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。

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a r X i v :c o n d -m a t /0503543v 3 [c o n d -m a t .s o f t ] 28 J u l 2005Europhysics Letters PREPRINT A model for the generic alpha relaxation of viscous liquids Jeppe C.Dyre Department of Mathematics and Physics (IMFUFA),“Glass and time”-Danish Na-tional Research Foundation Centre for Viscous Liquid Dynamics,Roskilde University,Postbox 260,DK-4000Roskilde,Denmark PACS.64.70.Pf –Glass transitions.PACS.77.22.Gm –Dielectric loss and relaxation.Abstract.–Dielectric measurements on molecular liquids just above the glass transition indicate that alpha relaxation is characterized by a generic high-frequency loss varying as ω−1/2,whereas deviations from this come from one or more low-lying beta processes [Olsen et al,Phys.Rev.Lett.86(2001)1271].Assuming that long-wavelength ﬂuctuations dominate the dynamics,a model for the dielectric alpha relaxation based on the simplest coupling between the density and dipole density ﬁelds is proposed here.The model,which is solved in second order perturbation theory in the Gaussian approximation,reproduces the generic features of alpha relaxation.Introduction.–Alpha relaxation is the slowest and the dominant relaxation process in viscous liquids,closely linked to the glass transition because it sets the time scale for equilibration.Two of the most important unsolved problems regarding viscous liquids concern properties of the alpha process.One problem [1]is to explain the temperature dependence of the alpha relaxation time τ,which for some liquids increases by more than a factor of 10when temperature is lowered by just 1%.Here we address the second problem:how can the observed shape of the alpha loss peak be explained?It was recognized already more than 40years ago that the dielectric alpha process is always asymmetric,with almost Debye behavior at frequencies lower than the loss-peak frequency whereas there is a non-trivial power law loss,ǫ′′(ω)∝ω−n ,above the loss peak [ǫ(ω)=ǫ′(ω)−iǫ′′(ω)is the frequency-dependent dielectric constant,ǫ′′the dielectric loss].In many cases n was found to be close to 1/2and over time several theories were proposed predicting n =1/2[2].As measurements accumulated apparently contradicting any such universality,it became popular to represent data by the stretched exponential dipole time-autocorrelation function (∝exp[−(t/τ)β]where t is time and β<1a parameter).Via the ﬂuctuation-dissipation theoremthis function reproduces the Debye behaviour at low frequencies,and it predicts an asymptotic power-law decay above the loss peak with n =β.Often,however,the high-frequency power law applies only just above the loss peak frequency and a “wing”appears at higher frequencies.Nagel and co-workers proposed that,when scaled in an unusual way,the alpha dielectric loss follows a universal curve which includes the wing [3].In 1997at the 3rd International Discussion Meeting on Relaxations in Complex Systems it was suggested by Olsen,however,cEDP Sciences2EUROPHYSICS LETTERS that beta processes may play a role at much lower frequencies than previously expected and that this could explain the wing[4].This was conﬁrmed in experiments by Lunkenheimer and co-workers and subsequently by other groups,showing that the wing indeed develops into a separate relaxation process after long-time annealing right below the glass transition [5].A publication from2001giving data for ten molecular liquids[6]presented indications that n=1/2is a universal high-frequency exponent in the following sense:Equilibrium viscous liquids approach this behaviour as temperature is lowered and eﬀects of beta processes gradually become negligible;in some cases,however,this may happen only when the alpha relaxation time is months or longer,making it impossible to verify or falsify the conjecture that n=1/2is universal.It is not generally accepted that n=1/2is a generic high-frequency exponent of alpha relaxation.Nevertheless,it makes sense to look for simple theories which can reproduce the proposed generic behaviour;the exponent n=1/2ought to be easier to explain than any other non-trivial exponent.For reasons not to be detailed here,the old models all have problems.In a recent paper it was suggested that a long-time-tail mechanism might be at the root of generic alpha relaxation[7].Two immediate questions arise if this were correct:1.The mechanism should apply a few decades above the alpha loss peak frequency,i.e.,at times shorter than the alpha relaxation time;2.the long-time tail of the relevant velocity autocorrelation function must be negative.The latter problem is solved by assuming stochastic dynamics,whereas a solution of the former is suggested by the solidity of viscous liquids[7].Reference[7]did not discuss a speciﬁc model for the alpha relaxation.This is done below,where a model implementing the long-time-tail mechanism is proposed and solved forǫ(ω)in the simplest approximation.Model.–Consider N molecules in volume V.If n is the normalized molecular dipole vector,the basic degrees of freedom are taken to be the densityﬁeldρ(r)and the dipole-densityﬁeld n(r).It is convenient to transform into k-space by deﬁning the following sums over all moleculesρk=1N j e i k·r jn k=1N j n j e i k·r j.(1)Ifβis the inverse temperature,the Hamiltonian(free energy)for a Gaussian model H0is given byβH0= kρkρ−k/2S(k)+ k n k·n−k/2ǫ(k)where S(k)is the static structure factor andǫ(k)the dimensionless wavevector-dependent static dielectric constant.The simplest interaction term is H int∝ n2(r)ρ(r)d r.We shall assume that the dynamics are dominated by the long-wavelength behaviour of theﬁelds.Denoting the k→0(bare)limits of S(k)and ǫ(k)by A and B respectively,in terms of a dimensionless coupling constantλthe Hamiltonian becomesβH=12B k n k·n−k+λN k,k′n k·n k′ρ−k−k′.(2)This deﬁnes an“ultra-local”ﬁeld theory,i.e.,one where equal-timeﬂuctuations are uncor-related in space.A more realistic Hamiltonian has gradient terms,but ignoring these is consistent with the assumption of long-wavelength dominance.Sinceρ0is always equal toJeppe C.Dyre:A model for the generic alpha relaxation of viscous liquids3√+ξk(t)∂ρ−k∂(βH)˙n k=−Γn(k)√√dt2 n k(0)·n−k(t) = n k(0)·n−k(t)4λ2B2+4EUROPHYSICS LETTERSwhich can be expressed in terms of the error function.In dimensionless units the dielectric√constant is the Laplace transform of−˙Φ,leading(Appendix)to(where C1=Λ√√1+iωτ +C21Jeppe C.Dyre :A model for the generic alpha relaxation of viscous liquids 5must appear,because otherwise there is no bulk volume relaxation on the alpha time scale and the glass would have the same compressibility as the equilibrium liquid.Thus a more accurate model would assume Γρ(k )=Γ0+Dk 2(which,however,only slightly perturbs model predictions if Γ0∼1/τas required by experiment).2.The Hamiltonian of eq.(2)is “bottomless”in the sense that it allows arbitrarily low energies [11],implying that the canonical distribution is not normalizable.This inconsistency is readily resolved by assuming that there are higher even order terms in n (r ).If these terms are small,they barely inﬂuence the equilibrium autocorrelation functions,but still ensure that the ﬁelds do not eventually run oﬀto inﬁnity.3.According to the model the generic ω−1/2high-frequency decay of the alpha process is speciﬁc to three dimensions.In four dimensions,for instance,the sum in eq.(6)varies with time as t −2,leading to an autocorrelation function which varies logarithmically with time as t →0.——————————————————–H B Nielsen,G Ruocco and T Schrøder are thanked for helpful comments.This work was supported by the Danish National Research Foundation.∗∗∗APPENDIXIn order to calculate the Laplace transform of −˙Φ,we ﬁrst note that the second term of eq.(8)leads to the C 2-term of eq.(9).The ﬁrst term requires calculating the following integral (where s ≡iωand tildes are left out for simplicity of notation):I = ∞0dt e −st ∞t cosh(t ′−t )e −t ′t ′−3/2dt ′=∞0dt e −st ∞te t ′−t +e t −t ′2 ∞0dt e −(s +1)t ∞tt ′−3/2dt ′+12 ∞0dt e −(s −1)t∞t (e −2t ′−1)t ′−3/2dt ′.Thus I = ∞0e −(s +1)t t −1/2dt + ∞e −(s −1)t t −1/2dt +(∗).Since ∞0e −t t −1/2dt =√π(1+s )−1/2+√2 1216EUROPHYSICS LETTERS(∗)=11−s∞0(1−e−2t)t−3/2dt− ∞0 e−(s−1)t−e−(s+1)t t−3/2dt ,or(∗)=12 ∞0 e−at−e−bt t−3/2dt.Calculating this function is straightforward:F(a,b)= (e−at−e−bt)(−t−1/2) t=∞t=0− ∞0 ae−at−be−bt t−1/2dt=φ(b)−φ(a), whereφ(x)= ∞0xe−xt t−1/2dt=√π.Summarizing,I π=(1+s)−1/2+(s−1)−1/2+12−√s+1+√√2−√1−s=11+s+12+√Jeppe C.Dyre:A model for the generic alpha relaxation of viscous liquids7 [3]Dixon P.K.,Wu L.,Nagel S.R.,Williams B.D.and Carini J.P.,Phys.Rev.Lett.,65(1990)1108;Leheny R.L.and Nagel S.R.,Europhys.Lett.,39(1997)447.[4]Olsen N.B.,J.Non-Cryst.Solids,235(1998)399.[5]Schneider U.,Brand R.,Lunkenheimer P.and Loidl A.,Phys.Rev.Lett.,84(2000)5560;Lunkenheimer P.,Wehn R.,Riegger T.and Loidl A.,J.Non-Cryst.Solids,307(2002)336;Blochowicz T.and R¨o ssler E.A.,Phys.Rev.Lett.,92(2004)225701;Hensel-Bielowka S.,Pawlus S.,Roland C.M.,Ziolo J.and Paluch M.,Phys.Rev.E,69(2004)050501.[6]Olsen N.B.,Christensen T.and Dyre J.C.,Phys.Rev.Lett.,86(2001)1271.[7]Dyre J.C.,Phys.Rev.E,72(2005)011501.[8]A related identity is proved on page168in Risken H.,The Fokker-Planck Equation(Springer,Berlin)1989.A simple proof proceeds as follows:Sinceξi(0)=˙Q i(0)+µ∂i H(0)andξi(t)=˙Q(t)+µ∂i H(t)are uncorrelated whenever t=0,one has(no sum over i)0= ˙Q i(0)˙Q i(t) + iµ2 ∂i H(0)∂i H(t) +two terms.The latter two terms add to zero by time-reversal invariance, leading to the required identity since ˙Q i(0)˙Q i(t) =−d2/dt2 Q i(0)Q i(t) .[9]Goldstein M.,J.Chem.Phys.,51(1969)3728;Heuer A.,Phys.Rev.Lett.,78(1997)4051;Sastry S.,Debenedetti P.G.and Stillinger F.H.,Nature,393(1998)554;DemichelisF.,VilianiG.and Ruocco G.,m.,9/01889A(1999);Schrøder T.B.,Sastry S.,Dyre J.C.and Glotzer S.C.,J.Chem.Phys.,112(2000)9834;Angelani L., Di Leonardo R.,Ruocco G.,Scala A.and Sciortino F.,Phys.Rev.Lett.,85(2000)5356;Broderix K.,Bhattacharya K.K.,Cavagna A.,Zippelius A.and Giardina I.,Phys.Rev.Lett.,85(2000)5360.[10]Fujara F.,Geil B.,Sillescu H.and Fleischer G.,Z.Phys.B,88(1992)195;StillingerF.H.and Hodgdon J.A.,Phys.Rev.E,50(1994)2064;Cicerone M.T.and Ediger M.D.,J.Chem.Phys.,104(1996)7210;Diezemann G.,Sillescu H.,Hinze G.and B¨o hmerR.,Phys.Rev.E,57(1998)4398;Douglas J.F.and Leporini D.,J.Non-Cryst.Solids,235 (1998)137;Swallen S.F.,Bonvallet P.A.,McMahon R.J.and Ediger M.D.,Phys.Rev.Lett.,90(2003)015901.[11]Greensite J.and Halpern M.B.,Nucl.Phys.B,242(1984)167.。