reentry vehicle travels along a“purely”ballistic trajectory in the atmosphere,hence the name ballistic RV(BRV).Such an endo-atmospheric motion is governed by atmospheric drag,the Earth’s gravity,and depending on the CS,the Coriolis and centrifugal forces.Knowing the important role the drag parameter plays in the drag,it should come as no surprise that BRV motion models can be classiﬁed into two groups:with and without knowledge of the drag parameter.4.2.1Kinematic Models(Known Drag Parameter)If the drag parameter(or the BC)is(assumed)known,obtaining motion models of a BRV is straightforward by using a coast models of Sec.3and an additional drag model(23).Since the drag is a function of the relative velocity,it is more convenient to use some Earth-ﬁxed CS(usually the ENU-CS),instead of the ECI-CS.Considerﬁrst the simplest case.Assuming aﬂat,nonrotating Earth model with a constant gravity9(i.e.,neglecting the Coriolis and centrifugal accelerations and treating the target as a mass-point with constant mass),it follows directly from(23) 8“Local”in the sense that the function is piecewise exponential,with parameters depending on the target height;that is,each piece corresponds to some layer of the atmosphere.9This could be acceptable for low-altitude targets.Proc. SPIE Vol. 4473567。

1310nmLD组件高低温循环寿命研究

第33卷第1期2007年1月光学技术O PT I CAL TECHN I GUEV o l.33N o.1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Jan.2007文章编号：1002-1582（2007）01-0077-021310n m LD组件高低温循环寿命研究"左日方1，苏美开2，武金刚1（1.北京科技大学信息工程学院，北京100083；2.北京理工大学光电工程系，北京100081）摘要：研究了半导体激光二极管（LD）组件的使用寿命。

模拟了在不同的环境条件下，对1310n m LD组件进行了高低温循环寿命实验，建立了循环寿命的数学模型。

结果表明：循环寿命与循环的温差、循环的速度成指数关系，通过测试LD组件在高温差、高循环速度条件下的循环寿命，外推器件正常工作条件下的循环寿命。

得到了LD组件可靠性数据，为工艺设计人员提出量化数据。

关键词：半导体激光器组件；高低温循环寿命；寿命数学模型中图分类号：TN312+.8文献标识码：AResearch on hi g h-low te m p erat ure c y cle lifeti m e of1310n m LD s m odulesZUO f an91，SU me i-kai2，W U ji n-9an93（1.I nf or m ation En g i neeri n g S choo l，U n ivers it y o f S cience and T echno lo gy B e i j i n g，B e i j i n g100083，Ch i na）（2.D e p art m ent o f O p to-e lectron ic En g i neeri n g，B e i j i n g I nstitute o f T echno lo gy，B e i j i n g100081，Ch i na）Abstract：L if eti m e o f se m iconductor laser d iode（LD）m odule w as researched.H i g h-lowte m p erature c y cle lif eti m e o f1310 n m LD m odules w as stud ied b y ex p eri m ent under d iff erent s i m ulation environ m ental cond itions.A m at he m atical m ode l o f t he c y cle lif eti m e o f laser d iode w as estab lished.T he result show s t hat c y cle lif eti m e has i ndex re lation to t he te m p erature d iff erence o f c y cle and c y cle s p eed.B y testi n g t he c y cle lif eti m e o f LD s under t he h i g h te m p erature d iff erence and h i g h c y cle rate cond i-tions，t he c y cle lif eti m e o f LD s under t he ord i nar y w orki n g cond itions can be esti m ated.R e liab ilit y data o f LD m odule is ob-tai ned.G uantit y data is p rovi ded f or techn i C ue des i g ner.K e y words：se m iconductor laser d iode m odule；h i g h-low te m p erature c y cle lif eti m e；m at he m atical m ode l o f lif eti m e0引言1310n m是光纤通信常用的工作波长，单模光纤在该处的能量损耗只有0.40dB。

Parametric Design Representationmooc课后章节答案期末考试题库

Parametric Design & Representation_北京理工大学中国大学mooc课后章节答案期末考试题库2023年1.The way of how to fulfill the command is depended on the known conditions.参考答案:正确2.For symmetrical extrusion, the input distance of the extrusion is the half ofthe feature size.参考答案:错误3.In the feature of extrusion, we should make as more profile in a sketch aspossible for higher modeling efficiency.参考答案:错误4.The profile in the sketch must be exactly the same as the shape of the featureof extrusion.参考答案:错误5.The size of the feature can be different with the dimension of the views as ifthe size of the model is the same.参考答案:错误6.The color of fully constrained sketch is dark blue.参考答案:正确7.In the modeling, we do not need to repeat identical feature, because we canduplicate it either in the sketch or in the feature level.参考答案:正确8.If original plane or surfaces of existing feature do not meet the requirment ofa sketch making on it, we have to create a work plane.参考答案:正确9.In the command of symmetry in sketch, if there’s no symmetrical line, weneed to make a construction symmetrical line first.参考答案:正确10.The construction line is part of the sketch but not part of the prufile, it can benot fully constrained.参考答案:正确。

国际贸易术语英汉对译

国际贸易单证术语英汉对照*分析证书certificate of an alysis*一致性证书cettificate of con formity*质量证书certificate of quality*测试报告test report«产品性能报告product performa nee report«产品规格型号报告product specificati on report«工艺数据报告process data report«首样测试报告first sample test report«价格/销售目录price / sales catalogue«参与方信息party in formati on«农产品加工厂证书mill certificate*邮政收据post receipt*重量证书weight certificate*重量单weight list*证书cerificate«价值与原产地综合证书comb ined certificate of value adn orig in «移动声明A. TR . 1 movement certificate A . TR . 1«数量证书certificate of qua ntity«质量数据报文quality data message«查询query«查询回复resp onse to query«订购单purchase order«制造说明manu facturi ng in struct ions*领料单stores requisition«产品售价单in voici ng data sheet«包装说明pack ing in struct ion«内部运输单internal tran sport order«统计及其他管理用内部单证statistical and oter admi nistrative in ternaldocu-me nts*直接支付估价申请direct payme nt valuatio n request*直接支付估价单direct payme nt valuati on*临时支付估价单rpovisi onal payme nt valuati on*支付估价单payme nt valuati on*数量估价单qua ntity valuati on request«数量估价申请qua ntity valuati on request*合同数量单con tract bill of qua ntities-BOQ*不祭价投标数量单unpriced ten der BOQ*标价投标数量单priced ten der BOQ*询价单enquiry*临时支付申请in terim applicati on for payme nt*支付协议agreeme nt to pay«意向书letter of intent4订单order«总订单blanket order*现货订单sport order«租赁单lease order*紧急订单rush order*修理单repair order«分订单call off order*寄售单con sig nment order«样品订单sample order«换货单swap order«订购单变更请求purchase order cha nge request«订购单回复purchase order resp onse«租用单hire order«备件订单spare parts order«交货说明delivery in struct ions*交货计划表delivery schedule*按时交货delivery just-i n-time«发货通知delivery release«交货通知delivery note«装箱单packing list«发盘/报价offer / quotation«报价申请request for quote«合同con tract*订单确认ack no wledgeme nt of order«形式发票proforma in voice«部分发票partial in voice«操作说明operati ng in struct ions«铭牌name /product plate«交货说明请求request for delivery in struct ions«订舱申请book ing request«装运说明shipp ing in structi ons*托运人说明书(空运)shipper's letter of in struct ion s(air) *短途货运单cartage order(local tran sport)*待运通知ready for despatch advice*发运单despatch order*发运通知despatch advice*单证分发通知advice of distrbuti on of docume nt . *商业发票commercial in voice*贷记单credit note*佣金单commissi on note«借记单debit note*更正发票corrected in voice *合并发票con solidated in voice预付发票prepayme nt in voice租用发票hire in voice税务发票tax in voice自用发票self-billed in voice保兑发票delcredere in voice代理发票factored in voice租赁发票lease in voice寄售发票con sig nment in voice代理贷记单factored credit note银行转帐指示in struct ions for bank transfer银行汇票申请书applicati on for ban ker's draft托收支付通知书collecti on payme nt advice跟单信用证支付通知书document . ry credit payment advice跟单信用证承兑通知书document . ry credit acceptanee advice 跟单信用证议付通知书document . ry credit negotiation advice 银行担保申请书applicati on for ban ker's g uara ntee银行担保ban ker's guara ntee跟单信用证赔偿单document . ry credit letter of indemnity信用证预先通知书preadvice of a credit托收单collect ion order单证提交单document . presentation form付款单payment order扩展付款单exte nded payme nt order多重付款单multiple payme nt order贷记通知书credit advice扩展贷记通知书exte nded credit advice借记通知书debit advice借记撤消reversal of debit贷记撤消reversal of credit跟单信用证申请书document . ry credit application跟单信用证document . ry credit跟单信用证通知书document . ry credit notification跟单信用证转让通知document . ry credit transfer advice跟单信用证更改通知书document . ry credit amendment notification 跟单信用证更改单document . ry credit amendment汇款通知remitta nee advice银行汇票banker's draft汇票bill of excha nge本票promissory note帐户财务报表finan cial stateme nt of acco unt帐户报表报文stateme nt of acco unt message保险赁证in sura nee certificate保险单in sura nee policy保险申报单(明纟田表)in sura nee declarati on sheet (bordereau)«保险人发票in surer's in voice*承保单cover note«货运说明forward ing in structi ons*货运代理给进口代理的通知forwarder's advice to import age nt«货运代理给出口商的通知forwarder's advice to exporter*货运代理发票forwarder's in voice*货运代理收据证明forwarder's certificate of receipt«托运单shipping note«货运代理人仓库收据forwarder's warehouse receipt«货物收据goods receipt«港口费用单port charges docume nt .«入库单warehouse warrant«提货单delivery order«装卸单handling order«通行证gate pass*运单waybill*通用(多用)运输单证universal (multipurpose) transport document .*承运人货物收据goods receipt, carriage«全程运单house waybill«主提单master bill of lading«提单bill of lad ing«正本提单bill of lading original«副本提单bill of lading copy«空集装箱提单empty container bill«油轮提单ta nker bill of ladi ng«海运单sea waybill*内河提单in la nd waterway bill of ladi ng«不可转让的海运单证(通用)non-negotiable maritime transport document(generic) «大畐U据mate's receipt«全程提单house bill of ladi ng«无提单提货保函letter of indemn ity for non-surre nder of bill of ladi ng«货运代理人提单forwarder's bill of ladi ng*铁路托运单(通用条款)rail con sig nment note (ge neric term)*陆运单road list-SMGS*押运正式确认escort official recog niti on*分段计费单证recharg ing docume nt .*公路托运单road cosig nment note*空运单air waybill*主空运单master air waybill*分空运单substitute air waybill*国人员物品申报crew's effects declarati on«乘客名单passe nger list*铁路运输交货通知delivery notice(rail tran sport)*由E递包裹投递单despatch note (post parcels)多式联运单证（通用）multimodal / combined transport document（generic）直达提单through bill of lading货运代理人运输证书forwarder's certificate of tran sport联运单证（通用）combined transport document（generic）多式联运单证（通用）multimodal transport document（generic）多式联运提单comb ined tran sport bill of ladi ng / multimoda bill of ladi ng订舱确认book ing con firmati on 要求交货通知call ing foward no tice运费发票freight in voice 货物到达通知arrival notice（goods）无法交货的通知no tice of circumsta nces preve nting delvery （goods）无法运货通知no tice of circumsta nces preve nti ng tran sport （goods）交货通知delivery no tice （goods）载货清单cargo manifest 载货运费清单freight ma nifest 公路运输货物清单bordereau集装箱载货清单container manifes （un it pack ing list）铁路费用单charges note托收通知advice of collecti on船舶安全证书safety of ship certificate无线电台安全证书safety of radio certificate设备安全证书safety of equipme nt certificate油污民事责任书civil liability for oil certificate载重线证书loadli ne docume nt . 免于除鼠证书derat docume nt .航海健康证书maritime declarati on of health船舶登记证书certificate of registry船用物品申报单ship's stores declarati on出口许可证申请表export lice nee, applicati on出口许可证export lice nee出口结汇核销单excha nge con trol declarati on, exprotT出口单证（海关转运报关单）（欧共体用）despatch note moder TT1出口单证（内部转运报关单）（欧共体用）despatch note model T1T2出口单证（原产地证明书）despatch note model T2T5管理单证（退运单证）（欧共体用）control document T5 铁路运输退运单re-se nding con sigme nt noteT2L出口单证（原产地证明书）（欧共体用）despatch note model T2L 出口货物报关单goods declaratio n for exportati on离港货物报关单cargo declarati on（ departure）货物监管证书申请表applicatio n for goods control certificate 货物监管证书申请表goods con trol certificate 植物检疫申请表applicati on for phytosa nitary certificate 植物检疫证书phytosa ni lary certificate 卫生检疫证书san itary certificate动物检疫证书veteri nary certifieate商品检验申请表applicati on for in spect ion certificate商品检验证书in specti on certificate原产地证书申请表certificate of origi n, applicati on for原产地证书certificate of origi n原产地申明declarati on of origi n地区名称证书regional appellati on certificate优惠原产地证书prefere nee certificate of orig in 普惠制原产地证书certificate oforigin form GSP 领事发票cosular in voice危险货物申报单dan gerous goods declarati on出口统计报表statistical doucume nt, export国际贸易统计申报单in trastat declarati on 交货核对证明delivery verificati oncertificate进口许可证申请表import lice nee, applicati on for 进口许可证import lice nee无商业纟田节的报关单customs declarati on without commercial detail有商业和项目纟田节的报关单customs declarati on with commercial and item detail无项目纟田节的报关单customs declaratio n without item detail有关单证related docume nt海关收据receipt (Customs)调汇申请applicati on for excha nge allocati on调汇许可foreign exchange permit进口外汇管理申报excha nge con trol declarati on (import)进口货物报关单goods declarati on for implortati on内销货物报关单goods declaratio n for home use海关即刻放行报关单customs immediate release declarati on海关放行通知customs delivery note到港货物报关单cargo declaratio n (arrival)货物价值申报清单value declaration海关发票customs in voice由E包报关单customs deciarati on (post parcels) 增值税申报单tax declaratio n (value added tax) 普通税申报单tax declaratio n (gen eral) 催税单tax dema nd禁运货物许可证embargo permit海关转运货物报关单goods declarati on for customs tran sitTIF国际铁路运输报关单TIF formTIR国际公路运输报关单TIR carnet欧共体海关转运报关单EC carnetEUR1 欧共体原产地证书EUR 1 certificate of origi n暂准进口海关文件ATA carnt欧共体统一单证sin gle adm ini strative docume nt .«海关一般回复gen eral resp onse (Customs)*海关公文回复docume nt response (Customs)«海关误差回复error resp onse (Customs)*海关一揽子回复packae response (Customs)«海关计税/确认回复tax calculati on / con firmati on response (Customs)*酉己额预分配证书quota prior allocati on certificate*最终使用授权书end use authorizati on«政府合同gover nment con tract«进口统计报表statistical docume nt . import«跟单信用证开证申请书application for document . ry credit«先前海关文件/报文previous Customs document . message。