A Comparison of LSA, WordNet and PMI-IR for Predicting User Click Behavior
Parametric high resolution techniques for radio astronomical imaging

higher sensitivity to observe faint objects results in high dynamic range requirements within the image, where strong sources can affect the imaging of the very weak sources. On the other hand, Moore’s law [3] together with recent advances in optimization theory [4] open the way to the application of more advanced computational techniques. In contrast to hardware implementation, these image formation algorithms, that are implemented in software can benefit from the continuing increase in computational power, even after the antennas and the correlators will be built. In this paper we extend the parametric deconvolution approach of [5] to obtain better power estimation accuracy and higher robustness to interference and modeling uncertainty. The algorithms presented here can also be used in conjunction with real-time interference mitigation techniques as described in [5] and [6]. We briefly describe the current status of radio astronomical imaging techniques. For a more extensive overview the reader is referred to [7], [8] or [9]. A good historic perspective can be found in [10], whereas [11] provides a very recent perspective. The principle of radio interferometry has been used in radio astronomy since 1946 when Ryle and Vonberg constructed a radio interferometer using dipole antenna arrays [12]. During the 1950’s several radio interferometers which use the synthetic aperture created by movable antennas have been constructed. In 1962 the principle of aperture synthesis using earth rotation has been proposed [13]. The basic idea is to exploit the rotation of the earth to obtain denser coverage of the visibility domain (spatial Fourier domain). The first instrument to use this principle was the five kilometer Cambridge radio telescope. During the 1970’s new instruments with large aperture have been constructed. Among these we find the Westerbork Synthesis Radio Telescope (WSRT) in the Netherlands and the Very Large Array (VLA) in the USA. Recently, the Giant Microwave Telescopes (GMRT) has been constructed in India and the Allen Telescope Array (ATA) in the US. Even these instruments subsample the Fourier domain, so that unique reconstruction is not possible without some further processing known as deconvolution. The deconvolution process uses some a-priori knowledge about the image to remove the effect of “dirty beam” side-lobes. Two principles dominate the astronomical imaging deconvolution. The first method was proposed by Hogbom [14] and is known as CLEAN. The CLEAN method is basically a sequential Least-Squares (LS) fitting procedure in which the brightest source location and power are estimated. The response of this source is removed from the image and then the process continues to find the next brightest source, until the residual image is noise-like. During the years it has been partially analyzed [15], [16] and [17].
cdmp p级题库

cdmp p级题库The problem at hand is related to the CDMP P-level question bank. To address this issue, it is important to consider various perspectives and provide a comprehensive response.From an educational standpoint, having a well-designed and comprehensive question bank is crucial for the success of any certification program. The CDMP P-level question bank plays a significant role in assessing the knowledge and skills of individuals seeking certification in data management. It is essential that the question bank covers a wide range of topics, including data governance, data quality, data modeling, and data integration, among others. By ensuring a diverse and comprehensive set of questions, the question bank can effectively evaluate the proficiency of candidates in various areas of data management.Moreover, the quality of the question bank is of utmost importance. The questions should be carefully crafted totest not only the candidates' theoretical knowledge but also their practical understanding and application of concepts. Each question should be clear, concise, and unambiguous to avoid any confusion or misinterpretation. Furthermore, the question bank should be regularly updated to keep up with the evolving field of data management. This can include incorporating new technologies, industry best practices, and emerging trends in data management.In addition to the educational perspective, it is crucial to consider the perspective of the candidates. The CDMP P-level certification is a significant milestone for individuals in their data management careers. Therefore, the question bank should provide a fair and accurate representation of the knowledge and skills required at this level. Candidates should feel confident that the questions are relevant, challenging, and aligned with the expectations of the industry. This will ensure that the certification holds value and serves as a credible recognition of their expertise in data management.Furthermore, it is important to address the emotionalaspect of the problem. Candidates invest significant time, effort, and resources in preparing for the CDMP P-level certification. They may experience various emotions, such as anxiety, stress, and excitement, throughout the preparation process. The question bank should be designedin a way that supports candidates' emotional well-being and motivates them to succeed. This can be achieved by providing clear instructions, offering practice questions, and providing feedback on performance. Additionally, the question bank should be accessible and user-friendly, allowing candidates to navigate through the questions easily.From a practical standpoint, the CDMP P-level question bank should be easily accessible to candidates. It should be available in various formats, such as online platforms or downloadable resources, to cater to different learning preferences. The question bank should also be accompanied by detailed explanations and references to relevant study materials. This will enable candidates to understand the rationale behind the correct answers and further enhance their knowledge in specific areas. Moreover, the questionbank should include a sufficient number of questions to adequately assess the candidates' proficiency level.In conclusion, the CDMP P-level question bank plays a crucial role in evaluating the knowledge and skills of individuals seeking certification in data management. It is essential to ensure that the question bank is comprehensive, of high quality, and regularly updated. Additionally, it should be designed to support candidates' emotional well-being and provide a fair representation of the knowledgeand skills required at the P-level. By considering these perspectives and addressing the various aspects of the problem, the CDMP P-level question bank can effectively serve its purpose and contribute to the success of data management professionals.。
生物信息学数据分析工具考核试卷

B. 3C
C. Capture-C
D. ChIP-Seq
三、填空题(本题共10小题,每小题2分,共20分,请将正确答案填到题目空白处)
1.生物信息学中,用于表示DNA序列的常见字母缩写是______。()
2.在基因组浏览器中,横轴通常表示______。()
3.基因组测序技术中,Sanger测序又称为______。()
A. BLAST
B. Clustal Omega
C. SAMtools
D. BEDTools
3.以下哪个不是生物信息学中常用的编程语言:( )
A. Python
B. R
C. Java
D. SQL
4.以下哪个工具用于基因组注释:( )
A. GATK
B. Bowtie2
C. TopHat2
D. Augustus
B. GenBank
C. UniProt
D. Ensembl
8.以下哪个软件用于RNA-seq数据分析:( )
A. Tophat
B. Cufflinks
C. HISAT2
D. All of the above
9.以下哪个概念用于描述序列相似性搜索中的统计显著性:( )
A. PID
B. E-value
10.人类基因组计划的目标是测定人类基因组的全部DNA序列。()
五、主观题(本题共4小题,每题5分,共20分)
1.请简述生物信息学中序列比对的基本原理,并列举三种常用的序列比对工具。()
2.描述RNA-seq数据分析的基本流程,包括数据质控、参考基因组比对、转录本组装和差异表达分析等步骤。()
3.请解释什么是系统发生树,并说明构建系统发生树的常见方法和软件。()
Coalgebras of words and phrases

a r X i v :m a t h /0408258v 1 [m a t h .Q A ] 19 A u g 2004COALGEBRAS OF WORDS AND PHRASESVLADIMIR TURAEVAbstract.We introduce two constructions of a coassociative comultiplication in the algebra of phrases in agiven alphabet.As a preliminary step we give two constructions of a pre-Lie comultiplication in the modulegenerated by words.1.Introduction We give two new constructions of non-commutative non-cocommutative Hopf algebras (of infinite rank).These constructions can be summarized as follows.A module M over a commutative ring gives rise to the tensor algebra W =T (M )=⊕m ≥0M ⊗m and to its positive subalgebra V =T +(M )=⊕m ≥1M ⊗m .Under certain additional assumptions,we define comultiplications in the tensor algebras P =T (V )and Q =T (W )so that they become Hopf algebras.A comultiplication ∆in a graded algebra ⊕m ≥0T m has a leading term which is the homomorphism T 1→T 1⊗T 1obtained by applying ∆and then projecting to T 1⊗T 1.This leading term yields the first approximation to ∆and is often interesting in itself.The leading terms of our comultiplications in P and Q are homomorphisms V →V ⊗V and W →W ⊗W .They turn out to be pre-Lie comultiplications in the sense explained below.In particular,dualizing and skew-symmetrizing them,we obtain Lie brackets in the dual modules V ∗and W ∗.The study of tensor algebras can be reformulated in terms of words and phrases.Suppose from now on that the module M is free with basis A .Consider the basis of V =T +(M )formed by the vectors A 1⊗A 2⊗...⊗A m where m ≥1and A 1,A 2,...,A m ∈A .Omitting the symbol ⊗we can identify these vectors with words in the alphabet A .A basis in P =T (V )is formed by finite sequences of basis vectors in V ,that is by sequences of words.We call such sequences phrases .Natural bases in W =T (M )and Q =T (W )are similarly interpreted as words and phrases with the difference that here we allow an empty word.This interpretation gives a pleasant linguistical flavour to the theory.It places the study of words and phrases in the setting of Hopf algebras,Lie algebras,and related algebraic objects.One well known construction of a comultiplication in W is based on the notion of a subword.For us,a word is a finite sequence of elements of A and a subword is any subsequence.Given a word w we can form the sum w ′⊗w/w ′where w ′runs over all subwords of w and the word w/w ′is obtained by deleting w ′from w .This yields the familiar shuffle comultiplication in W .It is coassociative and cocommutative.Our comultiplications are based on two different constructions.They are determined by certain additionaldata which should be fixed from the very beginning.The comultiplications in V and P =T (V )depend on a choice of a so-called stable set of words L in the alphabet A .We associate with a word w the sum w ′⊗w/w ′where w ′runs over all subwords of w formed by consecutive letters and belonging to L .This yields a pre-Lie comultiplication ρL in V .In a similar way we construct a coassociative comultiplication in P with leading term ρL .The second construction begins with fixing a mapping µfrom A×A to the ground ring.The corresponding pre-Lie comultiplication ρµin W is defined as follows.For a subword of length two a =AB of a word w ,set µa =µ(A,B )and denote by w ′a be the subword of w formed by the letters of w appearing between A and B .Let w ′′a be the word obtained by deleting both a and w ′a from w .Then ρµ(w )= a µa w ′a ⊗w ′′a .In a similar way we construct a coassociative comultiplication in Q =T (W )with leading term ρµ.Under a certain choice of µ,the latter comultiplication is closely related to the Connes-Kreimer comul-tiplication in the algebra of rooted trees,see [CK],or,more precisely,to its non-commutative version for planar rooted trees due to Foissy [Fo].Although we can directly define our comultiplications in P and Q ,we begin with a study of their leading terms.In Section 2we discuss relevant notions from the theory of pre-Lie multiplications and comultipli-cations.Sections 3and 4are concerned with the comultiplication in P derived from a stable set of words:12VLADIMIR TURAEVthe leading term is defined in Section3and the comultiplication itself is defined in Section4.Sections5 and6are concerned with the comultiplication in Q derived from a mappingµas above:the leading term is defined in Section5and the comultiplication itself is defined in Section6.At the end of Section6we discuss connections with the theory of planar rooted trees.Throughout the paper,wefix a commutative ring with unit R.The symbol⊗denotes the tensor product of R-modules over R.2.Pre-Lie coalgebras and left-handed bialgebras2.1.Pre-Lie algebras.By a multiplication in an R-module M we mean an R-bilinear mapping M×M→M. A(left)pre-Lie algebra over R is an R-module M endowed with a multiplication M×M→M,denoted⋆, such that for any f,g,h∈M,(2.1.1)f⋆(g⋆h)−(f⋆g)⋆h=g⋆(f⋆h)−(g⋆f)⋆h.The mapping⋆is called then a pre-Lie multiplication,see[Ge]and[Vi].For example,any associative multiplication is pre-Lie.A fundamental property of a pre-Lie multiplication⋆in M is that the formula [f,g]=f⋆g−g⋆f defines a Lie bracket in M.The Jacobi identity[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0for f,g,h∈M is a direct consequence of(2.1.1).In this way every pre-Lie algebra becomes a Lie algebra.There is a similar notion of right pre-Lie algebras.We will consider only left pre-Lie algebras and refer to them simply as pre-Lie algebras.Non-trivial examples of pre-Lie algebras can be obtained from derivations in algebras.A derivation in an associative R-algebra M is an R-linear homomorphism d:M→M such that d(fg)=d(f)g+f d(g)for any f,g∈M.An easy computation shows that for any a,b∈M,the formula f⋆g=afg+bfd(g)defines a pre-Lie multiplication in M.2.2.Pre-Lie coalgebras.A comultiplication in an R-module A is an R-linear homomorphismρ:A→A⊗A. We associate with suchρa homomorphism˜ρ:A→A⊗A⊗A sending a∈A to˜ρ(a)=(id A⊗ρ)ρ(a)−(ρ⊗id A)ρ(a).The comultiplicationρis coassociative if˜ρ=0.Dualizing Formula2.1.1,we obtain a notion of a pre-Lie ly,a pre-Lie coalgebra is an R-module A endowed with a comultiplicationρ:A→A⊗2=A⊗A such that for all a∈A,(2.2.1)P1,2˜ρ(a)=˜ρ(a).Here and below given a module U,we denote by P1,2the endomorphism of A⊗2⊗U permuting thefirst two tensor factors,i.e.,mapping a⊗b⊗u to b⊗a⊗u for a,b∈A,u∈U.A comultiplicationρsatisfying(2.2.1) is a pre-Lie comultiplication.Clearly,a coassociative comultiplication is pre-Lie.Given a pre-Lie coalgebra(A,ρ),consider the module A∗=Hom R(A,R)and the evaluation pairing , :A⊗A∗→R.The comultiplicationρinduces a pre-Lie multiplication⋆ρin A∗by(2.2.2) a,f⋆ρg = i a′i,f a′′i,g ∈Rfor any f,g∈A∗,a∈A and any(finite)expansionρ(a)= i a′i⊗a′′i∈A⊗2.In this way A∗becomes a pre-Lie algebra.odules over pre-Lie coalgebras.A(left)comodule over a pre-Lie coalgebra(A,ρ)is a pair(an R-module U,an R-homomorphismθ:U→A⊗U)such that˜θ=(id A⊗θ)θ−(ρ⊗id U)θ:U→A⊗2⊗U satisfies P1,2˜θ=˜θ.Given a comodule(U,θ)over(A,ρ),we define a right action of A∗on U byuf= i u′i,f u′′ifor any f∈A∗,u∈U,and any expansionθ(u)= i u′i⊗u′′i with u′i∈A,u′′i∈U.Lemma2.3.1.The action of A∗on U is a Lie algebra action.COALGEBRAS OF WORDS AND PHRASES3 Proof.We must prove that u[f,g]=(uf)g−(ug)f for all f,g∈A∗,u∈U where[f,g]=f⋆ρg−g⋆ρf.We shall use Sweedler’s notation for the expansionθ(u)= i u′i⊗u′′i and write it in the formθ(u)= (u)u′⊗u′′. Similarly,for a∈A,we writeρ(a)= (a)a′⊗a′′.By definition,(uf)g= (u) u′,f u′′g= (u),(u′′) u′,f (u′′)′,g (u′′)′′.Hence(uf)g−(ug)f= (u),(u′′) u′,f (u′′)′,g (u′′)′′− (u′′)′,f u′,g (u′′)′′.To compute u[f,g],observe that for a∈A,a,[f,g] = a,f⋆ρg−g⋆ρf = (a) a′,f a′′,g − a′′,f a′,g .Thereforeu[f,g]= (u) u′,[f,g] u′′= (u),(u′) (u′)′,f (u′)′′,g u′′− (u′)′′,f (u′)′,g u′′.The equality u[f,g]=(uf)g−(ug)f follows now from the formula(u′)′⊗(u′)′′⊗u′′−(u′)′′⊗(u′)′⊗u′′= (u),(u′′)u′⊗(u′′)′⊗(u′′)′′−(u′′)′⊗u′⊗(u′′)′′(u),(u′)which is a reformulation of the equality P1,2˜θ=˜θ.Lemma2.3.1implies that the formula fu=−uf with u∈U,f∈A∗defines a left Lie algebra action of A∗on U.It is clear that(A,θ=ρ:A→A⊗A)is a comodule over(A,ρ).Lemma2.3.1yields a Lie algebra action of A∗on A.2.4.Left-handed bialgebras.By a bialgebra we shall mean a pair(T,∆)where T is an associative unital R-algebra and∆:T→T⊗T is a coassociative algebra comultiplication in T.The words“algebra comulti-plication”mean that∆(1)=1⊗1and∆(ab)=∆(a)∆(b)for any a,b∈T.Here multiplication in T⊗T is defined by(a⊗a′)(b⊗b′)=ab⊗a′b′for a,b,a′,b′∈T.We do not require the existence of a counit T→R although in all our constructions of bialgebras there will be a counit.The comultiplication in a bialgebra(T,∆)induces an associative multiplication in T∗=Hom R(T,R) by fg(a)= (a)f(a′)g(a′′)for a∈T,f,g∈T∗,and any expansion∆(a)= (a)a′⊗a′′.This makes T∗into an associative algebra.If T has a counit,then T∗is a unital algebra.Dualizing multiplication in T we obtain a homomorphism from T∗to a certain completion of T∗⊗T∗.We call this homomorphism quasi-comultiplication.A bialgebra(T,∆)is graded if T splits as a direct sum of submodules T=⊕m≥0T m such that1∈T0and T m T n⊂T m+n for all m,n.Set T+=⊕m≥1T m⊂T.A graded bialgebra(T,∆)is left-handed if for any a∈T1,∆(a)−a⊗1−1⊗a∈T+⊗T1.The leading term of∆is then the homomorphism(π⊗π)∆|T1:T1→T1⊗T1whereπ:T→T1is the projection.We shall mainly apply these definitions in the case whereT=T(A)=⊕m≥0A⊗mis the tensor algebra of an R-module A.Here A⊗0=R,A⊗1=A,and A⊗m is the tensor product of m copies of A for m≥2.The next lemma relates left-handed bialgebras with pre-Lie coalgebras.Lemma 2.4.1.Let(T=⊕m≥0T m,∆)be a left-handed graded bialgebra such that T1generates T+=⊕m≥1T m.Then the leading termρ:T1→T1⊗T1of∆is a pre-Lie comultiplication in T1.Proof.For a∈T1,we can expand∆(a)=a⊗1+1⊗a+ i a′i⊗a′′i+ j a1j a2j⊗a3j(mod⊕m≥3T m⊗T1)4VLADIMIR TURAEVwhere i,j run overfinite sets of indices and a′i,a′′i,a1j,a2j,a3j∈T1.Clearly,ρ(a)= i a′i⊗a′′i.Our assump-tions imply that∆(T m)⊂⊕k+n≥m T k⊗T n for all puting(id⊗∆)∆(a)and(∆⊗id)∆(a)modulo ⊕k,n,r,k+n+r≥4T k⊗T n⊗T r and equating the resulting expressions we obtain that˜ρ(a)= j a1j⊗a2j⊗a3j+a2j⊗a1j⊗a3j.Therefore P1,2˜ρ(a)=˜ρ(a).Let(T,∆)be as in the lemma.A(left)comodule over(T,∆)is a pair(an R-module U,an R-homomorphism Θ:U→T⊗U)such that(∆⊗id U)Θ=(id T⊗Θ)ΘandΘ(u)−1⊗u∈T+⊗U for any u∈U.Then U becomes a right module over the algebra T∗by uf= (u)f(u′)u′′for f∈T∗,u∈U,and anyfinite expansionΘ(u)= (u)u′⊗u′′.For example,set U=T1and defineΘ:U→T⊗U byΘ(u)=∆(u)−u⊗1 for u∈U.The pair(U,Θ)is a comodule over(T,∆)(cf.the proof of Theorem4.2.1below).This makes U=T1into a right T∗-module.2.5.Remarks.1.Lemma2.4.1suggests to study when a given pre-Lie comultiplication is induced froma left-handed comultiplication in a graded bialgebra.For the pre-Lie comultiplications defined in the next sections this will be always the case.2.Consider a pre-Lie coalgebra(A,ρ)and a submodule B⊂A such thatρ(A)⊂B⊗A andρ(B)⊂B⊗B. Then(B,ρ|B)is a pre-Lie coalgebra and(A,θ=ρ:A→B⊗A)is a comodule over(B,ρ|B).The associated action of B∗on A is compatible with the action of A∗on A via the Lie algebra homomorphism A∗→B∗induced by the inclusion B⊂A.3.For a pre-Lie coalgebra(A,ρ),the homomorphismρ−P1,2ρ:A→A⊗2is a Lie cobracket(cf.,for instance[Tu]).In particular in Lemma2.4.1,ρ−P1,2ρis a Lie cobracket in T1.This still holds if the left-handedness assumption on∆is weakened to∆(a)−a⊗1−1⊗a∈T+⊗T+for all a∈T1.4.Lemma2.4.1extends to comodules as follows.Let(T,∆)be as in this lemma.The leading term ofa comodule(U,Θ)over(T,∆)is the homomorphismθ=(π⊗id)Θ:U→T1⊗U whereπ:T→T1is the projection.A direct computation shows that(U,θ)is a comodule over the pre-Lie coalgebra(T1,ρ= (π⊗π)∆|T1).Note also that if T0=R,then the projection T→T0=R is a counit of(T,∆).3.Coalgebra of words3.1.Words.For us,an alphabet is an arbitrary set and letters are its elements.Throughout the paper we fix an alphabet A.A word of length m≥1is a mapping from the set{1,2,...,m}to A.By definition,there is a unique word of length0called the empty word and denotedφ.To present a word w:{1,2,...,m}→A it is enough to write down the sequence of letters w(1)w(2)...w(m).For instance,the symbol ABA represents the word{1,2,3}→A sending1and3to A∈A and sending2to B∈A.Writing down consecutively the letters of two words w and x we obtain their concatenation wx.For instance,the concatenation of w=ABA and x=BB is the word wx=ABABB.A word x is a factor of a word w if w=yxz for certain words y,z.A factor x of w is proper if x=w.Given a word w of length m≥1and numbers1≤i≤j≤m+1,set w i,j=w(i)w(i+1)...w(j−1). This word of length j−i is a factor of w=w1,i w i,j w j,m+1.The word w i,j is empty iffi=j and proper iff(i,j)=(1,m+1).Let W=W(A)be the free R-module freely generated by the set of words in the alphabet A.A typical element of W is afinite formal linear combination of words with coefficients in R.Each word w represents a vector in W denoted also w.These vectors form a basis of W.Let V=V(A)be the submodule of W generated by non-empty words.Clearly,W=V⊕Rφ.3.2.Stable sets of words.A set L of non-empty words is stable if it satisfies the following condition:(∗)For any word w of length m≥1and any indices1≤i<j≤m+1such that(i,j)=(1,m+1)and w i,j∈L,the word w belongs to L if and only if the word w1,i w j,m+1belongs to L.The words belonging to a stable set L will be called L-words.The“only if”part of(∗)means that striking out from any L-word a proper factor belonging to L one obtains an L-word.The“if”part of(∗)means that inserting an L-word in an L-word one obtains an L-word.In particular,concatenation of L-words is an L-word.We give a few examples of stable sets of words.(1)The set of all non-empty words and the void set of words are stable.COALGEBRAS OF WORDS AND PHRASES5 To give the next example,denote by f A(w)the number of appearances of a letter A∈A in a word w.For instance,f A(ABA)=2.(2)Pick A∈A.The set of non-empty words w such that f A(w)=0is stable.Similarly,for any integer N,the set of non-empty words w such that f A(w)is divisible by N is stable.(3)Pick letters A1,...,A n∈A and elements g1,...,g n of a certain abelian group.The set of non-emptywords w such that f A1(w)g1+...+f An(w)g n=0is stable.Observefinally that the intersection of any family of stable sets of words is stable.A union of stable sets can be non-stable.3.3.Pre-Lie coalgebra of words.Wefix a stable set of words L and derive from it a pre-Lie comultipli-cationρL in the module V.A simple cut of a word w of length m≥1is a pair of indices1≤i<j≤m+1such that(i,j)=(1,m+1) and w i,j∈L.To indicate that(i,j)is a simple cut of w we write(i,j)≺w.SetρL(w)= (i,j)≺w w i,j⊗w1,i w j,m+1∈V⊗Vwhere the sum runs over all simple cuts of w.Note that the words w i,j and w1,i w j,m+1are necessarily non-empty.This definesρL on the basis of V and extends by linearity to a comultiplicationρL:V→V⊗V. For example,if A,B∈A and L is the set of all non-empty words,thenρL(A)=0,ρL(AB)=A⊗B+B⊗A,ρL(ABA)=A⊗BA+B⊗AA+A⊗AB+AB⊗A+BA⊗A.Theorem3.3.1.ρL is a pre-Lie comultiplication in V.Proof.For a word w of length m≥1,(id⊗ρL)ρL(w)= (i,j)≺w w i,j⊗ρL(w1,i w j,m+1).To compute w i,j⊗ρL(w1,i w j,m+1),we need to consider factors of w1,i w j,m+1.There are three kinds of them: (i)the factors of w1,i−1,(ii)the factors of w j+1,m+1,and(iii)the factors obtained by concatenation of some w i′,i with some w j,j′where i′≤i,j≤j′.Here and below i,j,i′,j′∈{1,2,...,m}.The factors of w1,i w j,m+1 of type(i)contribute to w i,j⊗ρL(w1,i w j,m+1)the expressionx i,j=(i′,j′)≺w,j′<iw i,j⊗w i′,j′⊗w1,i′w j′,i w j,m+1.The factors of w1,i w j,m+1of type(ii)contributey i,j=(i′,j′)≺w,j<i′w i,j⊗w i′,j′⊗w1,i w j,i′w j′,m+1.The factors of w1,i w j,m+1of type(iii)contributez i,j=i′≤i,j≤j′,(i′,j′)=(1,m+1),w i′,i w j,j′∈Lw i,j⊗w i′,i w j,j′⊗w1,i′w j′,m+1.Recall that w i,j∈L.By definition of a stable set of words,w i′,i w j,j′∈L iff(i′,j′)=(i,j)and w i′,j′∈L. Therefore the conditions on i′,j′in the latter sum are equivalent to the conditions i′≤i,j≤j′,(i′,j′)≺w,(i′,j′)=(i,j).We obtain(i,j)≺w z i,j=(i,j)≺w,(i′,j′)≺w,i′≤i,j≤j′,(i′,j′)=(i,j)w i,j⊗w i′,i w j,j′⊗w1,i′w j′,m+1 = (i′,j′)≺wρL(w i′,j′)⊗w1,i′w j′,m+1=(ρL⊗id)ρL(w).Hence,˜ρL(w)=(id⊗ρL)ρL(w)−(ρL⊗id)ρL(w)= (i,j)≺w x i,j+y i,j.It remains to observe thatP1,2( (i,j)≺w x i,j)= (i,j)≺w y i,j.6VLADIMIR TURAEVTherefore˜ρL(w)is invariant under P1,2.3.4.Word indicators.The elements of the module V∗=Hom R(V,R)are called word indicators.The module V∗admits a decreasingfiltration V∗=V∗(1)⊃V∗(2)⊃...where V∗(m)consists of the word indicators annihilating all non-empty words of length≤m−1.Clearly,∩m V∗(m)=0.We can consider infinite sums in V∗as follows.Let f1,f2,...∈V∗be a sequence of word indicators such that for any m≥1all terms of the sequence starting from a certain place belong to V∗(m).Then for any a∈V,the sum f(a)=f1(a)+f2(a)+···contains only afinite number of non-zero terms and the formula a→f(a):V→R defines a word indicator f=f1+f2+···.A similar construction shows that the natural homomorphism V∗→proj lim m(V∗/V∗(m)) is an isomorphism.By the general theory,the pre-Lie comultiplicationρL in V induces a pre-Lie multiplication⋆L and a Lie bracket[f,g]L=f⋆L g−g⋆L f in V∗.It is clear that V∗(m)⋆L V∗(n)⊂V∗(m+n)and[V∗(m),V∗(n)]L⊂V∗(m+n) for all m,n.Therefore V∗=proj lim m(V∗/V∗(m))is a projective limit of nilpotent Lie algebras.Recall the Lie algebra action of V∗on V induced byρL.For f∈V∗and a word w of length m≥1,fw=−wf=− (i,j)≺w w i,j,f w1,i w j,m+1.All word indicators annihilating L lie in the kernel of this action.The module V has an increasingfiltration0=V0⊂V1⊂V2⊂...where V m is generated by non-empty words of length≤m.It is clear that f V m⊂V m−1for all m and f∈V∗.Thus the action of V∗on V m is nilpotent for all m.The restriction ofρL:V→V⊗V to the module RL⊂V generated by L is a pre-Lie comultiplication in RL sinceρL(RL)⊂RL⊗RL.ClearlyρL(V)⊂RL⊗V so that V becomes a comodule over the pre-Lie coalgebra RL.The associated actions of(RL)∗and V∗on V are compatible via the Lie algebra homomorphism V∗→(RL)∗induced by the inclusion RL⊂V.This is a special case of Remark2.5.2.3.5.Example.Let the alphabet A consist of only one letter A.Denote by A m the word consisting of m letters A where m≥1.A stable set of words L is determined by a positive integer N and consists of all words A m with m divisible by N.The comultiplicationρL is computed byρL(A m)= k≥1,kN<m(m+1−kN)A kN⊗A m−kNfor any m≥1.We can identify a word indicator f with its generating function m≥1f(A m)t m.One easily computes that f⋆L g=(f|N)·(g+tg′)where for a formal power series f= m≥1a m t m,we set f|N= m≥1a mN t mN and f′= m≥1ma m t m−1.For N=1,we obtain f⋆L g=fg+tfg′.3.6.The group Exp L V∗.Assume in this subsection that R⊃Q.For f,g∈V∗,setfg=f+g+112([f,[f,g]L]L+[g,[g,f]L]L)+ (V)where the right-hand side is the Campbell-Baker-Hausdorffseries for log(e f e g).The resulting mapping V∗×V∗→V∗is a group multiplication in V∗.Here f−1=−f and0∈V∗is the group unit.The resulting group is denoted Exp L V∗.Heuristically,this is the“Lie group”with Lie algebra(V∗,[,]L).The equality V∗=proj lim m(V∗/V∗(m))implies that Exp L V∗is pro-nilpotent.The action of V∗on V induced byρL integrates to a group action of Exp L V∗on V.To see this,denote byϕ(f)the additive endomorphism a→fa of V determined by f∈V∗.Seteϕ(f)(a)= k≥01COALGEBRAS OF WORDS AND PHRASES7 3.7.Remarks.1.There is an embeddingδ:V֒→V∗mapping a word w into the word indicatorδw whose value on w is1and whose value on all other words is0.It is easy to check that the image ofδis closed under⋆L.This induces a pre-Lie multiplication◦L in V.For words w=A1A2...A m and x=B1B2...B n with m,n≥1,we have w◦L x=0if w/∈L andw◦L x=δ−1(δw⋆Lδx)=ni=0B1...B i A1...A m B i+1...B nif w∈L.In the case where L is the set of all non-empty words,this pre-Lie multiplication in V is essentially due to Gerstenhaber[Ge].2.With a slight modification,the notion of a stable set of words can be used to define a coassociative comultiplication in W=V⊕Rφ.Let us say that a set of words S is strongly stable ifφ∈S and for any word w and any its subword w′we have w′∈S⇒(w∈S⇔w/w′∈S)where w/w′is the word obtained by deleting w′from w.Examples of strongly stable sets of words can be obtained by adjoining the empty word to the stable sets of words in the examples in Section3.2.For a strongly stable set S and a word w, set∆S(w)= w′w′⊗w/w′where w′runs over all subwords of w belonging to S.It is easy to see that∆S extends to a coassociative algebra comultiplication in W.When S is the set of all words,this is the shuffle comultiplication mentioned in the introduction.4.Hopf algebra of phrasesThe results above suggest that there may exist a left-handed comultiplication in the tensor algebra T(V) with leading termρL.We construct such a comultiplication.4.1.Phrases.A phrase of length k≥1is a sequence of k words in the alphabet A.Some(or all)of these words may be empty.We also allow an empty phrase consisting of0words and denoted1.A more interesting example of a phrase:“NIHIL NOV I SUB SOLE”where A is the set of capital Latin letters.Here words are separated by blank spaces and the quotation marks indicate the beginning and the end of the phrase. These conventions,customary in ordinary texts,are not quite convenient for mathematical formulas.In the formulas we shall indicate the beginning and the end of a phrase by round brackets and separate words by vertical bars.Of course,we assume that the round brackets and the vertical bar are not letters in A.The same example can be re-written as(NIHIL|NOV I|SUB|SOLE).By abuse of notation,the phrase(w) consisting of one word w will be also denoted by w.Let Q=Q(A)be the free R-module freely generated by the set of phrases in the alphabet A.Concatenation of phrases defines a multiplication in Q by(w1|w2|...|w k)(x1|x2|...|x t)=(w1|w2|...|w k|x1|x2|...|x t)where k,t≥0and w1,...,w k,x1,...,x t are words.This makes Q into an associative algebra with unit1(the empty phrase).This algebra is graded,the grading being the length of phrases.In this section we shall focus on the subalgebra P=P(A)of Q additively generated by phrases formed by non-empty words.Thus(w1|w2|...|w k)∈P iffk=0or k≥1and all the words w1,...,w k are non-empty. Note that1∈P.The algebra P is graded:P=P0⊕P1⊕...where P k is the module additively generated by phrases of length k.The inclusion V=P1֒→P extends to an algebra isomorphism T(V)→P and we identify the tensor algebra T(V)with P along this isomorphism.Phrases in the alphabet A can be viewed as words in the extended alphabet A∐{|}obtained by adjoining the new letter|to A.However,multiplication of phrases is different from multiplication of words in this extended alphabet due to the additional symbol|between w k and x1in the formula above.ultiplication in P.Fix a stable set of words L.We define here a comultiplication∆=∆L in P. We begin with preliminary definitions.Let w be a word of length m≥1.A cut of w of length k≥0is a sequence of2k integers1≤i1<j1<i2<j2<...<i k<j k≤m+1such that the k words w i1,j1,...,w ik,j kbelong to L.For k=1,we additionallyrequire that(i1,j1)=(1,m+1).A cut of length1is nothing but a simple cut in terminology of Section3.3. Every word has a unique empty cut∅of length0.8VLADIMIR TURAEVTo indicate that c=(i1,j1,...,i k,j k)is a cut of w we write c≪w.We also write#c={1,2,...,k}.Foru∈#c,set w c u=w iu ,j u.The factors w c1,...,w c k of w are called c-factors.Finally we define a phrase l c(w)= u∈#c(w c u)=(w c1|w c2|...|w c k)and a wordr c(w)=w1,i1w j1,i2w j2,i3...w jk−1,i kw jk,m+1.To specify a cut of w of length k it is enough to specify k non-overlapping proper factors of w belonging to L and such that consecutive factors are separated by at least one letter.These factors form the phrase l c(w). Deleting them from w we obtain the word r c(w).Set∆(w)=w⊗1+ c≪w l c(w)⊗r c(w)where c runs over all cuts of w.Note that the term corresponding to the empty cut is1⊗w.The mapping w→∆(w)extends uniquely to an algebra homomorphism∆=∆L:P→P⊗2.For example,for a word w of length1,we have∆(w)=w⊗1+1⊗w.If L is the set of all non-empty words and A,B,C∈A,then∆(AB)=AB⊗1+1⊗AB+A⊗B+B⊗A,∆(ABC)=ABC⊗1+1⊗ABC+A⊗BC+B⊗AC+C⊗AB+AB⊗C+BC⊗A+(A|C)⊗B,∆(AB|C)=∆(AB)∆(C)=(AB|C)⊗1+C⊗AB+(A|C)⊗B+(B|C)⊗A+AB⊗C+1⊗(AB|C)+A⊗(B|C)+B⊗(A|C).Theorem4.2.1.The pair(P,∆)is a left-handed graded bialgebra with leading termρL.Proof.The left-handedness and the claim concerning the leading term follow directly from the definitions.The only non-obvious assertion is the coassociativity of∆.It suffices to prove that(id⊗∆)∆(w)=(∆⊗id)∆(w) for any word w.SetΘ(w)=∆(w)−w⊗1= c≪w l c(w)⊗r c(w).The mapping w→Θ(w)defines an R-linear homomorphismΘ:V→P⊗V.We have(∆⊗id)∆(w)=(∆⊗id)Θ(w)+∆(w)⊗1=(∆⊗id)Θ(w)+Θ(w)⊗1+w⊗1⊗1. Similarly,(id⊗∆)∆(w)=(id⊗∆)Θ(w)+(id⊗∆)(w⊗1)=(id⊗Θ)Θ(w)+Θ(w)⊗1+w⊗1⊗1. Comparing these expressions we conclude that it is enough to prove that(∆⊗id)Θ(w)=(id⊗Θ)Θ(w).If follows from the definitions that(id⊗Θ)Θ(w)= c≪w e≪r c(w)l c(w)⊗l e(r c(w))⊗r e(r c(w)).We compute the right-hand side as follows.For integers i<j,set[i,j)={s∈Z|i≤s<j}.For cuts d=(i1,j1,...,i k,j k),d′=(i′1,j′1,...,i′k′,j′k′)of w,write d⊂d′if∪k u=1[i u,j u)⊂∪k′v=1[i′v,j′v).Suppose that d⊂d′.We say that the index v∈{1,...,k′}is special if i′v=i u and j′v=j u for some u=1,...,k.If v is non-special,then clearly[i′v,j′v) ∪k u=1[i u,j u). We can obtain a cut˜d′of r d(w)by deleting all d-factors from the d′-factors of w numerated by non-special indices.That the˜d′-factors of r d(w)belong to L follows from the stability of L.The cut˜d′is empty iffd=d′. Moreover,the formula(d,d′)→(d,˜d′)establishes a bijective correspondence between pairs(d≪w,d′≪w) with d⊂d′and pairs(c≪w,e≪r c(w)).Therefore(id⊗Θ)Θ(w)=d≪w,d′≪w,d⊂d′l d(w)⊗l˜d′(r d(w))⊗r˜d′(r d(w))=d≪w,d′≪w,d⊂d′l d(w)⊗l˜d′(r d(w))⊗r d′(w).We claim that the right-hand side is equal to(∆⊗id)Θ(w).(The proof of this does not use the stability of L).If follows from the definitions that(∆⊗id)Θ(w)= c≪w u∈#c w c u⊗1+ e u≪w c u l e u(w c u)⊗r e u(w c u) ⊗r c(w)COALGEBRAS OF WORDS AND PHRASES9= c≪w I⊂#c{e u≪w c u}u∈#c−Iv∈Iw c v u∈#c−I l e u(w c u) ⊗ u∈#c−I r e u(w c u)⊗r c(w).Here all the products are ordered in accordance with the natural order in#c.For example,if#c={1,2,3} and I={1,3},then the term in the big round brackets on the right-hand side is w c1l e2(w c2)w c3.Given a cut c=(i1,j1,...,i k,j k)of w and u∈#c,every cut e u of w c u yields a cutˆe u of w by adding i u−1 to all terms of e u.A set I⊂#c gives rise to a cutˆI of w formed by the indices{i v,j v}v∈I.With a tuple (c≪w,I⊂#c,{e u≪w c u}u∈#c−I)we associate a pair(d,d′)where d′=c and d is the cut of w obtained as the union ofˆI with all{ˆe u}u∈#c−I.This establishes a bijective correspondence between such tuples and the pairs(d≪w,d′≪w)with d⊂d′.The corresponding terms in the expansions of(id⊗Θ)Θ(w)and (∆⊗id)Θ(w)are equal.Therefore(id⊗Θ)Θ(w)=(∆⊗id)Θ(w). Corollary4.2.2.The bialgebra(P,∆)is a Hopf algebra.Proof.The augmentationε:P→R sending all non-empty phrases to0and sending1to1is a counit of∆. It remains to show the existence of an antipode.Consider the unique R-linear endomorphism s of P such that s(1)=1,s(ab)=s(b)s(a)for all a,b∈P and the value of s on words is defined by induction on the length as follows:for a word w of length1,set s(w)=−w;for a word w of length≥2,sets(w)=−w− c≪w,c=∅l c(w)s(r c(w))∈Pwhere we use that r c(w)is shorter than w.These formulas imply thatµ(id⊗s)∆(w)=ε(w)whereµis multiplication in P.In other words,s is a right inverse of id:P→P with respect to the(associative) convolution product•in End R(P)defined by f•g=µ(f⊗g)∆for f,g∈End R(P).Similar inductive formulas show that id has a left inverse s′∈End R(P)and then s′=s′•(id•s)=(s′•id)•s=s.Therefore s is an antipode for(P,∆).4.3.Phrase indicators.By phrase indicators we mean R-linear homomorphisms P→R.By the general theory of bialgebras,the comultiplication∆L in P induces an associative multiplication◦L in the module of phrase indicators P∗=Hom R(P,R).The product f◦L g may distinguish phrases indistinguishable by f,g∈P∗.For example,let L consist of all non-empty words and letℓand f B be the phrase indicators counting the number of words in a phrase and the number of appearances of B∈A in a phrase,respectively. Then the values ofℓ◦L f B on the1-word phrases(ABC)and(ACB)(where A,B,C are distinct letters in A)are4and3,respectively.The additive homomorphismΘ:V→P⊗V constructed in the proof of Theorem4.2.1makes V into a comodule over the bialgebra(P,∆).This induces a right action of the algebra P∗on ing the antiautomorphism of P∗induced by the antipode in P,we can transform the right action of P∗into a left action.The right and left actions of a phrase indicator f∈P∗on V depend only on the values of f on phrases with all words in L.This can be formalized as follows.Let P L be the subalgebra of P additively generated by the phrases whose all words belong to L(including the empty phrase1).It is clear that∆L(P L)⊂P L⊗P L so that P L is a Hopf subalgebra of P.Clearly,Θ(V)⊂P L⊗V.In this way V acquires the structure of a comodule over P L.The actions of the algebras P∗,(P L)∗on P are compatible via the algebra homomorphism P∗→(P L)∗induced by the inclusion P L⊂P.4.4.Dual Hopf algebra.We can define a Hopf algebra dual to P.Consider the algebra P∗with multipli-cation induced by∆L and quasi-comultiplication induced by multiplication in P.Consider the embedding δ:P֒→P∗mapping a phrase p into the phrase indicatorδp whose value on p is1and whose value on all other phrases is0.It is easy to see thatδ(P)is a subalgebra of P∗.In this way the module P acquires a new associative multiplication◦L.The quasi-comultiplication in P∗induces a genuine comultiplication in P transforming a phrase(w1|...|w k)into k i=0(w1|...|w i)⊗(w i+1|...|w k).This makes the algebra(P,◦L) into a Hopf algebra.By its very definition,it is dual to(P,∆L).For completeness,we describe multiplication◦L in P explicitly.For a phrase p=(w1|...|w k)and a non-empty word y,set p∗y=0if at least one of the words w1,...,w k does not belong to L.If w1,...,w k∈L, setp∗y= y=x1···x k+1x1w1x2w2...w k x k+1∈V⊂P。
appraisal_theory

appraisal engagement graduation
Appraisal Theory, David Hope and Jonathon Read
June 22, 2006
Presentation outline
• Systemic Functional Linguistics
Appraisal… …elaborates on the notion of interpersonal meaning, describing how social relationships are negotiated through evaluations of self, others and artefacts. …is situated within discourse semantics; attitude is often realised across grammatical boundaries, and is dynamic throughout a text.
• explicitly, through the lexicogrammar (inscribed) or
• implicitly, through ideational meanings (invoked) that: are marked with attitudinal lexical items (flagged);
• Appraisal resources
Attitude Engagement
Graduation
• Proposed research • Preliminary experiments • Discussion
Appraisal Theory, David Hope and Jonathon Read
Knudsen Diffusion in Silicon Nanochannels

a r X i v :0802.1852v 1 [p h y s i c s .f l u -d y n ] 13 F eb 2008Knudsen Diffusion in Silicon NanochannelsSimon Gruener and Patrick Huber ∗Faculty of Physics and Mechatronics Engineering,Saarland University,D-66041Saarbr¨u cken,GermanyMeasurements on helium and argon gas flow through an array of parallel,linear channels of 12nmdiameter and 200µm length in a single crystalline silicon membrane reveal a Knudsen diffusion type transport from 102to 107in Knudsen number Kn.The classic scaling prediction for the transport diffusion coefficient on temperature and mass of diffusing species,D He ∝√m Ar /m He .Deviations of thechannels from a cylindrical form,resolved with electron microscopy down to subnanometer scales,quantitatively account for a reduced diffusivity as compared to Knudsen diffusion in ideal tubular channels.The membrane permeation experiments are described over 10orders of magnitude in Kn,encompassing the transition flow regime,by the unified flow model of Beskok and Karniadakis.PACS numbers:47.45.n,47.61.-k,47.56.+rKnudsen diffusion (KD)refers to a gas transport regime where the mean free path between particle-particle collisions λis significantly larger than at least one characteristic spatial dimension d of the system con-sidered [1].By virtue of the negligible mutual particle collisions transport in such systems takes place via a se-ries of free flights and statistical flight direction changes after collisions with the confining walls.Because of the dependency of λon pressure p and temperature T given by the kinetic theory of gases,λ∝T/p ,this transport regime is only observable in macroscopic systems at very low p or elevated T .By contrast,for transport in spa-tially nanoconfined systems with d =O (10nm),the Knudsen number Kn =λ/d ,which quantifies the gas rarefaction,is larger than 1even for ambient pressures and temperatures,e.g.λ(He)=118nm at p =1bar and T =297K.Thus,for many processes involving gas transport in restricted geometries,e.g.gas catalysis and storage[2]or equilibration phenomena via gas flow in meso-and nanopores[3,4],KD plays a crucial role.Albeit the phenomenon of KD has been known for al-most a hundred years,details on its microscopic mecha-nisms are still at debate,in particular how wall rough-ness down to atomic scales and particle-wall interaction parameters affect the statistics of the diffusion process and thus the value of the KD transport coefficient D [5].In the past,a comparison between theory and experi-ment has often been hampered by complex pore network topologies,tortuous transport paths and poorly charac-terized pore wall structures in available nanoporous ma-trices.Nowadays,membranes with better defined geome-tries,such as carbon nanotube bundles[6,7],porous alu-mina [8],and tailored pores in mesoporous silicon [3,9],allow us to address such fundamental questions in more detail.A better understanding of,and more in-depth in-formation on,transport in such systems is not only of academic interest,but also of importance for the archi-tecture and functionality in the emerging field of nano-electromechanical and nanofluidic systems [10,11].FIG.1:(color online)He pressure relaxations in R1and R2at T =297K for starting pressures p s =0.1mbar (stars)and 100mbar (triangles)without (a)and with SiNC mem-brane (b).(c)Pressure relaxation time τas determined from measurements without (squares)and with SiNC membrane (circles)versus Knudsen number in the capillaries Kn c (bot-tom axis)and in the SiNCs Kn (top axis).The lines in (a),(b)and (c)represent calculated p (t )and τ(Kn )values.Insets in (c):TEM cross section views of two SiNCs in comparison with circular perimeters of 12nm diameter (dashed lines).In this Letter,we report the first rigorous experimen-tal study on transport of simple rare gases,i.e.He and Ar through an array of parallel aligned,linear channels with ∼12nm diameter and 200µm length in single crys-talline Si over a wide temperature (40K <T <300K)and Kn -range (102<Kn <107).We explore the clas-sical scaling predictions given by Knudsen for D on T ,mass of diffusing species,m ,and explore its dependencyon Kn.Attention shall also be paid to the morphology of the channel walls,resolved with transmission electronmicroscopy(TEM),and how it affects the diffusion. Linear,non-interconnected channels oriented along the 100 Si crystallographic direction(SiNCs)are electro-chemically etched into the surface of a Si(100)wafer[12].After the channels reach the desired length L of 200±5µm,the anodization current is increased by a factor of10with the result that the SiNC array is de-tached from the bulk wafer[13].The crystalline Si walls call for irregular channel perimeters as one can see in the TEM pictures of Fig.1(c),recorded from a mem-brane part Ar ion milled to∼5nm thickness.Infirst approximation,however,the SiNC’s cross-section can be described as circular with a diameter of∼12nm,in ac-cordance with the analysis of an Ar sorption experiment at T=86K,which yields additionally a SiNCs density ofΦ=(1.5±0.1)·1011cm−2.Our experimental setup consists of a copper cell with an inlet and outlet opening[14].Inlet and outlet are connected via stainless steel capillaries of radius r c= 0.7mm and length l c=70cm with two gas reservoirs, R1and R2,of an all-metal gas handling.Additionally two pneumatic valves V1and V2are used to open and close the connections between the sample cell and R1and the sample cell and R2.Four thermostatted capacitive pressure gauges allow us to measure the gas pressures in R1and R2,p1and p2,resp.,over a wide pressure range(5·10−3mbar<p<1bar)with an accuracy of 10−3mbar.The cell is mounted in a closed-cycle He cryostat to control the temperature between40K and 300K with an accuracy of1mK.Our goal of studying gas transport dynamics over alarge Kn range necessitates a thorough understanding of the intrinsicflow characteristics of our apparatus.There-fore,we start with measurements on Heflow through the macroscopic capillaries and the empty sample cell at room temperature T=297K.We record the equilibra-tion of p1(t)and p2(t)towards a pressure p eq as a function of time t after initial conditions,p1=p s>p2=0mbar at t=0s.In Fig.1(a)p1(t)/p s and p2(t)/p s are plotted for starting pressures of p s=0.1and100mbar.We ob-serve pressure equilibrations towards p eq=0.4·p s.The value0.4is dictated by the volume ratio of R1to R2. From the p(t)-curves we derive characteristic relaxation timesτaccording to the recipe p1(t=τ)−p2(t=τ)≡1/10·p s.It is understood that p is monotonically decreas-ing downstream from R1to R2.In order to quantify the gas rarefaction we resort,therefore,to a calculation of a mean Knudsen number Kn c=λ(p=p eq.This simplification is justified by the analysis provided below and by theo-retical studies which indicate differences of less than1% betweenflow rates calculated with an exact and an av-eraged treatment for Kn[10,15].In Fig.1(c)we plot measuredτvalues versus Kn c corresponding to a p s vari-ation from0.01to100mbar.For Kn c<0.1,τincreases with increasing Kn c.In an intermediate Kn c range,0.1<Kn c<0.6,we observe a cross-over regime towardsa saturation plateau withτ∼140s that extends to the largest Kn c studied.These changes in the p relaxation and henceflow dynamics are reminiscent of the three distinct transport regimes known to occur for gases as a function of their rarefaction[10,16]:For Kn c<0.1the number of interparticle collisions still predominates over the number of particle-wall collisions.Hagen-Poiseuille’s law is valid and predicts a decreasingflow rate and hence increasingτdue to the1/Kn-scaling of the particle num-ber density in gasflows.For Kn c>1we enter the pure KD regime[1],where theory predicts a He KD transport coefficient D He c dependent,however a Kn c independent particleflow rate,˙n Kn=πr2ck B TD He c,(1)which is responsible for the plateau inτfor the larger Kn c investigated.In Eq.1p i,p o refers to the inlet and outlet pressure of the capillary considered,resp., and k B to the Boltzmann factor.In the intermediate Kn-range,the interparticle collisions occur as often as particle-wall collisions which gives rise to the cross-over behavior found forτ.We elucidate this behavior in more detail by divid-ing theflow path within our apparatus into twoflow segments(up-and downstream capillary)and calculate the particle number changes along theflow path and the resulting p1(t),p2(t)with a1ms resolution using the unifiedflow model of Beskok and Karniadakis(BK-model)[10,17,18]and a local Kn-number for eachflow segment,Kn l=Kn((p i+p o)/2):˙n BK=πr4c(p2o−p2i)µ(T) 1+4Kn ldictions with our measurements is evident and the ex-tracted parameters α1=10±0.2and β=0.5±0.05(D He c =0.412±0.06m 2/s)agree with BK-modelling of He gas flow as a function of rarefaction [10,19].An anal-ogous analysis for Ar gas flow yields α1=1.55±0.05andβ=0.1±0.02(D Arc =0.134±0.02m 2/s).FIG.2:(color online)Pressure relaxation time τfor He (cir-cles)and Ar (triangles)measured with built-in SiNC mem-brane at T =297K versus Knudsen number in the SiNCs Kn (bottom axis)and in the capillaries Kn c (top axis).The lines represent calculated τ(Kn )-curves.Inset:TEM cross section of a SiNC in comparison with a circular channel perimeter of 12nm diameter (dashed line).The SiNC’s perimeter is highlighted by a line.Having achieved a detailed understanding of the intrin-sic flow characteristics of our apparatus,we can now turn to measurements with the SiNC membrane.The mem-brane is epoxy-sealed in a copper ring [14]and special at-tention is paid to a careful determination of the accessible membrane area,A =0.79±0.016cm 2in order to allow for a reliable determination of the number of SiNCs in-serted into the flow path,N =Φ·A =(11.8±1)·1010.As expected the pressure equilibration is significantly slowed down after installing the membrane -compare panel (a)and (b)in Fig.1.Choosing selected p s within the range 0.005mbar to 100mbar,which corresponds to a variation of Kn in the SiNCs from 102to 107,we find an increase in τof ∼170s,when compared to the empty cell mea-surements -see Fig.1(c).From the bare offset in τfor Kn c >1,we could calculate D He in the SiNCs.We are,however,interested in the D He behavior in a wide Kn -range,therefore we modify our flow model by inserting a segment with KD transport mechanism characteristic of N tubular channels with d =12nm and L =200µm in between the capillary flow segments.Adjusting the single free parameter in our simulation,the value of the He diffusion coefficient D He in a single SiNC,we arrive at the p (t )-and τ(Kn )-curves presented as solid lines in Fig.1(b)and (c),resp.The agreement with our mea-surement is excellent and the model yields a Kn inde-pendent D He =3.76±0.8mm 2/s.It is worthwhile to compare this value with Knudsen’sprediction for D .In his seminal paper he derives an ex-pression for D with two contributions,a factor G ,charac-teristic of the KD effectivity of the channel’s shape,and a factor solely determined by the mean thermal velocity of the particles3G3Gπm.(3)Interestingly,by an analysis of the number of particles crossing a given section of a channel in unit time after completely diffuse reflection from an arbitrary element of wall surface and while assuming an equal collision acces-sibility of all surface elements,Knudsen derived an ana-lytical expression for G for a channel of arbitrary shape.Knudsen’s second assumption is,however,only strictly valid for circular channel cross-sections,as pointed out by v.Smoluchowski [1]and elaborated for fractal pore wall morphologies by Coppens and Froment [5].Given the roughly circular SiNCs’cross section shapes,we nev-ertheless resort to Knudsen’s formula,here quoted nor-malized to the KD form factor for a perfect cylinder:14L Lo (l )measurements,which confirms the prediction of Eq.3,DHe /DAr =T ,see inset in Fig.3,confirming Knudsen’sclassic result down to the lowest T investigated.FIG.3:(color online)Pressure relaxation time τfor He gas flow through SiNC membrane (symbols)and simulated τ(Kn)(lines)versus Knudsen number in the SiNC membrane Kn (bottom axis)and in the supply capillaries Kn c (top axis)at selected temperatures T .Inset:He diffusion coefficient D He in SiNCs in comparison with the√T .We find no hints of anomalous fast KD here,as was re-cently reported for a bundle of linear carbon nanotubes [7]and explained by an highly increased fraction of spec-ularly reflected particles upon wall collisions [1].The al-tered collision statistics was attributed to the crystalline structure and the atomical smoothness of the nanotube walls.The SiNC walls are also crystalline,however,not atomical flat,as can be seen from our TEM analysis.Along with the formation of a native oxide layer,typical of Si surfaces,which renders the near surface structure amorphous,silica like,this,presumably,accounts for the normal KD observed here contrasting the one found for the graphitic walls of carbon nanotubes.We presented here the first detailed study of gas trans-port in linear SiNCs.Our conclusions are drawn from a correct treatment of gas flow over 10orders of magnitude in gas rarefaction,which is,to the best of our knowledge,the largest Kn range ever explored experimentally.The characteristic properties of KD,an independency of D on Kn ,its scaling predictions on m ,T ,and on details of the channel’s structure,resolved with sub-nm resolution,are clearly exhibited by our measurements.We thank A.Beskok for helpful discussions and the German Research Foundation (DFG)for support within the priority program 1164,Nano-&Microfluidics (Hu 850/2).∗E-mail:p.huber@physik.uni-saarland.de[1]M.Knudsen,Ann.Phys.(Leipzig)333,75(1909),M.v.Smoluchowski,ibid.338,1559(1910).[2]J.K¨a rger and D.Ruthven,Diffusion in Zeolites and Mi-croporous Solids (Wiley &Sons,New York,1992),S.M.Auerbach,Int.Rev.Phys.Chem.19,155(2000).[3]D.Wallacher et al.,Phys.Rev.Lett.92,195704(2004).[4]P.Huber and K.Knorr,Phys.Rev.B 60,12657(1999);R.Paul and H.Rieger,J.Chem.Phys.123,024708(2005);R.Valiullin et al.,Nature 443,965(2006).[5]M.O.Coppens and G. F.Froment,Fractals 3,807(1995);A.I.Skoulidas et al.,Phys.Rev.Lett.89,185901(2002);G.Arya,H.C.Chang,and E.J.Maginn,ibid.91,026102(2003);S.Russ et al.,Phys.Rev.E 72,030101(R)(2005).[6]B.J.Hinds et al.,Science 303,62(2004).[7]J.K.Holt et al.,Science 312,1034(2006).[8]S.Roy et al.,J.Appl.Phys.93,4870(2003);K.J.Alvine et al..Phys.Rev.Lett.97,175503(2006).[9]C.C.Striemer et al.,Nature 445,749(2007).[10]G.Karniadakis,A.Beskok,and N.Aluru,Microflowsand Nanoflows (Springer,2005).[11]H.A.Stone, A.D.Stroock,and A.Ajdari,Ann.Rev.Fluid Mech.36,381(2004);P.Huber et al.,Eur.Phys.J.Spec.Top.141,101(2007);M.Whitby and N.Quirke,Nature Nanotechnology 2,87(2007).[12]V.Lehmann and U.G¨o sele,Appl.Phys.Lett.58,856(1991);V.Lehmann,R.Stengl,and A.Luigart,Mat.Sc.Eng.B 69-70,11(2000).[13]A.Henschel et al.,Phys.Rev.E 75,021607(2007);P.Kumar et al.,J.Appl.Phys.103,024303(2008).[14]See EPAPS Document No.E-PRLTAO-100-019807for a raytracing illustration of the gas flow appa-ratus employed for the membrane permeation mea-surements.For more information on EPAPS,see /pubservs/epaps.html.[15]F.M.Sharipov and V.D.Seleznev,J.Vac.Sc.Technol.A 12,2933(1994).[16]P.Tabeling Introduction to Microfluidics (Oxford Uni-versity Press,New York,2005).[17]A.Beskok and G.Karniadakis,Nanosc.Microsc.Ther-mophys.Eng.3,43(1999).[18]S.Gr¨u ner,Diploma thesis,Saarland University (2006).[19]S.Tison,Vacuum 44,1171(1993).。
Predicting the Out-ofVocabulary Rate and the Required Vocabulary Size for

Following statement can be made for the vocabulary size V(K train) : V(K train) ≤ V(K) (9)
such a sample, the average vocabulary size ϕ K, S(M) and the average OOV-rate ω K, S(M) are calculated. For a sufficient sample size2, it can be assumed: ϕ K(M) ≈ ϕ K, S (M) ω K(M) ≈ ω K, S (M) (14) (15)
The vocabulary size V(K) is the number of all at least once occurring words within the corpus K: V(K) = T (4)
3. DETERMINING THE OOV-RATE OF ANY SUB-CORPUS
m=1
(10)
An examined word chain W l m is considered as OOV-word chain, if at least one word of W l m is not included in the regarded vocabulary V(Ktrain): if one or more words of Wl m 1 , are not within V(K train) = W V ( K ) oov train lm if all words of Wl m 0, are within V(K train)
1 M
K train = K \ { W l , …, W l } and 1 M K test = { W l , …, W l } and 1 M
Lie algebroid structures and Lagrangian systems on affine bundles

a r X i v :m a t h /023178v1[mat h.DG ]18Mar22Lie algebroid structures and Lagrangian systems on affine bundles Eduardo Mart´ınez †,Tom Mestdag ‡and Willy Sarlet ‡†Departamento de Matem´a tica Aplicada Universidad de Zaragoza,Mar ´ıa de Luna 3,50015Zaragoza,Spain ‡Department of Mathematical Physics and Astronomy Ghent University,Krijgslaan 281,B-9000Ghent,Belgium Abstract As a continuation of previous papers,we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immer-sion of the affine bundle into its bidual.We pay particular attention to the prolongation and various lifting procedures,and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.1Introduction Since the book of Mackenzie [12],the mathematics of Lie algebroids (and groupoids)has been studied by many authors;for a non-exhaustive list of references,see for example [3,5,6,9,10,13,18].The potential relevance of Lie algebroids for applications in physics and other fields of applied mathematics has gradu-ally become more evident.In particular,contributions by Libermann [11]and Weinstein [21]have revealed the role Lie algebroids play in modelling certain problems in mechanics.The concept of ‘Lagrangian equations’on Lie algebroids certainly defines an interesting generalisation of Lagrangian systems as known from classical mechanics,if only because of the more general class of differentialequations it involves while preserving a great deal of the very rich geometrical structure of Lagrangian (and Hamiltonian)mechanics.One of us [16],in particular,has produced evidence of this rich structure by showing that one can prolong a Lie algebroid in such a way that the newly obtained space has all the features of tangent bundle geometry,which are im-portant for the geometrical construction of Lagrangian systems.That is to say,the prolonged Lie algebroid carries a Liouville-type section and a vertical endomorphism which enables the definition of a Poincar´e -Cartan type 1-form,associated to a function L ;the available exterior derivative then does the rest1for arriving at an analogue of the symplectic structure from which Lagrangian equations can be derived.In[20],we have started an investigation on the possible generalisation of the concept of Lie algebroids to affine bundles.Our principle motivation was to create a geometrical model which would be a natural environment for a time-dependent version of Lagrange equations on Lie algebroids,as discussed for example in[21]and[16].Since classical time-dependent mechanics is usually described on thefirst-jet bundle J1M of a manifold Mfibred over I R(see e.g. [4,14]),and J1M→M is an affine bundle,it looks natural to build up a time-dependent generalisation in such a way that J1M→M is the image bundle of the anchor map of a Lie algebroid structure on some affine bundle E→M. An additional indication that such a set-up is well suited came from a naive calculus of variations approach,which gives a clue on the analytical format one should expect for such time-dependent Lagrange equations(see[19]).Lie algebroids on vector bundles are known to give rise,among other things, to a linear Poisson structure on the dual bundle,as well as a coboundary opera-tor on its Grassmann algebra;in fact,these properties equivalently characterise the Lie algebroid structure.One of the features of the approach to affine al-gebroids adopted in[20]was our specific choice to develop,in a direct way, a consistent theory of forms on sections of an affine bundle and their exterior calculus.By contrast,however,in the context of briefly mentioning the related Poisson structure in the concluding remarks,we did announce a forthcoming different approach,which would be based on the fact that an affine bundle can be regarded as an affine sub-bundle of a vector bundle,namely the dual of its extended dual.This is the line of reasoning we will develop here;it could be termed‘indirect’because it makes use of an imbedding into a larger bundle, but it has some marked advantages,such as the fact that proving a number of properties becomes much easier and that new insights come to the forefront.As a matter of fact,one readily recognises via this approach that much(if not all) of the theory of affine Lie algebroids can be developed without needing an extra fibration of the base manifold over I R.We will accordingly start our present analysis in this more general set-up and briefly come back to the special case appropriate for time-dependent systems in the concluding remarks.Note:while the very last editing of this paper was being done,we have been informed of similar investigations on affine algebroids,which have been carried out by Grabowski et al[7].The reader mayfind it instructive to compare the two simultaneous developments,which are related to each other up to and including our Section6.The scheme of the paper is as follows.A fairly detailed description of purely algebraic aspects of the theory is given in Sections2to4;it involves the introduc-tion of the concept of a Lie algebra over an affine space and aspects of exterior calculus.A Lie algebroid structure on a general affine bundleτ:E→M is defined in Section5:essentially,it comes from a classical Lie algebroid on the dual˜E of the extended dual E†of E,with the property that the bracket of sections in the image of the inclusion i:E→˜E,lies in the image of the under-lying vector bundle.Equivalent characterisations of this property can be found2in the subsequent section on the exterior differential and the associated Poisson structure.Section7presents a number of simple examples of affine algebroids. The important concept of prolongation of an algebroid is discussed in Section8: starting from a general construction on vector algebroids,it is shown that the prolonged bundle inherits the affine structure coming from E when the vector algebroid is the one on˜E.For the specific case of interest,it is further shown (see Section9)that there is a canonical map which gives rise to a‘vertical en-domorphism’on sections of the prolonged bundle.Natural constructions which are then available are complete and vertical lifts;they play a role in the ge-ometric definition of Lagrangian systems on affine Lie algebroids,presented in Section10.2Immersion of an affine space in a vector space Let A be an affine space modelled on a vector space V,and let A†=Aff(A,I R) be the extended dual of A,that is,the vector space of all affine maps from A to the real line.We consider the bidual˜A of A,in the sense˜A=(A†)∗.It is well known that in the case of a vector space V,the bidual˜V=(V∗)∗is isomorphic to V.In the case of an affine space,the bidual includes‘a copy’of A,as it is shown in the following statement.Proposition1.The map i:A→˜A given by i(a)(ϕ)=ϕ(a)is an injective affine map,whose associated vector map is i:V→˜A given by i(v)(ϕ)=ϕ(v) Proof.If a∈A and v∈V,then for allϕ∈A†,i(a+v)(ϕ)=ϕ(a+v)=ϕ(a)+ϕ(v)=i(a)(ϕ)+i(v),from which it follows that i is an affine map whose associated linear map is i.To prove that i is injective,it suffices to prove that i is injective,which is obvious since if v is an element in the kernel of i then i(v)(ϕ)=ϕ(v)=0for allϕ∈V∗,hence v=0.The vector space˜A is foliated by hyperplanes parallel to the image of i. Every vector z∈˜A is either of the form z=i(v)for some v∈V or of the form z=λi(a)for someλ∈I R\{0}and a∈A.Moreover,λ,a and v are uniquely determined by z.The image of the map i consists of the points for whichλ=1. To understand this description in more detail,we will prove that we have an exact sequence of vector spaces0−→V i−→˜A−→I R−→0.To this end we consider the dual sequence.Proposition2.Let l:I R→A†be the map that associates toλ∈I R the con-stant functionλon A.Let k:A†→V∗the map that associates to every affine function on A the corresponding linear function on V.Then,the sequence of vector spaces0−→I R l−→A†k−→V∗−→0is exact.3Proof.Indeed,it is clear that l is injective,k is surjective and k◦l=0,so that Im(l)⊂Ker(k).Ifϕ∈A†is in the kernel of k,that is the linear part ofϕvanishes,then for every pair of points a and b=a+v we have thatϕ(b)=ϕ(a+v)=ϕ(a)+ϕ(v)=ϕ(a),that isϕis constant,and hence in the image of l.The dual map of k is i,since for v∈V we havek(ϕ),v = ϕ,v = ϕ,i(v)The dual map j of the map l is given by j(αi(a)+iv)=α.Indeed,for every λ∈I R we havej(z)λ= z,l(λ) = αi(a)+i(v),l(λ) =α i(a)),l(λ) + i(v),l(λ) =αλIt follows thatCorollary1.If A isfinite dimensional,then the sequence0−→V i−→˜A j−→I R−→0is exact.Note that in this way we can clearly identify the image of V as the hyperplane of˜A with equation j(z)=0,and the image of A as the hyperplane of˜A with equation j(z)=1,in other wordsi(V)=j−1(0)and i(V)=j−1(1).Note in passing that if we have an exact sequence0−→Vα−→W j−→I R−→0, then we can define A=j−1(1);it follows that A is an affine space modelled on the vector space V and W is canonically isomorphic to˜A.The isomorphism is the dual map ofΨ:W∗→A†,Ψ(φ)(a)=φ(i(a)),where i:A→W is the canonical inclusion.We now discuss the construction of a basis for A†.Let(O,{e i})be an affine frame on A.Thus every point a has a representation a=O+v i e i.The family of affine maps{e0,e1,...,e n}given bye0(a)=1e i(a)=v i,is a basis for A†.Ifϕ∈A†,and we putϕ0=ϕ(O)andϕi=ϕ(e i),then ϕ=ϕ0e0+ϕi e i.It is to be noticed that,contrary to e1,...,e n,the map e0does not depend on the frame we have chosen for A.In fact,e0coincides with the map j.Let now{e0,e1,...,e n}denote the basis of˜A dual to{e0,e1,...,e n}.Then the image of the canonical immersion is given byi(O)=e0i(e i)=e i4from which it follows that for a=O+v i e i,we have i(a)=e0+v i e i.If we denote by(x0,x1,...,x n)the coordinate system on˜A associated to the basis {e0,...,e n},then the equation of the image of the map i is x0=1,while the equation of the image of i is x0=0.Coordinates in˜A∗=A†associated to the basis above will be denoted by (µ0,µ1,...,µn),that isµα(ϕ)= eα,ϕ for everyϕ∈A†.3Lie algebra structure over an affine space Implicit in our previous paper[20]is the following definition of a Lie algebra over an affine space.Definition1.Let A be an affine space over a vector space V.A Lie algebra structure on A is given by•a Lie algebra structure[,]on V,and•an action by derivations of A on V,i.e.a map D:A×V→V,(a,v)→D a v with the propertiesD a(λv)=λD a vD a(v+w)=D a v+D a wD a[v,w]=[D a v,w]+[v,D a w],•satisfying the compatibility propertyD a+v w=D a w+[v,w].Incidentally,it is sufficient to require in thefirst item that the bracket on V is I R-bilinear and skew-symmetric,since the Jacobi identity then follows from the requirements on D a.If we use a bracket notation[a,v]≡D a v,then the conditions in the defini-tion above read[a,λv]=λ[a,v][a,v+w]=[a,v]+[a,w][a,[v,w]]=[[a,v],w]+[v,[a,w]][a+v,w]=[a,w]+[v,w].This allows us to define a bracket of elements of A by putting[v,a]=−[a,v]and,if b=a+v,then[a,b]=[a,v].This bracket is skew-symmetric by construction and also satisfies a Jacobi-type property.5Theorem1.A Lie algebra structure over an affine space A is equivalent to a Lie algebra extension of the trivial Lie algebra I R by V.Explicitly,it is equivalent to the exact sequence of vector spaces0−→V i−→˜A j−→I R−→0being an exact sequence of Lie algebras.Proof.If the exact sequence is one of Lie algebras,we of course have a Lie algebra structure on V and the map D determined by D a v=[i(a),i(v)]satisfies all requirements to define a Lie algebra structure on A.Conversely,assume we have a Lie algebra structure on the affine space A.If wefix an element a∈A,then every element z∈˜A can be written in the form z=λi(a)+i(v).We can define a bracket of two elements z1=λ1i(a)+i(v1) and z2=λ2i(a)+i(v2)by[z1,z2]=i([v1,v2]+λ1D a v2−λ2D a v1)This bracket is clearly bi-linear and skew-symmetric,and a straightforward cal-culation shows that it satisfies the Jacobi identity.Moreover,the definition does not depend on the choice of the point a;if a′is another point in A,then a′=a+w for some w∈V and the compatibility condition implies that the result is independent of that choice.Finally,it is obvious that the maps i and j then are Lie algebra homomorphisms.Notice that the only condition for a Lie algebra structure on˜A to be an extension of I R by V is that the bracket takes values in V,symbolically:[˜A,˜A]⊂V.Once we have chosen an affine frame on A,we have that the bracket on˜A is determined by the brackets of the associated basis elements.These must be of the form[e0,e0]=0[e0,e j]=C k0j e k[e i,e j]=C k ij e k,since all brackets must take values in the image of the map i.It is well known that a Lie algebra structure on a vector space defines, and is defined by,a linear Poisson structure on the dual vector space.In the light of the results of the previous section we have a Poisson bracket on V∗and one on A†.Furthermore,˜A being an extension of I R by V,we have that the Poisson structureΛA†is an extension byΛV∗ofΛI R=0(see[2]for the details on Poisson extensions and their relations to Lie algebra extensions). Therefore,once we havefixed a point a∈A,we have a splitting of the sequence 0−→V i−→˜A j−→I R−→0given by h(λ)=λi(a),and the Poisson tensorcan be written asΛA†=ΛV∗+X a∧X Da ,where X Dais the linear vectorfield associated to the linear map D a∈End(V),and X a is the constant vector corresponding to i(a).Moreover,we have that L XD aΛV∗=0.In the coordinatesµ0,µ1,...,µn on A†associated to the basis{eα},we have {µ0,µ0}=0{µ0,µj}=µk C k0j{µi,µj}=µk C k ij,6where C k0j,C k ij are the structure constants introduced above.Therefore,the Poisson tensor readsΛ†A=1∂µi∧∂∂µ0∧∂Then,conditions1and2in the definition above are trivially satisfied.More-over,if wefix a0∈A and write a i=a0+v i,then by skew-symmetry of˜ωwe haveω(a1,...,a k)=˜ω(i(a1),...,i(a k))=˜ω(i(a0)+i(v1),...,i(a0)+i(v k))=kj=1(−1)j+1˜ω(i(a0),i(v1),..., i(v j),...,i(v k))+˜ω(i(v1),...,i(v k))=kj=1(−1)j+1ω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k),which proves condition3.Conversely,assume we are given a k-formωon the affine space A with its associatedω0andω.Fixing a0∈A,we know that every point z∈˜A can be written in the form z=λi(a0)+i(v)forλ∈I R and v∈V.We define the map ˜ωby˜ω(z1,...,z k)=˜ω(λ1i(a0)+i(v1),...,λk i(a0)+i(v k))=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k).By virtue of1and3,it follows that˜ωis multi-linear and skew-symmetric, i.e.it is a k-form on˜A.Moreover,˜ω(z1,...,z k)is independent of the choice of the point a0.Indeed,if we choose a different point a′0=a0+w,then z j=λj i(a′0)+v′j with v′j=v j−λj w,and applying the definition above we get ˜ω(z1,...,z k)=˜ω(λ1i(a′0)+i(v′1),...,λk i(a′0)+i(v′k))=kj=1(−1)j+1λjω0(a′0,v′1,..., v′j,...,v′k)+ω(v′1,...,v′k)=kj=1(−1)j+1λjω0(a0+w,v1−λ1w,..., v j,...,v k−λk w) +ω(v1−λ1w,...,v k−λk w)=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k)where we have used the properties ofω0andω.The form˜ωis unique,since,if˜θis a k-form on˜A such that i∗˜θ=0,then it follows that the associatedθ0andθvanish from where we deduce that˜θ=0.Once a reference frame has beenfixed on A,a1-form˜ωon˜A is of the form ˜ω=ω0e0+ωi e i,and then the local representation ofω=i∗˜ωlooks exactly the same.8More generally,a p-form˜ωon˜A is of the form˜ω=1(p−1)!ω0i1···i p−1e0∧e i1∧···∧e i p−1+1Definition3.A Lie algebroid structure on E consists of a Lie algebra structure on the(real)affine space of sections of E together with an affine mapρ:E→T M(the anchor),satisfying the following compatibility conditionDσ(fζ)=ρ(σ)(f)ζ+fDσζ,for everyσ∈Sec(E),ζ∈Sec(V)and f∈C∞(M),and where Dσis the action σ→Dσof Sec(E)on Sec(V).The compatibility condition ensures that the associationσ→Dσ,which acts by derivations on the real Lie algebra Sec(V),also acts by derivations on the C∞(M)-module Sec(V).The anchor mapρextends to a linear map˜ρ:˜E→T M,which we will describe in more detail below.It is of interest,however,to observe now already that the map i:V→˜E is a morphism of Lie algebroids,since we have[i(η1),i(η2)]=i([η1,η2])and˜ρ◦i=ρ,whereρis the linear part ofρ.On the contrary,the map j:˜E→M×I R is not a morphism of Lie algebroids,since we have that j([ζ1,ζ2])=˜ρ(ζ1)f2−˜ρ(ζ2)f1, while[j(ζ1),j(ζ2)]=0since thefibres of M×I R are1-dimensional.The affine Lie algebroid structure we studied in[20]is the case that M is furtherfibred over the real lineπ:M→I R and the anchor mapρtakes values in J1M.Notice that such an extrafibration is not generally available,not even locally.For instance,if we take any affine bundleτ:E→M with the trivial Lie algebroid structure(null bracket and anchor)then there is nofibration over I R such that the image ofρis in J1M,since the vectors in i(J1M)are non-zero.The following result shows that one can alternatively define an affine Lie algebroid structure on E as a vector Lie algebroid structure([,],˜ρ)on˜E such that the bracket of two sections in the image of i belongs to the image of i. Theorem2.A Lie algebroid structure on the vector bundle˜τ:˜E→M which is such that the bracket of sections in the image of i lies in the image of i defines a Lie algebroid structure on the affine bundleτ:E→M,whereby the brackets and maps are determined by the following relations:i([η1,η2])=[i(η1),i(η2)]i(Dση)=[i(σ),i(η)]ρ(σ)=˜ρ(i(σ)).Conversely,a Lie algebroid structure on the affine bundleτ:E→M extends to a Lie algebroid structure on the vector bundle˜τ:˜E→M such that the bracket of sections in the image of i is in the image of i.If wefix a sectionσof E and write sectionsζof˜E(locally)in the formζ=fi(σ)+i(η)then the anchor and the bracket are given by˜ρ(ζ)=fρ(σ)+ρ(η)[ζ1,ζ2]= ˜ρ(ζ1)(f2)−˜ρ(ζ2)(f1) i(σ)+i [η1,η2]+f1Dση2−f2Dση1 .10Proof.The verification of the above statements is straightforward but rather lengthy.We limit ourselves to checking that the compatibility conditions be-tween brackets and anchors are satisfied.For thefirst part,wefindi Dσ(fη) =[i(σ),i(fη)]=[i(σ),f i(η)]=˜ρ(i(σ))(f)i(η)+f[i(σ),i(η)]=i ρ(σ)(f)η+fDσ(η) , form which it follows that Dσ(fη)=ρ(σ)(f)η+fDσ(η).For the converse,observe that[ζ1,fζ2]−f[ζ1,ζ2]=f2˜ρ(ζ1)(f)i(σ)+i f1ρ(σ)(f)η2+ρ(η1)(f)η2=˜ρ(ζ1)(f)f2i(σ)+(f1ρ(σ)+ρ(η1))(f)i(η2)=˜ρ(ζ1)(f)ζ2,which is the required compatibility condition.In coordinates,if x i denote coordinates on M and yαfibre coordinates on E with respect to some local frame(e0,{eα})of sections of E,then we haveρ(e0+yαeα)=(ρi0+ρiαyα)∂∂x i,and the bracket is determined by[e0,e0]=0[e0,eβ]=Cγ0βeγ[eα,eβ]=Cγαβeγ.As afinal remark we mention that the orbits of the Lie algebroidτ:V→M are subsets of the orbits of the algebroid˜τ:˜E→M and they are equal if and only if there exists a sectionσof E such thatρ(σ)is in the image ofρ.6Exterior differential and Poisson structure Now that we have proved that a Lie algebroid structure on an affine bundle is equivalent to a Lie algebroid structure on˜E,we can define the exterior differen-tial operator on E by pulling back the exterior differential on˜E.More precisely, given a k-formωon the affine bundle E we know that there exists a unique˜ωon˜E such thatω=i∗(˜ω).Then we define dωas the(k+1)-form given bydω=i∗(d˜ω).11It is easy to see that this definition is equivalent to the one given in[20](at least when M isfibred over I R).The definition given here has some clear advantages.For instance,the prop-erty d2=0which was rather difficult to prove in[20],becomes evident now:d2ω=d(d(i∗˜ω))=d(i∗d˜ω)=i∗(d2˜ω)=0.The differential d on the Lie algebroid V is also related to the differential on˜E;for every k-form˜ωon E we have thatdi∗˜ω=i∗(d˜ω),which in fact simply expresses that i is a morphism of Lie algebroids.Hence, one canfind the differential of a formωon V by choosing a k-form˜ωon˜E such that i∗˜ω=ωand then obtain dωas i∗(d˜ω).That this does not depend on the choice of˜ωis expressed in equivalent terms by the following result.Proposition4.The ideal I={˜ω|i∗˜ω=0}is a differential ideal,i.e.d I⊂I.Proof.If i∗˜ω=0,then the same is true for d˜ωsince i∗(d˜ω)=di∗(˜ω)=0.The ideal I is generated by the1-form e0,I={e0∧θ|θis a form on˜E} so that de0belongs to I.The following result shows that e0in fact is d-closed and that this property characterises affine structures.Theorem3.A Lie algebroid structure on˜E restricts to a Lie algebroid struc-ture on the affine bundle E if and only if the exterior differential satisfies de0=0.Proof.Indeed,taking two sectionsσ1andσ2of E we havede0(i(σ1),i(σ2))=˜ρ(σ1) e0,i(σ2) −˜ρ(σ2) e0,i(σ1) − e0,[i(σ1),i(σ2)]=− e0,[i(σ1),i(σ2)]It follows that[i(σ1),i(σ2)]is in Im(i)=Ker(e0)if and only if de0vanishes on the image of i,which spans˜E.In coordinates,the exterior differential operator is determined byd f=ρi0∂f∂x ieα,for f∈C∞(M)andde0=0,deγ=−Cγ0βe0∧eβ−1f cannot simultaneously vanish,hence f defines a localfibration and then for any sectionσof E we have thatρ(σ)f= d f,σ = e0,σ =1,which is the condition for the anchor being1-jet-valued.When we have a Lie algebroid structure on˜E,there is a Poisson bracket on the dual bundle˜E∗=E†.Any sectionζof˜E,and in particular any section of E,determines a linear function ζon E†byζ(ϕ)= ζm,ϕ for everyϕ∈E†m.Then the Poisson bracket is determined by the condition{ ζ1, ζ2}= [ζ1,ζ2],which for consistency(using linearity and the Leibnitz rule)requires that we put{ ζ,g}=˜ρ(ζ)(g),and{f,g}=0,for f and g functions on M.It is of some interest to mention yet another characterisation of the result described in Theorem2.The above Poisson bracket in fact is determined by the bracket of linear functions coming from sections of E,since these span the set of all linear functions on E†.But the bracket of sections of E is a section of V;it follows that the corresponding Poisson brackets are independent of the coordinateµ0,and therefore,∂∂µ0∧X0,with X0=ρ(e0).(Noticethatρ(e0)=ρ(σ)for any sectionσof E.)The other brackets vanish,so there are no further conditions.In coordinates,we have{x i,x j}=0{µ0,x i}=ρi0{µα,x i}=ρiα{µ0,µβ}=Cγ0βµγ{µα,µβ}=Cγαβµγ13and therefore the Poisson tensor isΛE†=ρiα∂∂x i+1∂µα∧∂∂µ0∧ ρi0∂∂µβ .7ExamplesThe canonical affine Lie algebroid The canonical example of a Lie alge-broid over an affine bundle is thefirst jet bundle J1M→M to a manifold M fibered over the real lineπ:M→I R.The elements of the manifold J1M are equivalence classes j1tγof sectionsγof the bundleπ:M→I R,where two sec-tions are equivalent if they havefirst order contact at the point t.It is an affine bundle whose associated vector bundle is Ver(π)the set of vectors tangent to M which are vertical over I R.In this case it is well-known that J1M†=T∗M, and thereforeJ1M=T M.The canonical inmersion is given byi(j1tγ)=˙γ(t),i.e.it maps the1-jet of the sectionγat the point t to the vector tangent toγat the point t.In coordinates,if j1tγhas coordinates(t,x,v)theni(t,x,v)=∂∂x i (t,x)An element w of the associated vector bundle Ver(π)is of the formw=w i∂∂t +X i(t,x)∂∂x iwhich is obviously a section of the vector bundle.Affine distributions An affine E sub-bundle of J1M is involutive if the bracket of sections of the sub-bundle is a section of the associated vector bundle. Therefore,taking as anchor the natural inclusion into T M and as bracket the restriction of the bracket in J1M to E we have an affine Lie algebroid structure on E.14Lie algebra structures on affine spaces We consider the case in which the manifold M reduces to one point M={m}.Thus our affine bundle is E={m}×A and the associated vector bundle is W≡{m}×V for some affine space A over the vector space V.Then,a Lie algebroid structure over the affine bundle E is just an affine Lie algebra structure over A.Indeed,every section of E and of W is determined by a point in A and V,respectively.The anchor must vanishes since T M={0m},so it does not carry any additional information. Trivial affine algebroids By a trivial affine space we mean just a point A={O},and the associated vector space is the trivial one V={0}.The extended affine dual of A is A†=I R since the only affine maps defined on a space of just a point are the constant maps.It follows that the extended bi-dual is˜A=I RGiven a manifold M,we consider the affine bundle E=M×{O}with associated vector bundle V=M×{0}.On V we consider the trivial bracket [,]=0and the anchorρ=0,and as derivation D O we also take D O=0.Now, to construct a Lie algebroid structure on E,we take an arbitrary vectorfield X0on M as given and define the mapρ:E→T M byρ(m,O)=X0(m).Then it follows thatρis compatible with D O.The extended dual of E is E†=M×I R and the extended bi-dual is˜E= M×I R.We therefore have one section e0spanning the set of sections of E†, and the dual element e0(which is just the image under the canonical immersion of the constant section of value0.)We want to study the associated exterior differential operator and Poisson bracket.For the exterior differential operator,since we have1-dimensionalfibre on E†it follows that de0=0.On functions f∈C∞(M)we have d f=ρ(e0)(f)e0= X0(f)e0.For the Poisson structure,since thefibre of E†is1-dimensional,it is deter-mined by the equation{ˆe0,f}=ρ(e0)(f)andˆe0=µ0.We have that the only non-trivial brackets are{µ0,f}=X0(f).Therefore,the Poisson tensor is∂Λ=sectionsσi(m)=(m,ξi),is the constant section corresponding to the bracket of the values[σ1,σ2](m)=(m,[ξ1,ξ2]A).If we consider the Lie algebra˜A then˜A acts also on the manifold M.The extension˜E is the Lie algebroid associated to the action of˜A.Poisson manifolds with symmetry Consider a Poisson manifold(M,Λ) and an infinitesimal symmetry Y∈X(M)ofΛ,that is L YΛ=0.Take E to be T∗M with its natural affine structure,where the associated vector bundle is V=T∗M itself.On V we consider the Lie algebroid structure defined by the canonical Poisson structure.For a sectionαof E(i.e.a1-form on M)we define the map Dα:Sec(V)→Sec(V)byDαβ=L Yβ+[α,β].Since Y is a symmetry ofΛ,Dαis a derivation and clearly satisfies the required compatibility condition.If we further consider the affine anchorρ:E→T M, determined byρ(αm)=Λ(αm)+Y m,then we have a Lie algebroid structure on the affine bundle E.In this case,since there is a distinguished section of E(the zero section),we have that E†=T M×I R and˜E=T∗M×I R.Jets of sections in a groupoid Let G be a Lie groupoid over a manifold Mwith sourceαand targetβ(the notation is as in[1]).Let TαG(0)G=ker Tα|G(0)be the associated Lie algebroid,that is,the set ofα-vertical vectors at points in G(0)(the set of identities).The anchor is the mapρ=Tβ.Assume that M is furtherfibred over the real line,π:M→I R and consider the bundleE=JαG(0)G of1-jets of sections ofπ◦βwhich areα-vertical,at points in G(0).This is an affine bundle whose associated vector bundle is(TαG(0)G)ver the set of(π◦β)-vertical vectors on TαG(0)G.If i is the natural inclusion of(TαG(0)G)ver intoTαG(0)G and we define the map j:TαG(0)G→M×I R by j(v)=(α(τG(v)),t(v))(where t=π◦β◦τG),then we have the exact sequence of vector bundles over M0−→(TαG(0)G)ver i−→TαG(0)G j−→M×I R−→0and j−1(M×{1})=JαG(0)G.Moreover,the bracket of two sections of JαG(0)G isvertical over I R from where it follows that the Lie algebroid structure of TαG(0)Grestricts to a Lie algebroid structure on the affine bundle JαG(0)G.8ProlongationIn this section we define the prolongation of afibre bundle with respect to a (vector)Lie algebroid.We are primarily interested in the prolongation of the bundle E→M,but we will describe explicitly a more general constructionfirst, since this does not introduce extra complications(see also[8]for generalities).16。
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A Comparison of LSA, WordNet and PMI-IRfor Predicting User Click BehaviorIshwinder Kaur and Anthony J. HornofComputer and Information ScienceUniversity of OregonEugene, OR 97403, USA{ishakaur, hornof}@AbstractA predictive tool to simulate human visual search behavior would help interface designers inform and validate their design. Such a tool would benefit from a semantic component that would help predict search behavior even in the absence of exact textual matches between goal and target. This paper discusses a comparison of three semantic systems—LSA, WordNet and PMI-IR—to evaluate their performance in predicting the link that people would select given an information goal and a webpage. PMI-IR best predicted human performance as observed in a user study. Categories & Subject Descriptors: H.5.2 User Interfaces – evaluation/methodology, graphical user interfaces (GUI), screen design; H.5.4Hypertext/Hypermedia – Navigation; H.1.2 User/Machine Systems – Human information processing; H.1.1 Systems and Information Theory – Information theory, value of informationGeneral Terms: Design, Experimentation, Human Factors, VerificationKeywords: PMI, LSA, WordNet, semantic relatedness, semantic similarity, computational linguistics INTRODUCTIONWeb users continually apply their semantic knowledge while searching the web. For example, finding “the new bestselling textbook published by a psychology professor” in the website shown in Figure 1 is a web task that taxes semantic knowledge.Visual interfaces are the predominant method of conveying information during human computer interaction (HCI). Predicting human behavior in tasks involving their use would be useful for the designers of the interfaces. Efforts have been made in the usability simulation and analysis of visual displays. Display Analysis Program (DAP [36]) predicts the mean search time of an alphanumeric display based on the spatial relationship of objects in a layout. UCIE (Understanding Cognitive Information Engineering [23]) evaluates the usability of graphs and tables by predicting the search times and optimal eye movements of the users given an information goal. Though these tools are useful only in specific domains, they are an important first step in using predictive tools to simulate human responses,hence informing the design process.Figure 1. Where would you click to find information about “the new bestselling textbook published by apsychology professor”?(, January 26th 04)The research discussed in this paper is part of a larger effort to build a comprehensive tool for predicting visual search behavior, a tool that user interface designers could use early in a design process to evaluate screen layouts. The tool will be based on cognitive models of visual search, such as those of Byrne [6] and Hornof [14]. All of these models assume that the user knows the exact text of the target. However, exact matches between information goals and link labels are perhaps rare in real world tasks. The vocabulary problem, for example, illustrates that for a single concept there can be no single word or description (however well chosen) that will be used by all people [13]. Semantics affect how visual interfaces are searched [4, 28] and models of visual search will ultimately need to be enhanced with a semantic system to predict when users will terminate a search based on an inexact match. The semantic system would assign a quantitative value to the similarity in meaning between words and phrases.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.CHI 2005, April 2–7, 2005, Portland, Oregon, USA.Copyright 2005 ACM 1-58113-998-5/05/0004...$5.00.Some models of broader user navigational behavior (such as WUFIS [9]) that guide research and design of web-based information systems and site architectures do incorporate semantics. WUFIS simulates web users with specific information goals, and uses an “information scent” based approach to semantically associate the information on the web page to the information need of the user. Such systems would benefit from an evaluation of a variety of candidate semantic systems.A number of tools that evaluate some parameters of web usability, such as ease of navigation or lack of confusability, have also incorporated semantics in some form [3, 10]. It would be useful in such cases to have empirical validation for the choice of a semantic system in the context of HCI tasks.Systems that attempt to model web-surfing behavior, but do not yet have semantic behavior incorporated (for example, the Max Model [25]), would especially benefit from the work presented in this paper, as the rules guiding navigational decisions could then be based on the semantic content of the website along with its structure.Some systems monitor where users click in real-time to dynamically tailor the information environment and to assist the user by trying to infer their information goal (such as IUNIS [9] and ScentTrails [27]). One such assistive system, Letizia [20],models the content of a document as a list of keywords. The author of Letizia specifically states that “natural language capabilities that can extract some grammatical and semantic information quickly… could greatly improve its [Letizia’s] accuracy,” [20] suggesting the need for a semantic subsystem in the process of user modeling and behavior prediction.We would like to evaluate existing semantic systems that could be plugged in and be the semantic component of the tools in question. Given a web page and an information goal, the ideal system would choose the same link label that people would. To promote this engineering approach to usability analysis, the various systems evaluated in the paper are used, as best as possible, “off the shelf” and without enhancements, modifications or extensions.This paper is organized as follows: First we discuss and compare available semantic systems. Then we examine how some of those semantic systems are already used in usability evaluation tools. We then present a survey conducted for independent comparison of three of the systems in the context of HCI and discuss our methods for applying the three systems to predict the human data. Finally, we compare the predictions of the three systems. SEMANTIC SYSTEMSThere are a number of approaches for computing semantic similarity that might be useful in a predictive tool. These systems are typically discussed in the context of computational linguistics. The approaches can broadly be classified as follows: 1.Statistical: Semantic relationships between terms arecaptured from the probability of their co-occurrence ina text corpus, which is a large collection of documents.Some of these statistical systems are based on a vector space representation of a language.2.Taxonomical: The contents and relations of a hierarchyof terms are used to derive a quantitative value of similarity between terms. The terms are organized by subsumption with more general concepts subsuming the more specific ones (for example, automobile subsumes car). Other relations are also defined (for example, steering wheel is a part of a car).3.Hybrid: This approach borrows from both of the aboveapproaches. It is based on a taxonomical representation of concepts enhanced by the statistical properties of a text corpus.StatisticalStatistical approaches to semantic relatedness that are relevant to the context of predictive tools include Latent Semantic Analysis (LSA [12]), Hyperspace Analogue to Language (HAL [24]), Point-wise Mutual Information using Information Retrieval (PMI-IR [37]) and Non Latent Similarity (NLS [7]).LSA and HAL each process a large textual corpus to create a multi-dimensional semantic space. As the corpora used to build the semantic space can be categorized according to cultural, regional or educational variations, the corpus selection can be used to classify the resulting semantic spaces.In HAL and LSA, terms are represented by vectors in a semantic space. Term vectors that are closer together in a multi-dimensional semantic space are deemed to have a higher semantic relatedness. Semantic relatedness can include similarity (full or partial synonymy), meronymy (between a part and the whole), hypernymy or IS-A relationships (dog IS-A animal), and cause-effect relationships. The distinction between similarity and relatedness is important. For example, dog and collar co-occur in many texts, but ideally a predictive system should determine that they are related but not similar.The statistical approaches that use semantic spaces have been shown to reflect human performance quite accurately [16]. LSA reflects human acquisition of semantics as it performs comparably to humans in vocabulary tests, subject matter scores, word-sorting tasks, category judgments and lexical priming tasks [17].Purely statistical approaches will have some inherent potential limitations: First, the type of semantic relatedness is not captured but might be relevant to a predictive tool (e.g., the IS-A relationship might be most useful on websites that hierarchically categorize products like ). Second, different “senses” of words are treated as one (e.g., a bank could refer to a financial institution or the edge of a river). Third, different wordforms are treated as separate words (such as condition, conditional and conditioning). Lastly, part of speech is not taken into account (e.g., dog can be a verb and purchase can be a noun).PMI-IR is a statistical approach but it is not based on the concept of semantic spaces. PMI-IR uses the results of information retrieval to compute relatedness between words or phrases in terms of their mutual information; that is, the degree of shared content as measured by the probability of co-occurrence versus independent occurrence of terms [11]. PMI has produced results 10% better than LSA on synonym tests [37]. PMI-IR has also out-performed other computational methods for quantifying similarity [34].A number of other systems have also attempted to improve on LSA’s basic approach. NLS, for example, uses a corpus to parse and derive relationships between all word pairs. NLS captures syntactic relationships between the terms, over and above simple co-occurrence. The first order matrix of word relationships is squared to capture transitive, second-order relationships between terms. For example, as glass and heart are both related to break, it implies that glass and heart have something in common. In a test to predict word associates, NLS performed comparably to LSA for verbs and modifiers, and better than LSA for nouns [7].TaxonomicalTaxonomical relations are used to derive a number of semantic similarity and relatedness measures. Perhaps the best-known word taxonomy is WordNet [26], which is essentially a lexical database that encodes the relationships of synonymy, hypernymy and meronymy between terms. CYC [19] is also a large knowledge base and common-sense reasoning engine with a taxonomy of concepts and other relations such as causality.Hand-coded lexical knowledge databases such as WordNet and CYC make it is possible to encode large amounts of otherwise inaccessible human knowledge in a computational format. However, the knowledge acquisition is tedious, subject to the vagaries of human judgement, and not easily scalable to new terms, domains and languages. Since WordNet is simply a taxonomy, some kind of a system is needed to produce a similarity rating between two words in WordNet. Even though the lengths of links between the various nodes in WordNet are “subjective and vary widely” [Miller, G. A., personal communication], a number of different measures have been developed to compute a quantitative value of semantic similarity between terms in the WordNet taxonomy. wup [39] is a node-based measure that computes semantic similarity between terms based on the number of nodes between the terms, their lowest common subsumer, and the root. lesk[2] is a measure of semantic relatedness based on word overlap between term definitions, hypernyms, and hyponyms. path simply counts the number of edges between two terms. Another taxonomical system is NetSerf [8]. NetSerf uses a combination of semantic relations from WordNet and relationships extracted automatically from an online Webster’s dictionary (mainly IS-A relationships). NetSerf was developed to locate the web archive relevant to a query. NetSerf achieved this by locating the archive description that semantically subsumes the query, such as the archive of electronics when looking for digital cameras. Results show improvement over unstructured and non-semantic retrieval.HybridHybrid systems include res, jcn, lin and lch, all of which are based on the WordNet taxonomy, but also use a corpus for calculating similarity and relatedness values.res [30] is a WordNet based measure that uses probability-based information content and has been demonstrated to perform better than path at predicting human similarity judgements between terms. The similarity between two terms is defined as the information content of their most specific subsumer (calculated by taking the negative logarithm of the probability of its occurrence). One limitation of such an approach is that all terms within a branch will have the same similarity value with another term as long as the subsumer remains the same. For example, dog and cat have the same similarity value as dog and panther because both pairs have the same subsumer, carnivore.lin [21] overcomes this limitation by taking the information content of the participating terms into account. The similarity is then defined as the information content of the most specific subsumer divided by the information content of the two terms. jcn [15] is based on lexical edge counting, like path. Unlike path, the edges are weighted by frequency-based information content. The edge strength is determined by comparing the conditional probability of the child node given that the parent exists. The WordNet taxonomy has a non-uniform network density and highly variable node depth, and this measure tries to counter the effects of these problems. lch[18] combines semantic information from WordNet with local syntactic information to expand the size of sparse training data.The next section discusses how two semantic systems have been directly applied to predicting the usability of human computer interfaces.SEMANTIC SYSTEMS IN USABILITY TOOLSThis section discusses user interface analysis tools that make use of some of the previously discussed semantic systems to predict usability of web interfaces. These interface analysis tools would likely benefit from a comparison of the underlying semantic systems, as this paper aims to provide.Bloodhound [10] is a usability tool that relies on an underlying semantic system. Bloodhound automatically analyzes the navigability of a website by building on thetheory of information foraging [29]. Information foraging is based on the concept of information scent, which is the perceived utility of an information source and is related to the degree of similarity between (a) proximal cues afforded by the distal information source and (b) the information goal. Bloodhound computes this information scent based on the spreading activation using the probabilities of contextual occurrences of terms in a corpus. The information scent estimate is very similar in theory to PMI, discussed earlier.The approach used in Bloodhound has limitations. Since the web navigation simulator built into Bloodhound is sensitive to the choice of task query keywords, an analyst is required to choose the “correct” keywords to get the system to work in the desired manner. Also, the corpus used by the semantic system is typically the content of the very web site being tested. This choice of corpus might be appropriate for predicting user navigational behavior if the semantic structure of the website is consistent with the expectations of the users. But since the site is still being evaluated, the corpus based on the website could possibly be more representative of the designer’s rather than the users’ semantic expectations. A general-purpose or domain-specific corpus that captures the semantic acquisition of the target population and that is independent of the search documents may be more useful.The Cognitive Walkthrough for the Web (CWW [3]) builds on the Cognitive Walkthrough technique to evaluate the usability of a website. CWW uses LSA to compute the information scent for the links on a webpage in comparison to the user goal. This methodology has been used to find and repair usability problems with webpage designs, such as the unfamiliarity or confusability of link labels and group headings.A potential problem with using LSA for predicting usability is that, because LSA has a tendency to overestimate similarity, some of the predicted problems need to be subjectively discounted. Also, the use of the thresholds in CWW (e.g. LSA term vector length of less than 0.8 for a section heading implies that it is unfamiliar and a cosine of more than 0.6 between two headings implies that they are confusable) would benefit from more empirical validation. These limitations might be overcome by using a semantic system that is demonstrated to predict what a user would select, not just what would be confusing. This paper attempts to identify such a system.The usability tools discussed in this section use statistical techniques for semantic input. Apart from these techniques, other semantic systems such as NLS, PMI, HAL, and WordNet-based measures have been studied in various experiments, especially in the field of information retrieval, to see how they compare to human ratings of similarity, synonymy, antonymy, analogy relationships and detection of malapropisms. Researchers have used these systems to quantify semantic similarity in a manner that is consistent with the human view to a lesser or greater extent but, quite surprisingly, they have not to our knowledge been compared in the context of predicting user behavior in HCI. SURVEY OF USER BEHAVIOURWe wanted to test a number of semantic systems to compare their performance in predicting user selection behavior. To collect human data that could be used to compare these systems, we conducted a survey in which we asked participants what link they would select on a webpage given an information goal. For example, we asked them where they would click to find a publication from a professor in the Psychology department on the webpage inFigure 2.Figure 2. A sample webpage(/ Jan 26th 04)The data collected in this survey were then used to determine (a) if people tend to click the same links given the same information goal on the same web page, and (b) which semantic system best predicts their choice. ParticipantsTwenty-four native English speakers participated in the survey as paid participants. The sample included 12 female and 12 male participants, aged 18 to 56 years (median 20 years) and having 4 to 12 years (median 8 years) of experience using the web. Each session took approximately 50 minutes and each participant was paid $10.ApparatusParticipants were given paper printouts of the home pages of 45 colleges and universities randomly selected from a list of nearly 2,000 [22]. We collected the 45 goal statements from an email survey of students and alumni who access college and university webpages daily. The query posed in the email survey was “if you were searching a college or a university webpage, what would you be trying to find?” Our choice of webpages was constrained by their size so that the whole screenshot would fit on single sheet of paper. We wanted to stress ecological validity in our experiment and hence we did not otherwise limit our selection based on the visual design features even though the design elementsmay affect the links selected, along with the underlying semantic structure.ProcedurePairing each of the 45 webpages with 3 randomly selected goal statements resulted in 135 webpage-goal combinations. These 135 combinations were divided into three sets such that each web page and each goal statement appeared only once in a set. Each of the 24 participants was given one of the three sets, resulting in 8 participants per set. Hence, each of the 135 webpage-goal combinations was processed by 8 of the 24 participants, leading to a mixed (between- and within-subjects) design.AnalysisFor each webpage-goal combination, the participants wrote down one or two link labels on the web page that they would click to find that goal. Participants’ responses were studied to find clustering , which we define as the measure of consistency in the participant responses. That is, for each goal and web page combination, did the eight people who searched for the goal on the webpage choose the same link? Clustering measures the extent to which multiple people would select the same link given the same goal and the same web page. It is an important measure for the purpose of evaluating the quality of prediction, because for a system to adequately predict the choice made by the user, we need to ensure that there is indeed a set of “correct” links that users pick. Figure 3 shows the responses of the participants plotted on the web pages for samples of high and low clustering webpages. It is evident that a very high agreement between participants exists in the first case, whereas in the second case, the responses are less clustered. Each of the participants who searched for a specific goal on a particular webpage had effectively one vote, which they could either choose to give to a single link, or split equally between two links. The measure of consistency, or theclustering C, for the webpage-goal pair is defined as the normalized root mean square (NRMS) vote:C =x ii =1" (1)In Equation 1, x i refers to the total, non-zero votes given to the i th link that appears on the list of n competing links answered by the participants. As an example, for the goalstatementG2 in Figure 3(a), all the participants chose the same response, hence x i are {x 1 = 8} whereas for the goal G2 in figure 3(b), the eight votes are split between five different links, and hence x i are {x 1 = 3.5, x 2 = 2.5, x 3 = 1, x 4 = 0.5, x 5 = 0.5}. Table 1 shows the clustering measures for the goals G1, G2 and G3 in Figures 3(a) and 3(b). Table 1: The clustering measures of the responses to webpage-goal combinations plotted in Figure 3Webpage GoalFigure 3(a) Figure 3(b) G1 1.000 0.223 G2 1.000 0.250 G31.0000.559Survey ResultsFor the 135 webpage-goal combinations, the clustering measure showed a mean value of 0.49 with a standard deviation of 0.22. Note, that the value of 0.5 (or greater) on the clustering measure implies that the users agree on two links or fewer as their choice for the given goal statement. The survey data show that people do tend to cluster at a link when looking for the same piece of information on the same web page. Figure 3 shows the user data for the highest and lowest clustering webpage-goal combinations.(a) (b)Figure 3: Participant responses spatially plotted for the (a) highest clustering and (b) lowest clustering webpage-goal combinations. The different data points refer to different goal statements. For example, the square shapes plot the responses of the eight participants who pursued goal G1 on each webpage. The half moons indicate split votes.- G1 - G2 - G3- G1 - G2 - G3It would be useful to identify a semantic system that could predict where this clustering will occur so that the system could be built into tools used by web designers and analysts. The clustering data will now be used to evaluate candidate systems for such applications.APPLYING THE SYSTEMS TO PREDICT USER BEHAVIORThis section explains how the systems were used to predict the user behavior reported in the previous section. The goal is to determine which system most accurately predicts the human link selection behavior. WordNet (jcn, path, wup, lch, lin, res), LSA and PMI were all applied to predict the links selected by humans. The study attempts to evaluate these three semantic systems, as best as possible, off the shelf, with as little tweaking or enhancement as possible. WordNet: Since the majority of the entries in WordNet are single words and not phrases (only 0.74% of the words in all the goal statements and link labels formed a part of the compound words found in WordNet), the measures based on WordNet do not directly return a quantitative value of similarity/relatedness between two phrases. On the other hand, all of the goal statements and many of the link labels in our survey are phrases, sometimes even complete sentences.There are a number of different approaches that could be used to calculate phrase similarity using the WordNet based measures. We considered three approaches: (a) Ambroziak and Woods’ [1] approach of organizing phrases in a hierarchy using generality and subsumption, (b) Richardson and Smeaton’s [31] approach of averaging over all the word similarity values in a phrase, and (c) Sutcliffe et al.’s [33] technique of finding the single best match for a word from each phrase. We used approach (c) because, unlike (a), the mechanism was defined in sufficient detail for independent implementation and unlike (b), it is not biased against longer link labels.WordNet measures were computed using the WordNet::Similarity package [38]. Recall that a few of the measures use a text corpus for computing frequency of occurrence and the related information content of terms. A potential problem with using a text corpus is that polysemous words (with multiple unrelated meanings) might get exaggerated frequency counts, thus skewing predictions. To help combat this problem, all WordNet measures use a corpus called SemCor that is hand tagged with WordNet senses [32] and is included in the standard distribution of the WordNet similarity package.LSA: LSA measures were computed using the University of Colorado, Boulder's AutoCWW server. The first semantic space used was based on the generic TASA corpus containing 37,651 documents from general texts up to first year of college. Each paragraph in the text was treated as a separate document. LSA provides a built-in method of computing the similarity between word phrases based on the underlying vector space model. Through empirical analysis, this approach was found to be superior to the indirect approach of computing phrase similarity from word similarity (as described previously in the section on WordNet). Hence the indirect approach was not used in the final evaluation of LSA.To explore how the choice and size of corpus affects the results from LSA, we also built our own domain-relevant semantic space using articles from the Chronicle of Higher Education [35] as our corpus. We created these additional semantic spaces for LSA in part because it is relatively easy to do so, thus showcasing an off-the-shelf feature of LSA. Two versions of the Chronicle corpora were created and used. One version contains 112 articles, (3,990 “documents”). The other set used 363 articles (12,669 “documents”).PMI-IR: PMI-IR similarities were obtained using the Waterloo Multi Text System (WMTS), which is a distributed information retrieval system with one terabyte of corpus data. Turney’s original study concerning PMI-IR used the AltaVista search engine to pose queries to the web in real time, but in the current study, a static corpus including approximately 77 million web pages collected by a web crawler in 2001 has been used.We used the PMI-IR system in two versions. The first version, PMI-w, computed word similarity for each of the words in the goal statement and the link label. It then used the phrase similarity measure discussed for the WordNet measures to compute effective similarities.The PMI window of reference of size 20 was used for the first version of the system (20 was the default window size in Turney’s 2001 experiment). The second version, PMI-p, computed similarity using exact string comparison of the goal statement and the link labels. Since the probability of occurrence of exact phrase strings is less than that of individual words, the window of reference in this case was chosen to be the whole document.LSA and PMI-IR were thus tested with different corpora. Even though it would be interesting to compare the two systems using the same underlying domain-specific corpus, the point of this research is not so much to evaluate the underlying theories, but rather existing systems built on those theories, and it is less straightforward to train PMI-IR on a new corpus than it is LSA.SYSTEM PREDICTIONSThis section compares the predictions of the various systems. For each of the 135 webpage-goal pairs, participants in the survey responded with a set of links they would click on. These human preferences are taken to be target responses. The target responses are weighted by the number of votes received by each of them in the survey. For the same webpage-goal pair, each semantic system was used to rank the link labels from most similar to least。