Direct decompositions of lattices, I

语言学考研真题和答案第一章语言学Fill in the blanks1. Human language is arbitrary. This refers to the fact that there is no logical or intrinsic connection between a particular sound and the _______it is associated with. （人大2007研）meaning 语言有任意性，其所指与形式没有逻辑或内在联系2. Human languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at the moment of communication. This quality is labeled as _______. （北二外2003研）displacement 移位性指人类语言可以让使用者在交际时用语言符号代表时间和空间上不可及的物体、事件和观点3. By duality is meant the property of having two levels of structures, such that units of the _______ level are composed of elements of the __________ level and each of the two levels has its own principles of organization. （北二外2006研）primary, secondary 双重性指拥有两层结构的这种属性，底层结构是上层结构的组成成分，每层都有自身的组合规则4. The features that define our human languages can be called _______ features. （北二外2006）design人类语言区别于其他动物交流系统的特点是语言的区别特征，是人类语言特有的特征。

a rXiv:086.3898v1[mat h.RA ]24J un28CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS M.Dokuchaev Instituto de Matem´a tica e Estat´ıstica Universidade de S˜a o Paulo,05508-090S˜a o Paulo,SP,Brasil E-mail:dokucha@p.br R.Exel Departamento de Matem´a tica,Universidade Federal de Santa Catarina,88040-900Florian´o polis,SC,Brasil E-mail:exel@mtm.ufsc.br J.J.Sim´o n Departamento de Matem´a ticas,Universidad de Murcia,30071Murcia,Espa˜n a E-mail:jsimon@um.es Abstract.For a twisted partial action Θof a group G on an (associative non-necessarily unital)algebra A over a commutative unital ring k,the crossed product A ⋊ΘG is proved to be associative.Given a G -graded k -algebra B =⊕g ∈G B g with the mild restriction of homogeneous non-degeneracy,a criteria is established for B to be isomorphic to the crossed product B 1⋊ΘG for some twisted partial action of G on B 1.The equality B g B g −1B g =B g (∀g ∈G )is one of the ingredients of the criteria,and if it holds and,moreover,B has enough local units,then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.1.Introduction In the Theory of Operator Algebras partial actions of groups appeared as a general approach to study C ∗-algebras generated by partial isometries,several relevant classesof which turned out to be crossed products by partial actions.Crossed products clas-sically are in the center of the rich interplay between dynamical systems and operator algebras,and the efforts in generalizing them produce structural knowledge on alge-bras underlied by the new constructions.Thus the notion of a C ∗-crossed product by a partial automorphism,given in [19],permitted to characterize the approximately finite C ∗-algebras,the Bunce-Deddence and the Bunce-Deddence-Toeplitz algebras as such crossed products [20],[21].Further generalizations and steps were made in [34],2CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS [24],[22],[23].In[27]a machinery was developed,based on the interaction between partial actions and partial representations,enabling to study representations and the ideal structure of partial crossed products,and including into consideration prominent examples such as the Toeplitz C∗-algebars of quasi-lattice ordered groups,as well as the the Cuntz-Krieger algebras.The technique was also used in[26]to define and study Cuntz-Krieger algebras for arbitrary infinite matrices,and recently these ideas were also applied to Hecke algebras defined by subnormal subgroups of length two[25], as well as to C∗-algebras generated by crystals and quasi-crystals[6].The general notion of a(continuous)twisted partial action of a locally compact group on a C∗-algebra(a twisted partial C∗-dynamical system)and the corresponding crossed products were introduced in[22],where the associativity of crossed products was proved by means of approximate identities.The construction permitted to show in particular that any second countable C∗-algebraic bundle1with stable unitfibre is a crossed product by a twisted partial action[22].Algebraic counterparts for some of the above mentioned notions were introduced and studied in[14]and[13],stimulating further investigations in[5],[8],[9],[10],[11], [15],[16],[17],[18],[28],[29]and[33].Interesting results have been obtained also in[36]and[38](with respect to the topological part of[36]see also[1]).In[36]the authors point out several old and more recent results in various areas of mathematics underlied by partial group actions,and,more generally,the importance of the partial symmetry is discussed in[37],while developing the theory of inverse semigroups as a natural approach to such symmetries.The goal of the present article is to pursue the algebraic part of the programm by introducing the twisted partial actions of groups on abstract algebras over commuta-tive rings and the corresponding crossed products,and comparing them with general graded algebras.The reader will note a change in our treatment compared to previous ones:partial actions on rings,as well as on other structures,were defined so far in terms of intersections of the involved domains,and in the present paper we prefer to do it in terms of products.Moreover,the domains are supposed to be idempotent and pairwise commuting ideals.Observefirstly that this does not mean a change with respect to C∗-algebras:in that context the domains are closed two-sided ideals, and a product of such two ideals coincides with their intersection,in particular,they commute and are idempotent.Secondly,in the majority of the algebraic situations considered so far,the domains are unital(or more generally s-unital)rings and,as a consequence,again the products coincide with the intersections.Thirdly,the“product definition”serves our treatment of graded rings,whereas it is not even clear to us what would be the“nicest”definition in terms of intersections.The definition of a twisted partial action incorporates a“partial2-cocycle equality”which is definitely an associativity ingredient.The above mentioned conditions on theCROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS3 domains,i.e.to be idempotent and pairwise commuting,also serve the associativity purpose.In Section2.4,after giving our main definitions,we prove the associativity of the crossed product in Theorem2.4.This is done by means of a commuting property of left and right multipliers(2),introduced in[13]to serve the non-twisted case.Our next question is to give a criteria for a graded algebra to be a twisted partial crossed product.It is an easy exercise that for a group G,a G-graded unital algebra B= g∈G B g is a global crossed product B1⋊G exactly when each B g contains an element which is invertible in B,i.e.∀g∈G∃u g∈B g,v g∈B g−1such that u g v g=v g u g=1B.The corresponding criteria with twisted partial actions is more involved:using some analogue of the u g’s and v g’s one needs to construct isomorphisms D g−1→D g between some ideals D g of B1,a“partial twisting”formed by certain multipliers and everything should be combined into a twisted partial actionΘ,so that the given G-graded algebra B= g∈G B g will be B1⋊G.The easy part is to determine the D g’s:simply looking at the partial crossed product,one readily comes to define:D g=B g B g−1={ a i b i: a i∈B g−1,b i∈B g}.Next a crucial point comes:each B g is a(D g,D g−1)-bimodule, and one immediately forms the surjective Morita context(D g,D g−1,B g,B g−1)in which the bimodule maps are determined by the product in B.The rings in the context are non-necessarily unital and if we start with an arbitrary G-graded B,they may be non-idempotent and even degenerate(in the sense of Section3).So a restriction which guarantees that the D g’are idempotent is needed and there is also a reasonable assumption designed to avoid degeneracy.The latter is provided by the homogeneous non-degeneracy condition(16)(see Section3)which we impose on our graded algebra2. As to thefirst one,D2g=D g is a direct consequence of the equality(1)B g B g−1B g=B g(∀g∈G),which is immediately seen to be true for twisted partial crossed products(see Section3). The latter equality also has several other useful consequences given in Lemma5.3,and it enters as one of the ingredients of our criteria.In particular,it implies that the D g’s pairwise commute.The other ingredient deals with the u g’s and v g’s we are looking for.Tofind them one takes the linking algebra(also called Morita context ring)C g=D g B gB g−1D g−1 ,and it turns out that the elements u g and v g are multipliers of C g withu g v g=e11= 1000 and v g u g=e22= 0001 .The matrices e11and e22are not elements of C g in general,but they can be easily interpreted as multipliers(see Section3).That crossed products by twisted partial4CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS actions possess this property is shown in Proposition3.1.The consequences of the existence of such multipliers in the linking algebra of a given Morita context is analyzed in Section4.An isomorphism between the rings in the context comes out as expected,as well as a couple of maps with a list of associa-tivity properties(see Proposition4.1)being used in further considerations.The next task is to get out from the constraints of a single Morita context in order to construct the multipliers which will form the partial twisting.This is done in Section5.Then the criteria,Theorem6.1,is proved in Section6with the restriction of homogeneous non-degeneracy of B.The condition of the existence of the u g’s and v g’s can be replaced by a more straightforward condition,which in practice is easier to check,if we assume that each D g is an s-unital ring(see Theorem6.5).The main result of[22](Theorem7.3)says that a second countable C∗-algebraic bundle B,which satisfies a certain regularity condition,can be obtained as a twisted partial C∗-crossed product.The regularity condition is based on the existence of a par-tial isometry u in the multiplier algebra of a certain linking algebra such that uu∗=e11 and u∗u=e22.If the unitfibre B1is stable,i. e.B1is isometrically isomorphic to B1⊗K,where K is the C∗-algebra of the compact operators on an infinite dimensional separable Hilbert space,then using[7],the regularity condition is guaranteed(see[22, Propositions5.2,7.1]).In[7]the authors prove the interesting fact that two strongly Morita equivalent C∗-algebras A and A′with countable approximate units are neces-sarily stably isomorphic,i.e.A⊗K∼=A′⊗K.It is also shown that this fails in the absence of countable approximate units.The condition of B to be second countable serves to ensure the existence of countable approximate unities(or equivalently the existence of strictly positive elements)which is needed when using[7].It is our next main goal to establish an algebraic version of Theorem7.3from[22]. Any C∗-algebra has approximate unities,which are“units in limit”,and it is an ana-lytic property.Our algebraic analogue of this are the local units(see Section7for the definitions).The property of having countable approximate unities then corresponds to that of having a countable set of local units.It is easily seen that such a set of idempotents can be orthogonalized(see Section7),and we shall work in the more general situation of rings with orthogonal local units.One also should note that any closed ideal in a C∗-algebra is again a C∗-algebra,so it possesses approximate units. For an abstract ring with local units the analogue is not true,a two-sided ideal in such a ring may not have local units.So we deal with a G-graded B with enough local units,which means that each D g has orthogonal local units.Further,K can be viewed as the C∗-direct(also called inductive)limit of the matrix algebras M n(C),in which M n(C)is embedded into the left-upper corner of M n+1(C).The algebraic direct limit of the M n(C)’s is the algebra FMatω(C)of allω×ω-matrices over C which have only finitely many non-zero entries,whereωstands for thefirst infinite ordinal.Since we do not impose restrictions on the cardinatlities of the orthogonal units,our analogue of K will be FMat X(k),where X is an appropriate infinite set of indices and k is theCROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS5 base ring.Our aim is to prove Theorem8.5which says that if B is a G-graded k-algebra with enough local units which satisfies(1),then FMat X(B),endowed with the grad-ing directly extended from B,is the crossed product by a twisted partial action of G on FMat X(B1).Our criteria permits us to work in a single Morita context (R,R′,R M R′,R′M′R).The main tool is the concept of afinitely determined map, defined in Section5.In the same section we build up severalfinitely determined maps in order to use them in the Eilenberg trick which results into afinitely deter-mined isomorphismψ:R(X)→M(X)(see Proposition7.6),where X is an appro-priate infinite index set.Then in Section8the main step is done in Theorem8.2, in the proof of which the mapψis interpreted as what we call a row and column summable X×X-matrix over Hom R(R,M),which can be used to define the multi-plier u(the analogue of the above u g)of the linking algebra of the Morita context (FMat X(R),FMat X(R′),FMat X(M),FMat X(M′)).The multiplier v can be defined by using the inverse ofψwhich is alsofinitely determined.In practice we need only to give two certain maps and Proposition6.2guarantees the existence of u and v. Theorem8.5is then a quick consequence of Theorem8.2thanks to the criteria.An algebraic version of the above mentioned stable isomorphism result from[7]is given in Corollary8.4,which is an immediate by product of Theorem8.2(see also[4]).In Section9we show by means of examples with uncountable local units that The-orem8.2and Theorem8.5fail if we abandon the orthogonality condition on local units.2.Associativity of crossed products by twisted partial actionsIn all what follows k will be a commutative associative unital ring,which will be the base ring for our algebras.The latter will be assumed to be associative and non-necessarily unital.Let A be such an algebra.We remind that the multiplier algebra M(A)of A is the setM(A)={(R,L)∈End(A A)×End(A A):(aR)b=a(Lb)for all a,b∈A}with component-wise addition and multiplication(see[13]or[30,3.12.2]for more de-tails).Here we use the right hand side notation for homomorphisms of left A-modules, while for homomorphisms of right modules the usual notation shall be used.In par-ticular,we write a→aR and a→La for R:A A→A A,L:A A→A A with a∈A. For a multiplier w=(R,L)∈M(A)and a∈A we set aw=aR and wa=La.Thus one always has(aw)b=a(wb)(a,b∈A).Thefirst(resp.second)components of the elements of M(A)are called right(resp.left)multipliers of A.Definition2.1.A twisted partial action of a group G on A is a tripleΘ=({D g}g∈G,{θg}g∈G,{w g,h}(g,h)∈G×G),where for each g∈G,D g is a two-sided ideal in A,θg is an isomorphism of k-algebras D g−1→D g,and for each(g,h)∈G×G,w g,h is an invertible element from M(D g·D gh),6CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS satisfying the following postulates,for all g,h and t in G:(i)D2g=D g,D g·D h=D h·D g;(ii)D1=A andθ1is the identity map of A;(iii)θg(D g−1·D h)=D g·D gh;(iv)θg◦θh(a)=w g,hθgh(a)w−1g,h,∀a∈D h−1·D h−1g−1;(v)w1,g=w g,1=1;(vi)θg(aw h,t)w g,ht=θg(a)w g,h w gh,t,∀a∈D g−1·D h·D ht.Some comments are needed concerning the above definition.It obviously follows from(i)that afinite product of ideals D g·D h·...is idempotent,andθg(D g−1·D h·D f)=D g·D gh·D gffor all g,h,f∈G,by(iii).Thus all multipliers in(vi)are applicable.Given an (associative)algebra I with I2=I,by[13,Prop.2.5]one has(2)(wx)w′=w(xw′)for any w,w′∈M(I),x∈I.This explains the absence of brackets in the right hand side of(iv).Observe also that(iii)impliesθ−1g(D g·D h)=D g−1·D g−1h,for all g,h∈G.Definition 2.2.Given a twisted partial actionΘof G on A,the crossed product A⋊ΘG is the direct sum:D gδg,g∈Gin which theδg’s are symbols.The multiplication is defined by the rule:(a gδg)·(b hδh)=θg(θ−1g(a g)b h)w g,hδgh.Here w g,h acts as a right multiplier onθg(θ−1g(a g)b h)∈θg(D g−1·D h)=D g·D gh.An element a in an algebra A obviously determines the multiplier(R a,L a)∈M(A) where xR a=xa and L a x=ax(x∈A).If I is a two-sided ideal in A then this multi-plier evidently restricts to one of I which shall be denoted by the same pair of symbols (R a,L a).Given an isomorphismα:I→J of algebras and a multiplier u=(R,L)of I,we have that uα=(α−1Rα,αLα−1)is a multiplier of J.Observe that the effectCROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS7 ofα−1Rαon x∈I is x·(α−1Rα)=α(α−1(x)·R),whereas(αLα−1)·x=α(L·α−1(x)). Before proving the associativity of A⋊ΘG we separate two technical equalities. Lemma2.3.We have the following two properties:(i)aθh(θ−1h (b)c)=θh(θ−1h(ab)c),for any a,c∈A,b∈D h and h∈G;(ii)[θ−1gh(w g,hθgh(x))]c=θ−1gh(w g,hθgh(xc)), for any x∈D h−1·D h−1g−1,g,h∈G and c∈A.Proof.(i)Sinceθh:D h−1→D h is an isomorphism,(θ−1h R cθh,θh L cθ−1h)is a mul-tiplier of D h,and applying(2)for the multipliers(θ−1h R cθh,θh L cθ−1h)and(R a,L a)we see that L a·(b·(θ−1h R cθh))=(L a·b)·(θ−1h R cθh).But this is precisely what(i)does say. (ii)By(iii)of Definition2.1,θgh,restricted to D h−1·D h−1g−1,gives an isomorphismD h−1·D h−1g−1→D g·D gh.Hence wθ−1 ghg,h is a multiplier of D h−1·D h−1g−1,and combiningit in(2)with the multiplier(R c,L c)we obtain[(θ−1gh w g,hθgh)·x]·R c=(θ−1gh w g,hθgh)·[x·R c],which is exactly(ii).2Theorem2.4.The crossed product A⋊ΘG is associative.Proof.Obviously,A⋊ΘG is associative if and only if(3)(aδg bδh)cδt=aδg(bδh cδt)for arbitrary g,h,t∈G and a∈D g,b∈D h,c∈D puting the left hand side of the above equality,we have(aδg bδh)cδt=θg(θ−1g(a)b)w g,hδgh cδt=θgh{θ−1gh[θg(θ−1g(a)b)w g,h]c}w gh,tδght.On the other hand,aδg(bδh cδt)=aδgθh(θ−1h(b)c)w h,tδht=θg[θ−1g(a)θh(θ−1h(b)c)w h,t]w g,htδght=θg[θ−1g(a)θh(θ−1h(b)c)]w g,h w gh,tδght,by the co-cycle equality(vi)of Definition2.1,taking into account thatθ−1g(a)θh(θ−1h(b)c)∈D g−1·θh(D h−1·D t)=D g−1·D h·D ht.8CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRASThe last equality is obtained by using(iii)of Defiparing the two sides of(3)we may cancel w gh,t as it is invertible.Observing also thatθ−1g(a)runs overD g−1when a runs over D g,we have that(3)holds if and only if(4)θgh{θ−1gh[θg(ab)w g,h]c}=θg[aθh(θ−1h(b)c)]w g,his verified for any g,h∈G and a∈D g−1,b∈D h,c∈A(take t=1to see that c may be arbitrary in A).Applying(i)of Lemma2.3to the right hand side of(4),we have θg[aθh(θ−1h(b)c)]w g,h=θg[θh(θ−1h(ab)c)]w g,h.Now y=ab lies in D g−1D h,and therefore it is enough to show that(5)θgh{θ−1gh [θg(y)w g,h]c}=θg[θh(θ−1h(y)c)]w g,his satisfied for arbitrary g,h∈G,y∈D g−1·D h,c∈A.Write x=θ−1h(y)∈θ−1h(D g−1·D h)=D h−1·D h−1g−1.Then by(iv)of Definition2.1the left hand side of(5)becomesθgh{θ−1gh[θg◦θh(x)w g,h]c}=θgh{θ−1gh[w g,hθgh(x)]c}.Applying(ii)of Lemma2.3we see that the latter equals toθgh{θ−1gh [w g,hθgh(xc)]}=w g,hθgh(xc).Thus taking z=xc,(5)becomesw g,hθgh(z)=θg[θh(z)]w g,hwith arbitrary g,h∈G,z∈D h−1·D h−1g−1,but this is(iv)of Definition2.123.Some multipliers related to crossed productsWefirst derive some consequences from the definition of a twisted partial actionΘ. Taking h=g−1,t=g in(vi)of Definition2.1we easily obtain that(6)θg(aw g−1,g)=θg(a)w g,g−1for any g∈G,a∈D g−1.Applying the multiplier w−1g,g−1to the both sides of the aboveequality and replacing a by aw−1g−1,g,one has(7)θg(aw−1g−1,g )=θg(a)w−1g,g−1with any g∈G,a∈D g−1.From(6)and(7)we also obtain (8)θ−1g(aw g,g−1)=θ−1g(a)w g−1,g∀g∈G,a∈D g; and(9)θ−1g(aw−1g,g−1)=θ−1g(a)w−1g−1,g∀g∈G,a∈D g.Item(iv)of Definition2.1impliesθg◦θg−1(x)=w g,g−1xw−1g,g−1CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS9 with x∈D g.Taking x=θg(a),a∈D g−1and using(7),one comes to(10)θg(w g−1,g a)=w g,g−1θg(a)∀g∈G,a∈D g−1.Similarly as above one has the following equalities:(11)θg(w−1g−1,g a)=w−1g,g−1θg(a)∀g∈G,a∈D g−1;(12)θ−1g(w g,g−1a)=w g−1,gθ−1g(a)∀g∈G,a∈D g; and(13)θ−1g(w−1g,g−1a)=w−1g−1,gθ−1g(a)∀g∈G,a∈D g.Using the equalityθg−1◦θg(x)=w g−1,g xw−1g−1,g(x∈D g−1),we have(14)θ−1g(a)=w−1g−1,gθg−1(a)w g−1,g∀g∈G,a∈D g. Given a G-graded algebra B= g∈G B g,the setC g= B g B g−1B gB g−1B g−1B gis evidently a k-algebra with respect to the usual matrix operations.Here B g B g−1 stands for the k-span of the products ab with a∈B g,b∈B g−1,and similarly B g−1B g is defined.Obviously C g is contained in the algebra of all2×2matrices over B,and therefore,if B is unital,the elementary matrices e11= 1000 and e22= 0001 can be seen as multipliers for C g.However even if B is not a unital algebra,one can define the multipliers e11and e22of C g in the natural way:x y y′x′ · 1000 = x0y′0 , 1000 · x y y′x′ = x y00 ,x y y′x′ · 0001 = 0y0x′ , 0001 · x y y′x′ = 00y′x′ ,where x∈B g B g−1,x′∈B g−1B g,y∈B g,y′∈B g−1.In the case of a crossed product B=A⋊ΘG the algebra C g is obviously of the formD gδ1D gδgD g−1δg−1D g−1δ1 .If we assume that each algebra D g is unital with the unity element denoted by1g,then10CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRASC g is also unital and,consequently,C g can be identified with its multiplier algebra. Then taking the elementsu g= 01gδg00 and v g= 00w−1g−1,gδg−10 ,we see that C g possesses multipliers u g and v g which satisfy the equalities u g v g=e11 and v g u g=e22.This however holds for general crossed products:Proposition3.1.Given a crossed product A⋊ΘG by a twisted partial actionΘ,for each g∈G there exist multipliers u g,v g∈M(C g)such that u g v g=e11and v g u g=e22. Proof.Define the multipliers u g and v g as follows.aδ1bδg cδg−1dδ1 u g= 0aδg 0cw g−1,gδ1 ,u g aδ1bδg cδg−1dδ1 = θg(c)w g,g−1δ1θg(d)δg00 ,aδ1bδg cδg−1dδ1 v g= bδ10dw−1g−1,gδg−10 ,v g aδ1bδg cδg−1dδ1 = 00θ−1g(aw−1g,g−1)δg−1θ−1g(b)δ1 .Checkfirst that u g is a multiplier.Writex= aδ1bδgcδg−1dδ1 and y= a′δ1b′δg c′δg−1d′δ1By(6)we easily have(xy)u g= 0(aa′+bθg(c′)w g,g−1)δg0(cθg−1(a′)+dc′)w g−1,gδ1 =x(yu g). Furthermore,for the(1,1)-entry of u g(xy),by(iv)of Definition2.1,one hasθg(c)θg◦θg−1(a′)w g,g−1+θg(dc′)w g,g−1=θg(c)w g,g−1a′w−1g,g−1w g,g−1+θg(d)θg(c′)w g,g−1,which is the(1,1)-entry of(u g x)y.On the other hand,by(6)the(1,2)-entry of u g(xy) equals(θg(c)θg◦θg−1(b′)w g,g−1+θg(dd′))δg=(θg(c)w g,g−1b′+θg(d)θg(d′))δg,using again(iv)of Definition2.1.It is easily seen that this is the(1,2)-entry of(u g x)y, implying u g(xy)=(u g x)y.One also hasx(u g y)= aθg(c′)w g,g−1δ1aθg(d′)δgcθg−1(θg(c′)w g,g−1)δg−1cθg−1(θg(d′))w g−1,gδ1 =(xu g)yCROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS11 applying once more(6)and(iv)of Definition2.1.This shows that u g is a multiplier of C g.Similarly,using(7)one has(xy)v g=x(yv g),and by(9)and(14)it is easily seen that(xv g)y=x(v g y).As to the equality v g(xy)=(v g x)y,the(2,1)-entry of v g(xy)isθ−1g(aa′+bθg(c′)w g,g−1)w−1δg−1=g−1,g+θ−1g(b)c′)δg−1,(θ−1g(a)θ−1g(a′)w−1g−1,gby(9).Applying(14)this equals to the(2,1)-entry of(v g x)ing(9)and(14)we also see that the the(2,2)-entries coincide,which shows that v g(xy)=(v g x)y.Thus v g∈M(C g).By easy calculations we have u g v g=e11and v g u g=e22.2 We need one more property of a crossed product A⋊ΘG by a twisted partial action, which is in fact an immediate consequence of its definition.More precisely,we easily see that D gδg·D g−1δg−1·D gδg=(D2g)w g,g−1δe·D gδg=D gδe·D gδg=D gδg,as D g is idempotent and w g,g−1is an invertible multiplier of D g.Thus the equality(15)D gδg·D g−1δg−1·D gδg=D gδgholds for any g∈G.We shall work with a mild restriction onΘ,supposing that each domain D g is left and right non-degenerate.Following[13]and[32]we say that an algebra I is right non-degenerate if for any0=x∈I one has x I={0},and similarly is defined the concept of a left non-degenerate algebra.If each D g is left and right non-degenerate then we evidently havex=0=⇒x·D g−1δg−1=0and D g−1δg−1·x=0∀x∈D gδg. For a general G-graded algebra B= g∈G B g this corresponds to the property (16)x=0=⇒x·B g−1=0and B g−1·x=0∀x∈B g.We shall say that a G-graded algebra B is homogeneously non-degenerate if B satis-fies(16).It turns out that among the homogeneously non-degenerate graded algebras the crossed products by twisted partial actions are distinguished by the property(15) and the existence of the multipliers like in Proposition3.1.The precise statement will be given in Theorem6.1.Given a graded algebra B,write D g=B g B g−1.Observe that B g is a(D g,D g−1)-bimodule,whereas B g−1is a(D g−1,D g)-bimodule,so that(D g,D g−1,B g,B g−1)together with the surjective maps B g⊗B g−1→D g and B g−1⊗B g→D g−1,determined by the multiplication in B,form a Morita context.In general we know nothing about the existence of an identity element in D g,however in the situation in which we are interested in,the D g’s are idempotent rings and the bimodules B g,B g−1are unital,12CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRASand so by a result of[32],D g−1and D g are Morita equivalent(see Section7for the precise definitions).The idempotence of the D g’s,as well as the property of the B g’s being unital modules,is an immediate consequence of the analogue of(15)for graded algebras:(17)B g·B g−1·B g=B g∀g∈G.The existence of multipliers of B,like in Proposition3.1,turns out to be a powerful tool:we shall have that D g−1and D g are isomorphic and,the obtained isomorphisms willfit into a twisted partial actionΘof G on B1so that B∼=B1⋊ΘG.In order to carry out this idea we need some technical tools,which we develop in the next sections.4.Morita context and multipliersLet(R,R′,M,M′,τ,τ′)be a Morita context in which R and R′are some(non-necessarily unital)algebras.We recall that this means that M is an R-R′-bimodule, M′is an R′-R-bimodule,τ:M⊗R′M′→R is an R-bimodule map,τ′:M′⊗R M→R′is an R′-bimodule map,such thatτ(m1⊗m′)m2=m1τ′(m′⊗m2),∀m1,m2∈M,m′∈M′,andτ′(m′1⊗m)m′2=m′1τ(m⊗m′2),∀m′1,m′2∈M′,m∈M.One can construct the linking algebra of the Morita context,which is the setC= R MM′R′ ,with the obvious addition of matrices and multiplication by scalars,and the matrix multiplication determined by the bimodule structures on M and M′and the mapsτandτ′,so that m·m′=τ(m⊗m′)and m′·m=τ′(m′⊗m).If we have(18)R2=R,(R′)2=R′,R M+M R′=M and R′M′+M′R=M′,it is easily verified that C2=C.Thus by[13,Prop.2.5](19)(wx)w′=w(xw′)∀w,w′∈M(C),x∈C.In the case of a graded algebra B with R=D g,R′=D g−1,M=B g and M′=B g−1, the linking algebra C evidently coincides with the algebra C g considered in the previous section.If the graded algebra B satisfies(17)then obviously(18)is also satisfied. Similarly,as it was done in the previous section for C g,one defines the multipliers e11and e22of C in the natural way.CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS AND GRADED ALGEBRAS13 We are going to examine the situation in which the linking algebra C of a Morita context contains multipliers with properties announced in Proposition3.1.More pre-cisely,suppose that we have a Morita context(R,R′,M,M′,τ,τ′)such that the linking algebra C is idempotent.Suppose furthermore that there exists u,v∈M(C)such that uv=e11and vu=e22.One may assume that e11u=ue22=u and e22v=ve11=v. For it is enough to replace u by e11ue22and v by e22ve11,as(e11ue22)(e22ve11)= (uv)4=e11and similarly(e22ve11)(e11ue22)=e22.Let x= r m m′r′ be arbitrary in C.Thenux=e11u(e22x)=e11u 00m′r′ = ∗∗00 .Moreover,by(19),u 00m′0 =u(xe11)=(ux)e11= ∗000 ,and the(1,1)-entry of the latter matrix depends only on m′.Denote it by u·m′.Thus we have a map of k-modules M′∋m′→u·m′∈R which we denote by L u.Similarly, u 000r′ =u(xe22)=(ux)e22= 0u·r′00 ,and we have a k-linear map R′∋r′→u·r′∈M,which we denote by the same symbol L u.Thus we haveu 00m′r′ = um′ur′00 .Analogously,u,as a right multiplier,determines a pair of mapsR u:R∋r→r·u∈M and R u:M′∋m′→m′·u∈R′so that r0m′0 u= 0ru0m′u .In a similar way the multiplier v gives rise to the pairs of k-linear maps:L v:R→M′,M→R′(r→v·r,m→v·m)andR v:M→R,R′→M′(m→m·v,r′→r′·v),so thatv r m00 = 00vr vm ,and 0m0r′ v=mv0r′v0 .。

a rX iv:mat h /72574v2[mat h.CT]25Fe b27ACTORS IN CATEGORIES OF INTEREST J.M .CASAS 1,T.DATUASHVILI 2AND DRA 31Dpto.Matem´a tica Aplicada I,Univ.de Vigo,36005Pontevedra,Spain,e-mail:jmcasas@uvigo.es 2A.Razmadze Math.Inst.,Alexidze str.1,0193Tbilisi,Georgia,e-mail:tamar@rmi.acnet.ge 3Departamento de ´Algebra,Universidad de Santiago de Compostela,Santiago de Compostela 15782,Spain,e-mail:ladra@usc.es Abstract.For an object A of a category of interest C we con-struct the group with operations B (A )and the semidirect product B (A )⋉A and prove that there exists an actor of A in C if and only if B (A )⋉A ∈C .The cases of groups,Lie,Leibniz,asso-ciative,commutative associative,alternative algebras,crossed and precrossed modules are considered.In particular,we give exam-ples of categories of interest,where always exist actors.The paper contains some results for the case Ω2={+,∗,∗◦}.1.Introduction The paper is dedicated to the question of the existence and con-struction of actors for the objects in categories of interest (see below Section 2for the definitions and examples).This kind of categories were introduced by G.Orzech [Orz72].Actions in algebraic categories were studied by G.Hochschild [Hoc47],S.Mac Lane [Mac58],A.S.-T.Lue [Lue68],K.Norrie [Nor90],J.-L.Loday [Lod93],vendhomme and Th.Lucas [LL96]and others.The authors were looking for the analogs of automorphisms of groups in associative algebras,rings,Lie algebras,crossed modules and Leibniz algebras.We see different ap-proaches to this question.Lue and Norrie (on the base of the results of Lue [Lue79]and Whitehead [Whi48]),to any object associate a certain type of object,the construction in the corresponding category,called2J.M.CASAS,T.DATUASHVILI AND DRAactor of this object[Nor90],with special properties,analogous to group automorphisms,under which is meant that the actorfits into a certain commutative diagram(see Section2,diagram(2.7)).In[LL96]Lavend-homme and Lucas introduce the notion of aΓ-algebra of derivations for an algebra A,which is the terminal object in the category of crossed modules under A.Recently F.Borceux,G.Janelidze and G.M.Kelly [BJK05]proposed a categorical approach to this question.They study internal object actions and introduce the notion of a representable ac-tion,which in the case of a category of interest is equivalent to the definition of an actor given in this paper(see Section3).Let C be a category of interest with a set of operationsΩ=Ω0∪Ω1∪Ω2and a set of identities E.We define a general category of groups with operations C G with the same set of operations and a set of identities E G֒→E for which C֒→C G.We introduce the notions of an actor and of a general actor object for the objects of C.For any object A∈C we give a construction of the universal algebra B(A)with the operations fromΩ.We show that in general B(A)is an object of C G. For any A∈C we define an action of B(A)on A,which is a B(A)-structure on A in C G(i.e.the derived action appropriate to C G,see Section2below for the definitions).In a well-known way we define the universal algebra B(A)⋉A which is an object of C G.We define the homomorphism A−→B(A)in C G,which turned out to be a crossed module in C G.We show that the general actor object always exists and B(A)=GActor(A)(Theorem3.7).The main theorem states that an object A from C has an actor in C if and only if B(A)⋉A is an object in C and in this case A−→B(A)is an actor of A in C(Theorem3.6). From the results of[BJK05,Theorem6.3]and from Theorem3.6of this paper we conclude that a category of interest C has representable object actions in the sense of[BJK05]if and only if B(A)⋉A∈C for any A∈C,and if it is the case the corresponding representing objects are B(A),A∈C.We consider separately the caseΩ2={+,∗,∗◦}.In the case of groups (Ω2={+})we obtain that B(A)≈Aut(A),A∈G r.In the case of Lie algebras(Ω2={+,[,]})for A∈L ie we obtain B(A)≈Der(A). In the case of Leibniz algebras we have B(A)∈L eib,for any A∈L eib; B(A)has a derived set of actions on A if and only if for any B,C∈L eib which have a derived action on A we have[c,[a,b]]=−[c,[b,a]],for any a∈A,b∈B,c∈C,(which we call Condition1and it is equivalent to the existence of an Actor(A)).In this case B(A)=Actor(A)(Theorem 4.5).We give examples of such Leibniz algebras.In particular Leibniz algebras A with Ann(A)=(0),where Ann(A)denotes the annulator of A,and perfect Leibniz algebras(i.e.A=[A,A])satisfy ConditionACTORS IN CATEGORIES OF INTEREST3 1.We have an analogous picture for associative algebras.In this case B(A)is always an associative algebra,but the action of B(A)on A defined by us is not a derived action on A.Here we introduce Con-dition2:for any B and C∈A ss,which have a derived action on A we have c∗(a∗b)=(c∗a)∗b,for any a∈A,b∈B,c∈C, where∗denotes the action.The action of B(A)on A is a derived action if and only if A satisfies Condition2and it is equivalent to the existence of an Actor(A).In this case B(A)=Actor(A)(Proposi-tion4.6).Associative algebras with conditions Ann(A)=(0)or with A2=A satisfy Condition2.These kind of associative algebras are considered in[LL96,Mac58]).The cases of modules over some ring, commutative associative algebras,alternative algebras,crossed mod-ules and precrossed modules in the category of groups are discussed. Note,that the construction and the results given in our work enabled us to prove the existence of an actor in the category of precrossed mod-ules.We consider the case whereΩ2={+,∗,∗◦}.The necessary and sufficient conditions for the existence of an actor is determined in the case where E contains only all identities from E G and Axiom1and Ax-iom2,and Axiom2has not consequence identities(Theorem4.11).We consider separately the algebras of the same type as in Theorem4.11 with additional commutativity or anticommutativity condition.We obtain the necessary and sufficient conditions for the existence of an actor in the corresponding category(Theorem4.12)and give examples of such categories.The paper contains a comment to the formulation of Proposition1.1of[Dat95],which we apply in this paper.The re-sults obtained here enabled us tofind the construction of an actor of a precrossed module,which will be the subject of the forthcoming paper. The results of the paper are included in the Doctoral dissertation of the second author[Dat06]Chapter V.In Section2we present the main definitions and results which are used in what follows.In Section3we give the main construction of the object B(A)and the corresponding results.In Section4we consider the case of groups,Lie algebras,associative algebras and Leibniz algebras. For the special types of objects in A ss and L eib it is proved that B(A)≈Bim(A),B(A)≈Bider(A)respectively,(Propositions4.7,4.8),where Bim(A)denotes the associative algebra of bimultipliers defined by G. Hochschild and by S.Mac Lane for rings(called bimultiplications in [Mac58]and multiplications in[Hoc47]from where the notion comes) and Bider(A)denotes the Leibniz algebra of biderivations of A defined in Section2,which is isomorphic for these special types of Leibniz algebras to the biderivation algebra defined by J.-L.Loday in[Lod93]. At the end of the section we summarize for the special type of categories4J.M.CASAS,T.DATUASHVILI AND DRAof interest the results of Sections3and4and obtain the necessary and sufficient conditions for the existence of an actor.2.Preliminary definitions and resultsThis section contains well-known definitions and results which will be used in what follows.Let C be a category of groups with a set of operationsΩand with a set of identities E,such that E includes the group laws and the following conditions hold.IfΩi is the set of i-ary operations inΩ,then:(a)Ω=Ω0∪Ω1∪Ω2;(b)The group operations(written additively:0,−,+)are elements ofΩ0,Ω1andΩ2respectively.LetΩ′2=Ω2\{+},Ω′1=Ω1\{−}and assume that if∗∈Ω2,thenΩ′2contains∗◦defined by x∗◦y=y∗x. Assume further thatΩ0={0};(c)for each∗∈Ω′2,E includes the identity x∗(y+z)=x∗y+x∗z;(d)for eachω∈Ω′1and∗∈Ω′2,E includes the identitiesω(x+y)=ω(x)+ω(y)andω(x)∗y=ω(x∗y).We formulate more axioms on C(Axiom(7)and Axiom(8)of [Orz72]).If C is an object of C and x1,x2,x3∈C:Axiom1.x1+(x2∗x3)=(x2∗x3)+x1,for each∗∈Ω′2.Axiom2.For each ordered pair(∗,∗x3=W(x1(x2x3),x1(x3x2),(x2x3)x1,(x3x2)x1,x2(x1x3),x2(x3x1),(x1x3)x2,(x3x1)x2),where each juxtaposition represents an operation inΩ′2.We will write the right side of Axiom2as W(x1,x2;x3;∗,ACTORS IN CATEGORIES OF INTEREST5 below),takeΩ′2=([,],[,]◦),(here[a,b]◦=[b,a]).It is easy to see that all these algebras are categories of interest.In the example of groupsΩ′2=∅.As it is mentioned in[Orz72]Jordan algebras do not satisfy Axiom2.Definition2.1.[Orz72]Let C∈C.A subobject of C is called an ideal if it is the kernel of some morphism.Theorem2.2.[Orz72]Let A be a subobject of B in C.Then A is an ideal of B if and only if the following conditions hold;(i)A is a normal subgroup of B;(ii)For any a∈A,b∈B and∗∈Ω′2,we have a∗b∈A.Definition2.3.[Orz72]Let A,B∈C.An extension of B by A is a sequence(2.1)0E p0in which p is surjective and i is the kernel of p.We say that an exten-sion is split if there is a morphism s:B−→E such that ps=1B. Definition2.4.[Orz72]A split extension of B by A is called a B-structure on A.As in[Orz72],for A,B∈C we will say we have“a set of actions of B on A”,whenever there is a set of maps f∗:B×A−→A,for each ∗∈Ω2.A B-structure induces a set of actions ofB on A corresponding to the operations in C.If(2.1)is a split extension,then for b∈B,a∈A and∗∈Ω2′we haveb·a=s(b)+a−s(b),(2.2)(2.3)b∗a=s(b)∗a.(2.2)and(2.3)are called derived actions of B on A in[Orz72]and split derived actions in[Dat95],since we considered there actions derived from non split extensions too when A is a singular object.Given a set of actions of B on A(one for each operation inΩ2),let B⋉A be a universal algebra whose underlying set is B×A and whose operations are(b′,a′)+(b,a)=(b′+b,a′+b′·a),(b′,a′)∗(b,a)=(b′∗b,a′∗a+a′∗b+b′∗a).Theorem2.5.[Orz72]A set of actions of B on A is a set of derived actions if and only if B⋉A is an object of C.6J.M.CASAS,T.DATUASHVILI AND DRATogether with the description of the set of derived actions given inthe Theorem above,we will need the identities which satisfies a set of derived actions in case A,B∈C G and which guarantee that the set ofactions is a set of derived actions in C G.Proposition2.6.[Dat95]A set of actions in C G is a set of derivedactions if and only if it satisfies the following conditions:1.0·a=a,2.b·(a1+a2)=b·a1+b·a2,3.(b1+b2)·a=b1·(b2·a),4.b∗(a1+a2)=b∗a1+b∗a2,5.(b1+b2)∗a=b1∗a+b2∗a,6.(b1∗b2)·(a1∗a2)=a1∗a2,7.(b1∗b2)·(a∗b)=a∗b,8.a1∗(b·a2)=a1∗a2,9.b∗(b1·a)=b∗a,10.ω(b·a)=ω(b)·ω(a),11.ω(a∗b)=ω(a)∗b=a∗ω(b),12.x∗y+z∗t=z∗t+x∗y,for eachω∈Ω′1,∗∈Ω2′,b,b1,b2∈B,a,a1,a2∈A and for x,y,z,t∈A∪B whenever each side of12has a sense.Note that in the formulation of Proposition1.1in[Dat95]we meanthat the set of identities of the category of groups with operationscontains only identities from E G,but it is not mentioned there.Thesame concerns to some other statements of[Dat95].In the case where we have a category C with the set of identities E,the conditions1-12of the above proposition are necessary conditions.Of course it ispossible according to other identities included in E to write down thecorresponding conditions for derived actions which will be necessary and sufficient for the set of actions to be a set of derived actions(i.e. for B⋉A∈C).Denote all these identities by E G and E respectively. If the addition is commutative in C,then E(resp. E G)consists of the same kind of identities that we have in E(resp.in E G),written down forthe elements from the set A∪B,whenever each identity has a sense.Axiom2the identities for the action in C,which We will denote bycorrespond to Axiom2(see[Dat95]).In the category of groups,Lie, associative,Leibniz algebras derived actions are called simply actions. We will use this terminology in these special cases;we will also say“an action in C”,if it is a derived action,and we will say a set of actions is not an action in C if this set is not a set of derived actions.Recall that a left action of a group B on A is a mapε:B×A−→A,whichACTORS IN CATEGORIES OF INTEREST7 we denote byε(b,a)=b·a,with conditions(b1+b2)·a=b1·(b2·a),0·a=a,b·(a1+a2)=b·a1+b·a2.The right action is defined in an analogous way.All algebras below are considered over a commutative ring k with unit.In the case of associative algebras an action of B on A is a pair of bilinear maps(2.4)B×A−→A,A×B−→Awhich we denote respectively as(b,a)→b∗a,(a,b)→a∗b,with conditions(b1∗b2)∗a=b1∗(b2∗a),a∗(b1∗b2)=(a∗b1)∗b2,(b1∗a)∗b2=b1∗(a∗b2),b∗(a1∗a2)=(b∗a1)∗a2,(a1∗a2)∗b=a1∗(a2∗b),a1∗(b∗a2)=(a1∗b)∗a2.Here the associative algebra operation is denoted by∗(resp.a1∗a2) and the corresponding action by the same sign∗(resp.b∗a).For Lie algebras an action of B on A is a bilinear map B×A−→A, which we denote by(b,a)→[b,a],with conditions[[b1,b2],a]=[b1,[b2,a]]−[b2,[b1,a]],[b,[a1,a2]]=[a1,[b,a2]]+[[b,a1],a2].Note that we actually have above again two bilinear maps(2.4):(b,a)→[b,a],(a,b)→[a,b]with conditions[b,a]=−[a,b],[a,[b1,b2]]+[b1,[b2,a]]+[b2,[a,b1]]=0,[a1,[b,a2]]+[b,[a2,a1]]+[a2,[a1,b]]=0.Recall from[Lod93]that a Leibniz algebra L over a commutative ring k with unit is a k-module equipped with a bilinear map[−,−]:8J.M.CASAS,T.DATUASHVILI AND DRAL×L→L which satisfies the following identity,called the Leibniz identity:[x,[y,z]]=[[x,y],z]−[[x,z],y]for all x,y,z∈L.Obviously,when[x,x]=0for all x∈L,the Leibniz bracket is skew-symmetric,therefore the Leibniz identity comes down to the Jacobi identity,and a Leibniz algebra is then just a Lie algebra.For Leibniz algebras an action of B on A is a pair of bilinear maps (2.4),which we denote again by(b,a)→[b,a],(a,b)→[a,b]with conditions:[a1,[a2,b]]=[[a1,a2],b]−[[a1,b],a2],[a1,[b,a2]]=[[a1,b],a2]−[[a1,a2],b],[b,[a1,a2]]=[[b,a1],a2]−[[b,a2],a1],[a,[b1,b2]]=[[a,b1],b2]−[[a,b2],b1],[b1,[a,b2]]=[[b1,a],b2]−[[b1,b2],a],[b1,[b2,a]]=[[b1,b2],a]−[[b1,a],b2].Recall[Ser92]that a derivation for a Lie algebra A over a ring k is a k-linear map D:A−→A withD[a1,a2]=[D(a1),a2]+[a1,D(a2)].The set of all derivations Der(A)of A,with the operation defined by[D,D′]=DD′−D′Dis a Lie algebra.We recall the construction of the k-algebra Bim(A)of bimultipliers of an associative k-algebra A[Hoc47],[Mac58].An element of Bim(A) is a pair f=(f∗,∗f)of k-linear maps from A to A withf∗(a∗a′)=(f∗a)∗a′,(a∗a′)∗f=a∗(a′∗f),a∗(f∗a′)=(a∗f)∗a′.We prefer to use the notation∗f instead of f∗◦.We denote by f∗a (resp.a∗f)the value f∗(a)(resp.∗f(a)).Bim(A)is a k-module in an obvious way.The operation in Bim(A)is defined byf∗f′=(f∗f′∗,∗f∗f′),and Bim(A)becomes a k-algebra.Note that here we use different nota-tions than in[Mac58]and[LL96].Here as above∗denotes an operationACTORS IN CATEGORIES OF INTEREST9 in associative algebra,and f∗f′∗,∗f∗f′denote the compositions of maps.Thus(f∗f′∗)(a)=f∗(f′∗a),(∗f∗f′)(a)=(a∗f)∗f′.For the addition we havef+f′=((f∗)+f′∗,∗f+(∗f′)),where((f∗)+f′∗)(a)=f∗a+f′∗a,(∗f+(∗f′))(a)=a∗f+a∗f′.For a Leibniz k-algebra A we define the k-algebra Bider(A)of bideriva-tions in the following way.An element of Bider(A)is a pairϕ= ([,ϕ],[ϕ,])of k-linear maps A−→A with[a1,a2],ϕ = a1,[a2,ϕ] + [a1,ϕ],a2 ,ϕ,[a1,a2] = [ϕ,a1],a2 − [ϕ,a2],a1 ,a1,[a2,ϕ] =− a1,[ϕ,a2] .We used above the notation:[ϕ,](a)=[ϕ,a],[,ϕ](a)=[a,ϕ].Bideriva-tions where defined by Loday in[Lod93],where another notation is used;biderivation is a pair(d,D),where according to our definition [ϕ,]=D,[,ϕ]=−d and instead of the third condition we have in [Lod93][a1,d(a2)]=[a1,D(a2)].The operation in Bider(A)is defined by:[ϕ,ϕ′]= [,[ϕ,ϕ′]],[[ϕ,ϕ′],] ,wherea,[ϕ,ϕ′] = [a,ϕ],ϕ′ − [a,ϕ′],ϕ ,(2.5.1)[ϕ,ϕ′],a = ϕ,[ϕ′,a] + [ϕ,a],ϕ′ .(2.5.2)Note that we could define[[ϕ,ϕ′],]by(2.5.2’) [ϕ,ϕ′],a =− ϕ,[a,ϕ′] + [ϕ,a],ϕ′ .To avoid confusions we forget about∗◦in special cases,e.g.for the [,]operation.The above given both operations define a Leibniz al-gebra structure on Bider(A).It is easy to see that the second def-inition(2.5.1),(2.5.2’)gives the algebra which is isomorphic to the biderivation algebra defined in[Lod93];according to this definition [(d,D),(d′,D′)]=(dd′−d′d,Dd′−d′D).10J.M.CASAS,T.DATUASHVILI AND DRAWe have a set of actions of Der(A),Bim(A)and Bider(A)on A. These actions are defined by[D,a]=D(a),f∗a=f∗(a),a∗f=∗f(a),[ϕ,a]=[ϕ,](a),[a,ϕ]=[,ϕ](a),where a∈A,D∈Der(A),f=(f∗,∗f)∈Bim(A),ϕ=([,ϕ],[ϕ,])∈Bider(A)and A is a Lie algebra,an associative algebra and a Leibniz algebra respectively.In the case of Lie algebras the action of Der(A)on A is a set of derived actions,thus this action satisfies the corresponding conditions of an action in L ie,but for the case of associative and Leibniz algebras these actions do not satisfy all the conditions given above respectively for the action in A ss and L eib.Note that for the case of Leibniz al-gebras if[ϕ,[ϕ′,a]]=−[ϕ,[a,ϕ′]]for any a∈A andϕ,ϕ′∈Bider(A), then above two ways of defining operations in Bider(A)are equal and also the action of Bider(A)becomes a derived action(see below Propo-sition4.8).We have an analogous situation for associative algebras.The action of Bim(A)on A is not a derived action because the condition(2.6)(f∗a)∗f′=f∗(a∗f′)fails.So if we would have the condition for associative algebra A that for any two bimultipliers is fulfilled(2.6),then the action of Bim(A)onA defined above is a set of derived actions on A(see below Proposition4.7).An alternative algebra A over afield F is an algebra which satisfies the identitiesx2y=x(xy)andyx2=(yx)xfor all x,y∈A.These identities are known respectively as the left and right alternative laws.We denote the corresponding category of alternative algebras by A lt.Clearly any associative algebra is alterna-tive.The class of8-dimensional Cayley algebras is an important class of alternative algebras which are not associative[Sch66].The axioms above for alternative algebras are equivalent to the fol-lowing:x(yz)=(xy)z+(yx)z−y(xz)ACTORS IN CATEGORIES OF INTEREST11 and(xy)z=x(yz)−(xz)y+x(zy)We consider these conditions as Axiom2and consequently alternative algebras can be interpreted as categories of interest.For alternative algebras over afield F an action of B on A is a pair of bilinear maps(2.4),which we denote again by(b,a)→ba,(a,b)→ab with conditions:b(a1a2)=(ba1)a2+(a1b)a2−a1(ba2),(a1a2)b=a1(a2b)−(a1b)a2+a1(ba2),(ba1)a2=b(a1a2)−(ba2)a1+b(a2a1),a1(a2b)=(a1a2)b+(a2a1)b−a2(a1b),(b1b2)a=b1(b2a)−(b1a)b2+b1(ab2),a(b1b2)=(ab1)b2+(b1a)b2−b1(ab2),(ab1)b2=a(b1b2)−(ab2)b1+a(b2b1),b1(b2a)=(b1b2)a+(b2b1)a−b2(b1a).A crossed module in C is a triple(C0,C1,∂),where C0,C1∈C,C0 acts on C1(i.e.we have a derived action in C)and∂:C1−→C0is a morphism in C with conditions:(i)∂(r·c)=r+∂(c)−r;(ii)∂(c)·c′=c+c′−c;(iii)∂(c)∗c′=c∗c′;(iv)∂(r∗c)=r∗∂(c),∂(c∗r)=∂(c)∗rfor any r∈C0,c,c′∈C1,and∗∈Ω2′.A morphism between two crossed modules(C0,C1,∂)−→(C′0,C′1,∂′) is a pair of morphisms(T0,T1)in C,T0:C0−→C′0,T1:C1−→C′1, such thatT0∂(c)=∂′T1(c),T1(r·c)=T0(r)·T1(c),T1(r∗c)=T0(r)∗T1(c)for any r∈C0,c∈C1and∗∈Ω2′.Definition 2.7.For any object A in C an actor of A is a crossed module∂:A−→Actor(A),such that for any object C of C and an action of C on A there is a unique morphismϕ:C−→Actor(A)with c·a=ϕ(c)·a,c∗a=ϕ(c)∗a for any∗∈Ω2′,a∈A and c∈C. See the equivalent Definition3.9.in Section3.12J.M.CASAS,T.DATUASHVILI AND DRAFrom this definition it follows that an actor object Actor(A),for the object A∈C,with this properties is a unique object up to an isomorphism in C.Note that according to the universal property of an actor object,for any two elements x,y in Actor(A)from x·∗a=y·∗a,(here we mean equalities for the dot action and the action∗,for any∗∈Ω2′and any a∈A)and(w1···w n x)·a=(w1···w n y)·a,w1···w n∈Ω′1,it follows that x=y.It is well-known that for the case of groups Actor(G)=Aut(G);the corresponding crossed module is∂:G−→Aut(G),where∂sends any g∈G to the inner automorphism of G defined by g(i.e.∂(g)(g′)= g+g′−g,g′∈G).For the case of Lie algebras Actor(A)=Der(A), A∈L ie,and the operator homomorphism∂:A−→Der(A)is defined by∂(a)=[a,],so∂(a)(a′)=[a,a′].As we have seen above,in general,in A ss and L eib the objects Bim(A)and Bider(A)do not have derived actions on A in the cor-responding categories.So the obvious homomorphisms A−→Bim(A), A−→Bider(A)do not define crossed modules in A ss and L eib for any A from A ss and L eib respectively.It is well-known[Nor90]that for the case of groups if N is a normal subgroup of G andτ:N−→Inn(N)is the homomorphism sending any element n to the corresponding inner automorphism(τ(n)(n′)= n+n′−n),since G acts on N by conjugation,we have a unique homomorphismθ:G−→Actor(N),withθ(g)·n=g·n.Inn(N)is a normal subgroup of Actor(N),θextendsτand we have a commutative diagramτθ(2.7)0Actor(N)0. According to the work of vendhomme and Th.Lucas[LL96] in the categories G r,L ie the actor crossed modules A−→Actor(A) are terminal objects in the categories of crossed modules under A.If Ann(A)=(0)or A2=A then Bim(A)acts on A and the corresponding crossed module A−→Bim(A)is a terminal object in the category of crossed modules under A.It is easy to see that in this caseBim(A)=Actor(A).Definition2.8.A general actor object GActor(A)for A,A∈C,is an object from C G,which has a set of actions on A,which is a set of derived actions in C G,i.e.satisfies conditions of Proposition2.6,thereACTORS IN CATEGORIES OF INTEREST13 is a morphism d:A−→GActor(A)in C G which defines a crossed module in C G and for any object C∈C and a derived action of C on A,there exists in C G a unique morphismϕ:C−→GActor(A)such that c·∗a=ϕ(c)·∗a,for any c∈C,a∈A,∗∈Ω2′.It is easy to see that Bim(A)and Bider(A)are general actor objects for A∈A ss,A∈L eib respectively.These constructions satisfy the existence of the commutative diagram like(2.7).3.The main constructionIn this section C will denote a category of interest with a set of op-erationsΩand with a set of identities E.Let C G be the corresponding general category of groups with operations.According to the definition given in Section2,C G is a{Ω,E G}-algebra,whereΩis the same set of operations as we have in C,and E G includes group laws and identities (c)and(d)from the definition of Section2.Thus we have E G֒→E and C is the full subcategory of C G,C֒→C G.Let A∈C;consider all split extensions of A in CE j:0C j p j0,j∈J.Note that it may happen that B j=B k=B,for j=k,then these extensions will correspondent to different actions of B on A.Let {b j·,b j∗|b j∈B j,∗∈Ω′2}be the corresponding set of derived actions for j∈J.For any element b j∈B j denote b j={b j·,b j∗,∗∈Ω′2}.Let B={b j|b j∈B j,j∈J}.Thus each element b j∈B,j∈J is a special type of function b j:Ω2−→Maps(A→A),b j(∗)=b j∗−:A−→A.According to Axiom2from the definition of a category of interest, we define∗operation,b i∗b k,∗∈Ω′2,for the elements of B by the equalities(b i∗b k)∗),(b i∗b k)·(a)=a.We define the operation of addition by(b i+b k)·(a)=b i·(b k·a),(b i+b k)∗(a)=b i∗a+b k∗a.For a unary operationω∈Ω′1we defineω(b k)·(a)=ω(b k)·(a),ω(b k)∗(a)=ω(b k)∗(a),14J.M.CASAS,T.DATUASHVILI AND DRAω(b∗b′)=ω(b)∗b′and we will haveω(b)∗b′=b∗ω(b′),ω(b1+···+b n)=ω(b1)+···+ω(b n),(−b k)·(a)=(−b k)·a,(−b)·(a)=a(−b k)∗(a)=−(b k∗a),(−b)∗(a)=−(b∗(a)),−(b1+···+b n)=−b n−···−b1,where b,b′,b1,...,b n are certain combinations of star operations on theelements of B,i.e.the elements of the type b i1∗1···∗n−1b in,n>1.We do not know if the new functions defined by us are again in B. Denote by B(A)the set of functions(Ω2−→Maps(A→A))obtained by performing all kind of above defined operations on elements of B and new obtained elements as the results of operations.Note that b=b′in B(A)means that b·∗a=b′·∗a,w1...w n b·a=w1...w n b′·a for any a∈A,∗∈Ω′2,w1...w n∈Ω′1and for any n.It is an equivalence relation and under B(A)we mean the corresponding quotient object. Proposition3.1.B(A)is an object of C G.Proof.Direct easy checking of the identities. As above,we will write for simplicity b·(a)and b∗(a)instead of (b(+))(a)and(b(∗))(a)for b∈B(A)and a∈A.Define the set of actions of B(A)on A in a natural way.For b∈B(A)we defineb·a=b·(a),b∗a=b∗(a),∗∈Ω′2.Thus if b=b i1∗1···∗n−1b in,where we mean certain brackets,we haveb∗(a),b·a=a.The right side of the equality is defined inductively according to Axiom2.For b k∈B k,k∈J,we haveb k∗a=b k∗(a)=b k∗a,b k·a=b k·(a)=b k·a.Also(b1+b2+···+b n)∗a=b1∗(a)+···+b n∗(a),for b i∈B(A),i=1,···,n (b1+b2+···+b n)·a=b1·(b2···(b n·(a))···),ACTORS IN CATEGORIES OF INTEREST15b i∈B(A),i=1,···,nω(b)·a=a if b=b1∗···∗b n,b i∈B(A),i=1,···,nω(b k)·a=ω(b k)·a,k∈J,b k∈B k.Proposition3.2.The set of actions of B(A)on A is a set of derived actions in C G.Proof.The checking shows that the set of actions of B(A)on A satisfies conditions of Proposition2.6,which proves that it is a set of derived actions in C G. Define the map d:A−→B(A)by d(a)=a,where a={a·,a∗,∗∈Ω′2}.Thus we have by definitiond(a)·a′=a+a′−a,d(a)∗a′=a∗a′,∀a,a′∈A,∗∈Ω′2.Lemma3.3.d is a homomorphism in C G.Proof.We have to show that d(ωa)=ωd(a)for anyω∈Ω′1.For this we need to show thatd(ωa)·(a′)=(ωd(a))·(a′)ω′(d(ωa))·a′=ω′(ωd(a))·a′,for anyω′∈Ω′1d(ωa)∗(a′)=(ωd(a))∗(a′),for any∗∈Ω′2.We haved(ωa)·a′=ωa+a′−ωa,ωd(a)·a′=ω(a)·a′=ωa+a′−ωa,The second equality follows form thefirst one.For the third equality we haved(ωa)∗a′=(ωa)∗a′,(ωd(a))∗a′=ω(a)∗a′=ω(a)∗a′forω=−we have to show d(−a)·(a′)=(−(da))·a′and(d(−a))∗a′= (−d(a))∗a′.The checking of these equalities is an easy exercise. Now we will show that d(a1+a2)=d(a1)+d(a2).The direct com-putation of both sides for each a∈A givesd(a1+a2)·(a)=a1+a2+a−a2−a1,d(a1)+d(a2) ·(a)=d(a1)· d(a2)·a ,which shows that the desired equality holds for the dot action.The proof ofω(d(a1+a2))·a=ω(d(a1)+d(a2))·a is based on thefirst16J.M.CASAS,T.DATUASHVILI AND DRAequality,the property of unary operations to respect the addition and the fact that d commutes with unary operations.For any∗∈Ω′2we shall show thatd(a1+a2)∗(a)= d(a1)+d(a2) ∗(a).We haved(a1+a2)∗(a)=(a1+a2)∗a=a1∗a+a2∗a,d(a1)+d(a2) ∗(a)=d(a1)∗a+d(a2)∗a=a1∗a+a2∗awhich proves the equality.The next equality we have to prove is d(a1∗a2)=d(a1)∗d(a2).For this we need to show that d(a1∗a2)·(a)=(d(a1)∗d(a2))·(a),ω(d(a1∗a2))·a=ω(d(a1)∗d(a2))·a and d(a1∗a2)∗(a), for any∗(a)=(a1∗a2)∗),d(a1)∗d(a2) ∗ .These two expressions on the right sides of above equalities are equal, by the type of the word W in Axiom2and the definition of d. Proposition3.4.d:A−→B(A)is crossed module in C G.Proof.We have to check conditions(i)-(iv)from the definition of a crossed module given in Section2.(i)condition states that d(b·a)=b+d(a)−b for a∈A,b∈B(A);so we have to show that d(b·a)·∗a′=(b+da−b)·∗a′and ω1...ωn(d(b·a))·a′=ω1...ωn(b+da−b)·a′.Below we compute each side for the dot action of thefirst equality:d(b·a)·a′=b·a+a′−b·a,(b+d(a)−b)·a′=b·(d(a)·(−b·a′))=b·(a−b·a′−a)=b·a+a′−b·a.The second equality is proved in a similar way.Now we compute each side of thefirst equality for the∗action.d(b·a)∗a′=(b·a)∗a′=a∗a′by Proposition2.6;(b+da−b)∗a′=b∗a′+d(a)∗a′−b∗a′=b∗a′+a∗。

Draft:Deep Learning in Neural Networks:An OverviewTechnical Report IDSIA-03-14/arXiv:1404.7828(v1.5)[cs.NE]J¨u rgen SchmidhuberThe Swiss AI Lab IDSIAIstituto Dalle Molle di Studi sull’Intelligenza ArtificialeUniversity of Lugano&SUPSIGalleria2,6928Manno-LuganoSwitzerland15May2014AbstractIn recent years,deep artificial neural networks(including recurrent ones)have won numerous con-tests in pattern recognition and machine learning.This historical survey compactly summarises relevantwork,much of it from the previous millennium.Shallow and deep learners are distinguished by thedepth of their credit assignment paths,which are chains of possibly learnable,causal links between ac-tions and effects.I review deep supervised learning(also recapitulating the history of backpropagation),unsupervised learning,reinforcement learning&evolutionary computation,and indirect search for shortprograms encoding deep and large networks.PDF of earlier draft(v1):http://www.idsia.ch/∼juergen/DeepLearning30April2014.pdfLATEX source:http://www.idsia.ch/∼juergen/DeepLearning30April2014.texComplete BIBTEXfile:http://www.idsia.ch/∼juergen/bib.bibPrefaceThis is the draft of an invited Deep Learning(DL)overview.One of its goals is to assign credit to those who contributed to the present state of the art.I acknowledge the limitations of attempting to achieve this goal.The DL research community itself may be viewed as a continually evolving,deep network of scientists who have influenced each other in complex ways.Starting from recent DL results,I tried to trace back the origins of relevant ideas through the past half century and beyond,sometimes using“local search”to follow citations of citations backwards in time.Since not all DL publications properly acknowledge earlier relevant work,additional global search strategies were employed,aided by consulting numerous neural network experts.As a result,the present draft mostly consists of references(about800entries so far).Nevertheless,through an expert selection bias I may have missed important work.A related bias was surely introduced by my special familiarity with the work of my own DL research group in the past quarter-century.For these reasons,the present draft should be viewed as merely a snapshot of an ongoing credit assignment process.To help improve it,please do not hesitate to send corrections and suggestions to juergen@idsia.ch.Contents1Introduction to Deep Learning(DL)in Neural Networks(NNs)3 2Event-Oriented Notation for Activation Spreading in FNNs/RNNs3 3Depth of Credit Assignment Paths(CAPs)and of Problems4 4Recurring Themes of Deep Learning54.1Dynamic Programming(DP)for DL (5)4.2Unsupervised Learning(UL)Facilitating Supervised Learning(SL)and RL (6)4.3Occam’s Razor:Compression and Minimum Description Length(MDL) (6)4.4Learning Hierarchical Representations Through Deep SL,UL,RL (6)4.5Fast Graphics Processing Units(GPUs)for DL in NNs (6)5Supervised NNs,Some Helped by Unsupervised NNs75.11940s and Earlier (7)5.2Around1960:More Neurobiological Inspiration for DL (7)5.31965:Deep Networks Based on the Group Method of Data Handling(GMDH) (8)5.41979:Convolution+Weight Replication+Winner-Take-All(WTA) (8)5.51960-1981and Beyond:Development of Backpropagation(BP)for NNs (8)5.5.1BP for Weight-Sharing Feedforward NNs(FNNs)and Recurrent NNs(RNNs)..95.6Late1980s-2000:Numerous Improvements of NNs (9)5.6.1Ideas for Dealing with Long Time Lags and Deep CAPs (10)5.6.2Better BP Through Advanced Gradient Descent (10)5.6.3Discovering Low-Complexity,Problem-Solving NNs (11)5.6.4Potential Benefits of UL for SL (11)5.71987:UL Through Autoencoder(AE)Hierarchies (12)5.81989:BP for Convolutional NNs(CNNs) (13)5.91991:Fundamental Deep Learning Problem of Gradient Descent (13)5.101991:UL-Based History Compression Through a Deep Hierarchy of RNNs (14)5.111992:Max-Pooling(MP):Towards MPCNNs (14)5.121994:Contest-Winning Not So Deep NNs (15)5.131995:Supervised Recurrent Very Deep Learner(LSTM RNN) (15)5.142003:More Contest-Winning/Record-Setting,Often Not So Deep NNs (16)5.152006/7:Deep Belief Networks(DBNs)&AE Stacks Fine-Tuned by BP (17)5.162006/7:Improved CNNs/GPU-CNNs/BP-Trained MPCNNs (17)5.172009:First Official Competitions Won by RNNs,and with MPCNNs (18)5.182010:Plain Backprop(+Distortions)on GPU Yields Excellent Results (18)5.192011:MPCNNs on GPU Achieve Superhuman Vision Performance (18)5.202011:Hessian-Free Optimization for RNNs (19)5.212012:First Contests Won on ImageNet&Object Detection&Segmentation (19)5.222013-:More Contests and Benchmark Records (20)5.22.1Currently Successful Supervised Techniques:LSTM RNNs/GPU-MPCNNs (21)5.23Recent Tricks for Improving SL Deep NNs(Compare Sec.5.6.2,5.6.3) (21)5.24Consequences for Neuroscience (22)5.25DL with Spiking Neurons? (22)6DL in FNNs and RNNs for Reinforcement Learning(RL)236.1RL Through NN World Models Yields RNNs With Deep CAPs (23)6.2Deep FNNs for Traditional RL and Markov Decision Processes(MDPs) (24)6.3Deep RL RNNs for Partially Observable MDPs(POMDPs) (24)6.4RL Facilitated by Deep UL in FNNs and RNNs (25)6.5Deep Hierarchical RL(HRL)and Subgoal Learning with FNNs and RNNs (25)6.6Deep RL by Direct NN Search/Policy Gradients/Evolution (25)6.7Deep RL by Indirect Policy Search/Compressed NN Search (26)6.8Universal RL (27)7Conclusion271Introduction to Deep Learning(DL)in Neural Networks(NNs) Which modifiable components of a learning system are responsible for its success or failure?What changes to them improve performance?This has been called the fundamental credit assignment problem(Minsky, 1963).There are general credit assignment methods for universal problem solvers that are time-optimal in various theoretical senses(Sec.6.8).The present survey,however,will focus on the narrower,but now commercially important,subfield of Deep Learning(DL)in Artificial Neural Networks(NNs).We are interested in accurate credit assignment across possibly many,often nonlinear,computational stages of NNs.Shallow NN-like models have been around for many decades if not centuries(Sec.5.1).Models with several successive nonlinear layers of neurons date back at least to the1960s(Sec.5.3)and1970s(Sec.5.5). An efficient gradient descent method for teacher-based Supervised Learning(SL)in discrete,differentiable networks of arbitrary depth called backpropagation(BP)was developed in the1960s and1970s,and ap-plied to NNs in1981(Sec.5.5).BP-based training of deep NNs with many layers,however,had been found to be difficult in practice by the late1980s(Sec.5.6),and had become an explicit research subject by the early1990s(Sec.5.9).DL became practically feasible to some extent through the help of Unsupervised Learning(UL)(e.g.,Sec.5.10,5.15).The1990s and2000s also saw many improvements of purely super-vised DL(Sec.5).In the new millennium,deep NNs havefinally attracted wide-spread attention,mainly by outperforming alternative machine learning methods such as kernel machines(Vapnik,1995;Sch¨o lkopf et al.,1998)in numerous important applications.In fact,supervised deep NNs have won numerous of-ficial international pattern recognition competitions(e.g.,Sec.5.17,5.19,5.21,5.22),achieving thefirst superhuman visual pattern recognition results in limited domains(Sec.5.19).Deep NNs also have become relevant for the more generalfield of Reinforcement Learning(RL)where there is no supervising teacher (Sec.6).Both feedforward(acyclic)NNs(FNNs)and recurrent(cyclic)NNs(RNNs)have won contests(Sec.5.12,5.14,5.17,5.19,5.21,5.22).In a sense,RNNs are the deepest of all NNs(Sec.3)—they are general computers more powerful than FNNs,and can in principle create and process memories of ar-bitrary sequences of input patterns(e.g.,Siegelmann and Sontag,1991;Schmidhuber,1990a).Unlike traditional methods for automatic sequential program synthesis(e.g.,Waldinger and Lee,1969;Balzer, 1985;Soloway,1986;Deville and Lau,1994),RNNs can learn programs that mix sequential and parallel information processing in a natural and efficient way,exploiting the massive parallelism viewed as crucial for sustaining the rapid decline of computation cost observed over the past75years.The rest of this paper is structured as follows.Sec.2introduces a compact,event-oriented notation that is simple yet general enough to accommodate both FNNs and RNNs.Sec.3introduces the concept of Credit Assignment Paths(CAPs)to measure whether learning in a given NN application is of the deep or shallow type.Sec.4lists recurring themes of DL in SL,UL,and RL.Sec.5focuses on SL and UL,and on how UL can facilitate SL,although pure SL has become dominant in recent competitions(Sec.5.17-5.22). Sec.5is arranged in a historical timeline format with subsections on important inspirations and technical contributions.Sec.6on deep RL discusses traditional Dynamic Programming(DP)-based RL combined with gradient-based search techniques for SL or UL in deep NNs,as well as general methods for direct and indirect search in the weight space of deep FNNs and RNNs,including successful policy gradient and evolutionary methods.2Event-Oriented Notation for Activation Spreading in FNNs/RNNs Throughout this paper,let i,j,k,t,p,q,r denote positive integer variables assuming ranges implicit in the given contexts.Let n,m,T denote positive integer constants.An NN’s topology may change over time(e.g.,Fahlman,1991;Ring,1991;Weng et al.,1992;Fritzke, 1994).At any given moment,it can be described as afinite subset of units(or nodes or neurons)N= {u1,u2,...,}and afinite set H⊆N×N of directed edges or connections between nodes.FNNs are acyclic graphs,RNNs cyclic.Thefirst(input)layer is the set of input units,a subset of N.In FNNs,the k-th layer(k>1)is the set of all nodes u∈N such that there is an edge path of length k−1(but no longer path)between some input unit and u.There may be shortcut connections between distant layers.The NN’s behavior or program is determined by a set of real-valued,possibly modifiable,parameters or weights w i(i=1,...,n).We now focus on a singlefinite episode or epoch of information processing and activation spreading,without learning through weight changes.The following slightly unconventional notation is designed to compactly describe what is happening during the runtime of the system.During an episode,there is a partially causal sequence x t(t=1,...,T)of real values that I call events.Each x t is either an input set by the environment,or the activation of a unit that may directly depend on other x k(k<t)through a current NN topology-dependent set in t of indices k representing incoming causal connections or links.Let the function v encode topology information and map such event index pairs(k,t)to weight indices.For example,in the non-input case we may have x t=f t(net t)with real-valued net t= k∈in t x k w v(k,t)(additive case)or net t= k∈in t x k w v(k,t)(multiplicative case), where f t is a typically nonlinear real-valued activation function such as tanh.In many recent competition-winning NNs(Sec.5.19,5.21,5.22)there also are events of the type x t=max k∈int (x k);some networktypes may also use complex polynomial activation functions(Sec.5.3).x t may directly affect certain x k(k>t)through outgoing connections or links represented through a current set out t of indices k with t∈in k.Some non-input events are called output events.Note that many of the x t may refer to different,time-varying activations of the same unit in sequence-processing RNNs(e.g.,Williams,1989,“unfolding in time”),or also in FNNs sequentially exposed to time-varying input patterns of a large training set encoded as input events.During an episode,the same weight may get reused over and over again in topology-dependent ways,e.g.,in RNNs,or in convolutional NNs(Sec.5.4,5.8).I call this weight sharing across space and/or time.Weight sharing may greatly reduce the NN’s descriptive complexity,which is the number of bits of information required to describe the NN (Sec.4.3).In Supervised Learning(SL),certain NN output events x t may be associated with teacher-given,real-valued labels or targets d t yielding errors e t,e.g.,e t=1/2(x t−d t)2.A typical goal of supervised NN training is tofind weights that yield episodes with small total error E,the sum of all such e t.The hope is that the NN will generalize well in later episodes,causing only small errors on previously unseen sequences of input events.Many alternative error functions for SL and UL are possible.SL assumes that input events are independent of earlier output events(which may affect the environ-ment through actions causing subsequent perceptions).This assumption does not hold in the broaderfields of Sequential Decision Making and Reinforcement Learning(RL)(Kaelbling et al.,1996;Sutton and Barto, 1998;Hutter,2005)(Sec.6).In RL,some of the input events may encode real-valued reward signals given by the environment,and a typical goal is tofind weights that yield episodes with a high sum of reward signals,through sequences of appropriate output actions.Sec.5.5will use the notation above to compactly describe a central algorithm of DL,namely,back-propagation(BP)for supervised weight-sharing FNNs and RNNs.(FNNs may be viewed as RNNs with certainfixed zero weights.)Sec.6will address the more general RL case.3Depth of Credit Assignment Paths(CAPs)and of ProblemsTo measure whether credit assignment in a given NN application is of the deep or shallow type,I introduce the concept of Credit Assignment Paths or CAPs,which are chains of possibly causal links between events.Let usfirst focus on SL.Consider two events x p and x q(1≤p<q≤T).Depending on the appli-cation,they may have a Potential Direct Causal Connection(PDCC)expressed by the Boolean predicate pdcc(p,q),which is true if and only if p∈in q.Then the2-element list(p,q)is defined to be a CAP from p to q(a minimal one).A learning algorithm may be allowed to change w v(p,q)to improve performance in future episodes.More general,possibly indirect,Potential Causal Connections(PCC)are expressed by the recursively defined Boolean predicate pcc(p,q),which in the SL case is true only if pdcc(p,q),or if pcc(p,k)for some k and pdcc(k,q).In the latter case,appending q to any CAP from p to k yields a CAP from p to q(this is a recursive definition,too).The set of such CAPs may be large but isfinite.Note that the same weight may affect many different PDCCs between successive events listed by a given CAP,e.g.,in the case of RNNs, or weight-sharing FNNs.Suppose a CAP has the form(...,k,t,...,q),where k and t(possibly t=q)are thefirst successive elements with modifiable w v(k,t).Then the length of the suffix list(t,...,q)is called the CAP’s depth (which is0if there are no modifiable links at all).This depth limits how far backwards credit assignment can move down the causal chain tofind a modifiable weight.1Suppose an episode and its event sequence x1,...,x T satisfy a computable criterion used to decide whether a given problem has been solved(e.g.,total error E below some threshold).Then the set of used weights is called a solution to the problem,and the depth of the deepest CAP within the sequence is called the solution’s depth.There may be other solutions(yielding different event sequences)with different depths.Given somefixed NN topology,the smallest depth of any solution is called the problem’s depth.Sometimes we also speak of the depth of an architecture:SL FNNs withfixed topology imply a problem-independent maximal problem depth bounded by the number of non-input layers.Certain SL RNNs withfixed weights for all connections except those to output units(Jaeger,2001;Maass et al.,2002; Jaeger,2004;Schrauwen et al.,2007)have a maximal problem depth of1,because only thefinal links in the corresponding CAPs are modifiable.In general,however,RNNs may learn to solve problems of potentially unlimited depth.Note that the definitions above are solely based on the depths of causal chains,and agnostic of the temporal distance between events.For example,shallow FNNs perceiving large“time windows”of in-put events may correctly classify long input sequences through appropriate output events,and thus solve shallow problems involving long time lags between relevant events.At which problem depth does Shallow Learning end,and Deep Learning begin?Discussions with DL experts have not yet yielded a conclusive response to this question.Instead of committing myself to a precise answer,let me just define for the purposes of this overview:problems of depth>10require Very Deep Learning.The difficulty of a problem may have little to do with its depth.Some NNs can quickly learn to solve certain deep problems,e.g.,through random weight guessing(Sec.5.9)or other types of direct search (Sec.6.6)or indirect search(Sec.6.7)in weight space,or through training an NNfirst on shallow problems whose solutions may then generalize to deep problems,or through collapsing sequences of(non)linear operations into a single(non)linear operation—but see an analysis of non-trivial aspects of deep linear networks(Baldi and Hornik,1994,Section B).In general,however,finding an NN that precisely models a given training set is an NP-complete problem(Judd,1990;Blum and Rivest,1992),also in the case of deep NNs(S´ıma,1994;de Souto et al.,1999;Windisch,2005);compare a survey of negative results(S´ıma, 2002,Section1).Above we have focused on SL.In the more general case of RL in unknown environments,pcc(p,q) is also true if x p is an output event and x q any later input event—any action may affect the environment and thus any later perception.(In the real world,the environment may even influence non-input events computed on a physical hardware entangled with the entire universe,but this is ignored here.)It is possible to model and replace such unmodifiable environmental PCCs through a part of the NN that has already learned to predict(through some of its units)input events(including reward signals)from former input events and actions(Sec.6.1).Its weights are frozen,but can help to assign credit to other,still modifiable weights used to compute actions(Sec.6.1).This approach may lead to very deep CAPs though.Some DL research is about automatically rephrasing problems such that their depth is reduced(Sec.4). In particular,sometimes UL is used to make SL problems less deep,e.g.,Sec.5.10.Often Dynamic Programming(Sec.4.1)is used to facilitate certain traditional RL problems,e.g.,Sec.6.2.Sec.5focuses on CAPs for SL,Sec.6on the more complex case of RL.4Recurring Themes of Deep Learning4.1Dynamic Programming(DP)for DLOne recurring theme of DL is Dynamic Programming(DP)(Bellman,1957),which can help to facili-tate credit assignment under certain assumptions.For example,in SL NNs,backpropagation itself can 1An alternative would be to count only modifiable links when measuring depth.In many typical NN applications this would not make a difference,but in some it would,e.g.,Sec.6.1.be viewed as a DP-derived method(Sec.5.5).In traditional RL based on strong Markovian assumptions, DP-derived methods can help to greatly reduce problem depth(Sec.6.2).DP algorithms are also essen-tial for systems that combine concepts of NNs and graphical models,such as Hidden Markov Models (HMMs)(Stratonovich,1960;Baum and Petrie,1966)and Expectation Maximization(EM)(Dempster et al.,1977),e.g.,(Bottou,1991;Bengio,1991;Bourlard and Morgan,1994;Baldi and Chauvin,1996; Jordan and Sejnowski,2001;Bishop,2006;Poon and Domingos,2011;Dahl et al.,2012;Hinton et al., 2012a).4.2Unsupervised Learning(UL)Facilitating Supervised Learning(SL)and RL Another recurring theme is how UL can facilitate both SL(Sec.5)and RL(Sec.6).UL(Sec.5.6.4) is normally used to encode raw incoming data such as video or speech streams in a form that is more convenient for subsequent goal-directed learning.In particular,codes that describe the original data in a less redundant or more compact way can be fed into SL(Sec.5.10,5.15)or RL machines(Sec.6.4),whose search spaces may thus become smaller(and whose CAPs shallower)than those necessary for dealing with the raw data.UL is closely connected to the topics of regularization and compression(Sec.4.3,5.6.3). 4.3Occam’s Razor:Compression and Minimum Description Length(MDL) Occam’s razor favors simple solutions over complex ones.Given some programming language,the prin-ciple of Minimum Description Length(MDL)can be used to measure the complexity of a solution candi-date by the length of the shortest program that computes it(e.g.,Solomonoff,1964;Kolmogorov,1965b; Chaitin,1966;Wallace and Boulton,1968;Levin,1973a;Rissanen,1986;Blumer et al.,1987;Li and Vit´a nyi,1997;Gr¨u nwald et al.,2005).Some methods explicitly take into account program runtime(Al-lender,1992;Watanabe,1992;Schmidhuber,2002,1995);many consider only programs with constant runtime,written in non-universal programming languages(e.g.,Rissanen,1986;Hinton and van Camp, 1993).In the NN case,the MDL principle suggests that low NN weight complexity corresponds to high NN probability in the Bayesian view(e.g.,MacKay,1992;Buntine and Weigend,1991;De Freitas,2003), and to high generalization performance(e.g.,Baum and Haussler,1989),without overfitting the training data.Many methods have been proposed for regularizing NNs,that is,searching for solution-computing, low-complexity SL NNs(Sec.5.6.3)and RL NNs(Sec.6.7).This is closely related to certain UL methods (Sec.4.2,5.6.4).4.4Learning Hierarchical Representations Through Deep SL,UL,RLMany methods of Good Old-Fashioned Artificial Intelligence(GOFAI)(Nilsson,1980)as well as more recent approaches to AI(Russell et al.,1995)and Machine Learning(Mitchell,1997)learn hierarchies of more and more abstract data representations.For example,certain methods of syntactic pattern recog-nition(Fu,1977)such as grammar induction discover hierarchies of formal rules to model observations. The partially(un)supervised Automated Mathematician/EURISKO(Lenat,1983;Lenat and Brown,1984) continually learns concepts by combining previously learnt concepts.Such hierarchical representation learning(Ring,1994;Bengio et al.,2013;Deng and Yu,2014)is also a recurring theme of DL NNs for SL (Sec.5),UL-aided SL(Sec.5.7,5.10,5.15),and hierarchical RL(Sec.6.5).Often,abstract hierarchical representations are natural by-products of data compression(Sec.4.3),e.g.,Sec.5.10.4.5Fast Graphics Processing Units(GPUs)for DL in NNsWhile the previous millennium saw several attempts at creating fast NN-specific hardware(e.g.,Jackel et al.,1990;Faggin,1992;Ramacher et al.,1993;Widrow et al.,1994;Heemskerk,1995;Korkin et al., 1997;Urlbe,1999),and at exploiting standard hardware(e.g.,Anguita et al.,1994;Muller et al.,1995; Anguita and Gomes,1996),the new millennium brought a DL breakthrough in form of cheap,multi-processor graphics cards or GPUs.GPUs are widely used for video games,a huge and competitive market that has driven down hardware prices.GPUs excel at fast matrix and vector multiplications required not only for convincing virtual realities but also for NN training,where they can speed up learning by a factorof50and more.Some of the GPU-based FNN implementations(Sec.5.16-5.19)have greatly contributed to recent successes in contests for pattern recognition(Sec.5.19-5.22),image segmentation(Sec.5.21), and object detection(Sec.5.21-5.22).5Supervised NNs,Some Helped by Unsupervised NNsThe main focus of current practical applications is on Supervised Learning(SL),which has dominated re-cent pattern recognition contests(Sec.5.17-5.22).Several methods,however,use additional Unsupervised Learning(UL)to facilitate SL(Sec.5.7,5.10,5.15).It does make sense to treat SL and UL in the same section:often gradient-based methods,such as BP(Sec.5.5.1),are used to optimize objective functions of both UL and SL,and the boundary between SL and UL may blur,for example,when it comes to time series prediction and sequence classification,e.g.,Sec.5.10,5.12.A historical timeline format will help to arrange subsections on important inspirations and techni-cal contributions(although such a subsection may span a time interval of many years).Sec.5.1briefly mentions early,shallow NN models since the1940s,Sec.5.2additional early neurobiological inspiration relevant for modern Deep Learning(DL).Sec.5.3is about GMDH networks(since1965),perhaps thefirst (feedforward)DL systems.Sec.5.4is about the relatively deep Neocognitron NN(1979)which is similar to certain modern deep FNN architectures,as it combines convolutional NNs(CNNs),weight pattern repli-cation,and winner-take-all(WTA)mechanisms.Sec.5.5uses the notation of Sec.2to compactly describe a central algorithm of DL,namely,backpropagation(BP)for supervised weight-sharing FNNs and RNNs. It also summarizes the history of BP1960-1981and beyond.Sec.5.6describes problems encountered in the late1980s with BP for deep NNs,and mentions several ideas from the previous millennium to overcome them.Sec.5.7discusses afirst hierarchical stack of coupled UL-based Autoencoders(AEs)—this concept resurfaced in the new millennium(Sec.5.15).Sec.5.8is about applying BP to CNNs,which is important for today’s DL applications.Sec.5.9explains BP’s Fundamental DL Problem(of vanishing/exploding gradients)discovered in1991.Sec.5.10explains how a deep RNN stack of1991(the History Compressor) pre-trained by UL helped to solve previously unlearnable DL benchmarks requiring Credit Assignment Paths(CAPs,Sec.3)of depth1000and more.Sec.5.11discusses a particular WTA method called Max-Pooling(MP)important in today’s DL FNNs.Sec.5.12mentions afirst important contest won by SL NNs in1994.Sec.5.13describes a purely supervised DL RNN(Long Short-Term Memory,LSTM)for problems of depth1000and more.Sec.5.14mentions an early contest of2003won by an ensemble of shallow NNs, as well as good pattern recognition results with CNNs and LSTM RNNs(2003).Sec.5.15is mostly about Deep Belief Networks(DBNs,2006)and related stacks of Autoencoders(AEs,Sec.5.7)pre-trained by UL to facilitate BP-based SL.Sec.5.16mentions thefirst BP-trained MPCNNs(2007)and GPU-CNNs(2006). Sec.5.17-5.22focus on official competitions with secret test sets won by(mostly purely supervised)DL NNs since2009,in sequence recognition,image classification,image segmentation,and object detection. Many RNN results depended on LSTM(Sec.5.13);many FNN results depended on GPU-based FNN code developed since2004(Sec.5.16,5.17,5.18,5.19),in particular,GPU-MPCNNs(Sec.5.19).5.11940s and EarlierNN research started in the1940s(e.g.,McCulloch and Pitts,1943;Hebb,1949);compare also later work on learning NNs(Rosenblatt,1958,1962;Widrow and Hoff,1962;Grossberg,1969;Kohonen,1972; von der Malsburg,1973;Narendra and Thathatchar,1974;Willshaw and von der Malsburg,1976;Palm, 1980;Hopfield,1982).In a sense NNs have been around even longer,since early supervised NNs were essentially variants of linear regression methods going back at least to the early1800s(e.g.,Legendre, 1805;Gauss,1809,1821).Early NNs had a maximal CAP depth of1(Sec.3).5.2Around1960:More Neurobiological Inspiration for DLSimple cells and complex cells were found in the cat’s visual cortex(e.g.,Hubel and Wiesel,1962;Wiesel and Hubel,1959).These cellsfire in response to certain properties of visual sensory inputs,such as theorientation of plex cells exhibit more spatial invariance than simple cells.This inspired later deep NN architectures(Sec.5.4)used in certain modern award-winning Deep Learners(Sec.5.19-5.22).5.31965:Deep Networks Based on the Group Method of Data Handling(GMDH) Networks trained by the Group Method of Data Handling(GMDH)(Ivakhnenko and Lapa,1965; Ivakhnenko et al.,1967;Ivakhnenko,1968,1971)were perhaps thefirst DL systems of the Feedforward Multilayer Perceptron type.The units of GMDH nets may have polynomial activation functions imple-menting Kolmogorov-Gabor polynomials(more general than traditional NN activation functions).Given a training set,layers are incrementally grown and trained by regression analysis,then pruned with the help of a separate validation set(using today’s terminology),where Decision Regularisation is used to weed out superfluous units.The numbers of layers and units per layer can be learned in problem-dependent fashion. This is a good example of hierarchical representation learning(Sec.4.4).There have been numerous ap-plications of GMDH-style networks,e.g.(Ikeda et al.,1976;Farlow,1984;Madala and Ivakhnenko,1994; Ivakhnenko,1995;Kondo,1998;Kord´ık et al.,2003;Witczak et al.,2006;Kondo and Ueno,2008).5.41979:Convolution+Weight Replication+Winner-Take-All(WTA)Apart from deep GMDH networks(Sec.5.3),the Neocognitron(Fukushima,1979,1980,2013a)was per-haps thefirst artificial NN that deserved the attribute deep,and thefirst to incorporate the neurophysiolog-ical insights of Sec.5.2.It introduced convolutional NNs(today often called CNNs or convnets),where the(typically rectangular)receptivefield of a convolutional unit with given weight vector is shifted step by step across a2-dimensional array of input values,such as the pixels of an image.The resulting2D array of subsequent activation events of this unit can then provide inputs to higher-level units,and so on.Due to massive weight replication(Sec.2),relatively few parameters may be necessary to describe the behavior of such a convolutional layer.Competition layers have WTA subsets whose maximally active units are the only ones to adopt non-zero activation values.They essentially“down-sample”the competition layer’s input.This helps to create units whose responses are insensitive to small image shifts(compare Sec.5.2).The Neocognitron is very similar to the architecture of modern,contest-winning,purely super-vised,feedforward,gradient-based Deep Learners with alternating convolutional and competition lay-ers(e.g.,Sec.5.19-5.22).Fukushima,however,did not set the weights by supervised backpropagation (Sec.5.5,5.8),but by local un supervised learning rules(e.g.,Fukushima,2013b),or by pre-wiring.In that sense he did not care for the DL problem(Sec.5.9),although his architecture was comparatively deep indeed.He also used Spatial Averaging(Fukushima,1980,2011)instead of Max-Pooling(MP,Sec.5.11), currently a particularly convenient and popular WTA mechanism.Today’s CNN-based DL machines profita lot from later CNN work(e.g.,LeCun et al.,1989;Ranzato et al.,2007)(Sec.5.8,5.16,5.19).5.51960-1981and Beyond:Development of Backpropagation(BP)for NNsThe minimisation of errors through gradient descent(Hadamard,1908)in the parameter space of com-plex,nonlinear,differentiable,multi-stage,NN-related systems has been discussed at least since the early 1960s(e.g.,Kelley,1960;Bryson,1961;Bryson and Denham,1961;Pontryagin et al.,1961;Dreyfus,1962; Wilkinson,1965;Amari,1967;Bryson and Ho,1969;Director and Rohrer,1969;Griewank,2012),ini-tially within the framework of Euler-LaGrange equations in the Calculus of Variations(e.g.,Euler,1744). Steepest descent in such systems can be performed(Bryson,1961;Kelley,1960;Bryson and Ho,1969)by iterating the ancient chain rule(Leibniz,1676;L’Hˆo pital,1696)in Dynamic Programming(DP)style(Bell-man,1957).A simplified derivation of the method uses the chain rule only(Dreyfus,1962).The methods of the1960s were already efficient in the DP sense.However,they backpropagated derivative information through standard Jacobian matrix calculations from one“layer”to the previous one, explicitly addressing neither direct links across several layers nor potential additional efficiency gains due to network sparsity(but perhaps such enhancements seemed obvious to the authors).。

第17卷第4期计算机集成制造系统Vol.17No.42011年4月Computer Integrated Manufacturing SystemsApr.2011文章编号:1006-5911(2011)04-0826-06收稿日期:2010-03-16;修订日期:2010-08-24。

Received 16Mar.2010;accepted 24Aug.2010.基金项目:国家自然科学基金资助项目(50705076,50705077);国家863计划资助项目(2007AA04Z187);陕西省自然科学基础研究计划资助项目(2009JQ9002)。

Fo undatio n items:Project supported by th e Nation al Natural Science Foun dation,China (No.50705076,50705077),the Nation al H igh -Tech.R&D Program,Ch ina(No.2007AA04Z187),and the Natu ral Science Basic Research Program in Sh aanxi Province,Chin a(No.2009JQ9002).大规模作业车间的瓶颈分解调度算法翟颖妮,孙树栋,王军强,郭世慧(西北工业大学系统集成与工程管理研究所现代设计与集成制造技术教育部重点实验室,陕西 西安 710072)摘 要:针对大规模作业车间生产调度问题,提出一种基于瓶颈工序分解的调度算法。

该算法采用正交试验进行瓶颈设备的识别,在设备层分解的基础上进一步进行工序级的分解,将大规模调度问题分解为瓶颈工序集调度、上游非瓶颈工序集调度和下游非瓶颈工序集调度三个子问题,通过子问题的求解和协调获得原问题的解。

该算法遵循约束理论中/瓶颈机主导非瓶颈机0的原则,抓住调度问题的关键因素,采用分而治之的调度策略,不仅较大程度地降低了原问题的计算规模和复杂度,还兼顾了求解的质量。

ity to air pollution among different population subgroups. (Am. J. Respir. Crit. Care Med. .197,155, ,68-76) Dioxin trends in fish Concentrations of PCBs, polychlori-nated dibenzo-p-dioxins, and poly-chlorinated dibenzofurans are often gauged in terms of toxic equivalence factors. S.Y. Huestis and co-workers reported temporal (1977-93) and age-related trends in concentration and toxic equivalencies of these compounds in Lake Ontario lake trout. Analysis of the stored frozen fish tissue used a single analysis pro-tocol, which allowed improved com-parison of data from different time periods. Results showed that contami-nant levels have declined substantially since 1977 but concentration levels have stabilized approaching a steady state or a very slow decline The pro-portion of the total toxic equivalency ascribed to each compound has changed little in each of the three sets examined (Environ Toxicol Chem 1997 16(2) 154-64) Herbicide extractionEfficient extraction and analysis of phenoxyacid herbicides are difficult because of their high polarity and low volatility. T. S. Reighard and S. V Olesik reported the use of methanol/ C02 mixtures at elevated tempera-tures and pressures to extract these herbicides from household dust. The experiments were done at conditions covering both subcritical and super-critical regimes. They found that the highest recoveries (between 83% and 95% for the four herbicides studied) were observed at 20 mol % methanol and at temperatures of 100 °C or 150 °C. In addition, when a 200-uLvolume of hexane was added to the1-g dust sample, a preextraction withC02 and no methanol removed muchof the extraneous matrix. These ma-trix compounds, when present, cre-ate a more complex chromatogramand require more reagent. (Anal.Chem. 1997, 69, 566-74)Overcoming NIMBYPublic participation programs canhelp citizens get past "not-in-my-backyard" (NIMBY) responses to thesiting of hazardous waste facilities.J. J. Duffield and S. E Depoe de-scribed the effects of citizen partici-pation in the storage of 2.4 millioncubic yards of low-level radioactivewaste from the Fernald, Ohio, nu-clear weapons complex. Among theparticipants were labor representa-tives, academicians, 8X63. residents,and activists. Because the task forcehad the opportunity to questiontechnical experts and dispute evi-dence a democratic formatated for two-way communicationbetween officials and citizens (RiskPol Rev 1997 3(2) 31-34)RISKProbabilistic techniquesEfforts to improve risk characteriza-tion emphasize the use of uncer-tainty analyses and probabilistictechniques. K. M. Thompson andJ. D. Graham describe how MonteCarlo analysis and other probabilis-tic techniques can be used to im-prove risk assessments. A probabilis-tic approach to risk assessmentincorporates uncertainty and vari-ability, calculating risk with variablesthat include resources expended andpolicy mandates. Despite these ad-vantages, there are significant barri-ers to its widespread use, includingrisk managers' inexperience withprobabilistic risk assessment resultsand general suspicion of themethod. The authors describe waysto promote the proper use of proba-bilistic risk assessment. (Hum. Ecol.Risk Assess. 1996,2(4), 1008-34)Uncertainty modelingMonte Carlo modeling is a powerfulmathematical tool with many advan-tages over traditional point estimatesfor assessing uncertainty and vari-ability with hazardous waste site ex-posure and risk. However, a lack ofvariability information hindersMonte Carlo modeling. As a solu-tion, W. J. Brattin and colleaguesproposed running repeated MonteCarlo simulations using differentcombinations of uncertainty param-eters. The amount of variationamong the different simulationsshows how certain or uncertain anyindividual estimate of exposure orrisk may be An example of thisproach is provided including an es-timation of the average exposure toradon daughter products in indoorair (Hum Ecol Risk Assess .1992(4) 820-40)SOILDecomposition modelClay has a stabilizing effect on or-ganic matter in soil and thus reducesthe rate of decomposition. Currentcomputer simulation models, how-ever, do not adequately describe theprotection of organic matter by clay.J. Hassink and A. P. Whitmore devel-oped and tested a model that pre-dicts the preservation of added or-ganic matter as a function of theclay fraction and the degree of or~ganic matter saturation of the soil. Itclosely followed the buildup and de-cline of organic matter in 10 soils towhich organic matter was addedBetter than conventional modelsthis model was able to predict theaccumulation and degradation oforganic matter in soils of differenttextures and the contents of initialorganic matter (Soil Sci Soc Am J1997 61 131-39)Tracing trace metal complexationChemical reactions determine the fate of trace metals released into aquatic envi-ronments. J. M. Gamier and colleagues investigated the kinetics of trace metal complexation by monitoring chemical reactions in suspended matter from a French river. Radiotracer experiments on Mn, Co, Fe, Cd, Zn, Ag, and Cs identified sea-sonal variations in the predominant species. In summer, Cd and Zn were com-plexed by specific natural organic matter ligands. Cs remained in an inorganicform, whereas Fe and Ag were either organic complexes or colloidal species. In winter, a two-step process occurred for Mn and Co. They were rapidly complexed by weak ligands, followed by slower complexation by stronger ligands. The authors conclude that low concentrations of natural ligands may control the speciation of trace elements. (Environ. Sci. Techno!..,his issue, pp. .597-1606)2 60 A • VOL. 31, NO. 6, 1997 / ENVIRONMENTAL SCIENCE & TECHNOLOGY / NEWS。

重要哲学术语英汉对照——转载自《当代英美哲学概论》a priori瞐 posteriori distinction 先验-后验的区分abstract ideas 抽象理念abstract objects 抽象客体ad hominem argument 谬误论证alienation/estrangement 异化，疏离altruism 利他主义analysis 分析analytic瞫ynthetic distinction 分析-综合的区分aporia 困惑argument from design 来自设计的论证artificial intelligence (AI) 人工智能association of ideas 理念的联想autonomy 自律axioms 公理Categorical Imperative 绝对命令categories 范畴Category mistake 范畴错误causal theory of reference 指称的因果论causation 因果关系certainty 确定性chaos theory 混沌理论class 总纲、类clearness and distinctness 清楚与明晰cogito ergo sum 我思故我在concept 概念consciousness 意识consent 同意consequentialism 效果论conservative 保守的consistency 一致性，相容性constructivism 建构主义contents of consciousness 意识的内容contingent瞡ecessary distinction 偶然-必然的区分 continuum 连续体continuum hypothesis 连续性假说contradiction 矛盾（律）conventionalism 约定论counterfactual conditional 反事实的条件句criterion 准则，标准critique 批判，批评Dasein 此在，定在deconstruction 解构主义defeasible 可以废除的definite description 限定摹状词deontology 义务论dialectic 辩证法didactic 说教的dualism 二元论egoism 自我主义、利己主义eliminative materialism 消除性的唯物主义 empiricism 经验主义Enlightenment 启蒙运动（思想）entailment 蕴含essence 本质ethical intuition 伦理直观ethical naturalism 伦理的自然主义eudaimonia 幸福主义event 事件、事变evolutionary epistemology 进化认识论expert system 专门体系explanation 解释fallibilism 谬误论family resemblance 家族相似fictional entities 虚构的实体first philosophy 第一哲学form of life 生活形式formal 形式的foundationalism 基础主义free will and determinism 自由意志和决定论 function 函项（功能）function explanation 功能解释good 善happiness 幸福hedonism 享乐主义hermeneutics 解释学（诠释学，释义学）historicism 历史论（历史主义）holism 整体论iconographic 绘画idealism 理念论ideas 理念identity 同一性illocutionary act 以言行事的行为imagination 想象力immaterical substance 非物质实体immutable 不变的、永恒的individualism 个人主义（个体主义）induction 归纳inference 推断infinite regress 无限回归intensionality 内涵性intentionality 意向性irreducible 不可还原的Leibniz餾 Law 莱布尼茨法则logical atomism 逻辑原子主义logical positivism 逻辑实证主义logomachy 玩弄词藻的争论material biconditional 物质的双向制约materialism 唯物论（唯物主义）maxim 箴言，格言method 方法methodologica 方法论的model 样式modern 现代的modus ponens and modus tollens 肯定前件和否定后件 natural selection 自然选择necessary 必然的neutral monism 中立一无论nominalism 唯名论non睧uclidean geometry 非欧几里德几何non瞞onotonic logics 非单一逻辑Ockham餜azor 奥卡姆剃刀omnipotence and omniscience 全能和全知ontology 本体论（存有学）operator 算符(或算子)paradox 悖论perception 知觉phenomenology 现象学picture theory of meaning 意义的图像说pluralism 多元论polis 城邦possible world 可能世界postmodernism 后现代主义prescriptive statement 规定性陈述presupposition 预设primary and secondary qualities 第一性的质和第二性的质 principle of non瞔ontradiction 不矛盾律proposition 命题quantifier 量词quantum mechanics 量子力学rational numbers 有理数real number 实数realism 实在论reason 理性，理智recursive function 循环函数reflective equilibrium 反思的均衡relativity (theory of) 相对（论）rights 权利rigid designator严格的指称词Rorschach test 相对性（相对论）rule 规则rule utilitarianism 功利主义规则Russell餾 paradox 罗素悖论sanctions 制发scope 范围，限界semantics 语义学sense data 感觉材料，感觉资料set 集solipsism 唯我论social contract 社会契约subjective瞣bjective distinction 主客区分 sublation 扬弃substance 实体，本体sui generis 特殊的，独特性supervenience 偶然性syllogism 三段论things瞚n瞭hemselves 物自体thought 思想thought experiment 思想实验three瞯alued logic 三值逻辑transcendental 先验的truth 真理truth function 真值函项understanding 理解universals 共相，一般，普遍verfication principle 证实原则versimilitude 逼真性vicious regress 恶性回归Vienna Circle 维也纳学派virtue 美德注释计量经济学中英对照词汇(continuous)2007年8月23日，22:02:47 | mindreader计量经济学中英对照词汇(continuous)K-Means Cluster逐步聚类分析K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least Squared Criterion，最小二乘方准则Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Leveage Correction，杠杆率校正Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形MSC（多元散射校正）Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal P-P, 正态概率分布图Normal Q-Q, 正态概率单位分布图Normal ranges, 正常范围Normal value, 正常值Normalization 归一化Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Pareto, 直条构成线图（又称佩尔托图）Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式PCA（主成分分析）Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 构成图,饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡PLS（偏最小二乘法）Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal axis factoring,主轴因子法Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Pro, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和residual variance (剩余方差)Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system , SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequence, 普通序列图Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significant Level, 显著水平Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Test(检验)Test of linearity, 线性检验Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Unweighted least squares, 未加权最小平方法Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换注释。

托福阅读物理学学科词汇梳理考生假如对所考文章的学科词汇一窍不通，文章读起来必定是云里雾里难以连续，因此适当把握一些学科词汇会使我们更有效备考。

下面我就和大家共享托福阅读物理学学科词汇梳理,盼望能够关心到大家，来观赏一下吧。

托福阅读物理学学科词汇梳理托福阅读核心词汇(物理学专业词汇)1. physics 物理39. electric voltage 电压2. mechanics 力学40. electric shock 触电3. thermodynamics 热力学41. electric appliance 电器4.acoustics 声学42. conductor 导体5. electromagnetism 电磁学43. insulator 绝缘体6. optics 光学44. semiconductor 半导体7. dynamics 动力学45. battery (cell) 电池8. force 力46. dry battery 干电池9. velocity 速度47. storage battery 蓄电池10. acceleration 加速度48. electronics 电子学11. equilibrium 平衡49. electronic 电子的12. statics 静力学50. electronic component (part) 电子零件13. motion 运动51. integrated circuit 集成电路14. inertia 惯性52. chip 集成电器片，集成块15. gravitation 引力53. electron tube 电子管16. relativity 相对54. vacuum tube 真空管17. gravity 地心引力55. transistor 晶体管18. vibration 震惊56. amplification ( 名词) 放大19. medium (media) 媒质57. amplify （动词）放大20. frequency 频率58. amplifier 放大器，扬声器21. wavelength 波长59. oscillation 震荡22. pitch 音高60. optical 光（学）的23. intensity 强度61. optical fiber 光学纤维24. echo 回声62. lens 透镜，镜片25. resonance 回声，嘹亮63. convex 凸透镜26. sonar 声纳64. concave 凹透镜27. ultrasonics 超声学65. microscope 显微镜28. electricity 电66. telescope 望远镜29. static electricity 静电67. magnifier 放大镜30. magnetism 磁性，磁力68. spectrum 光谱31. magnet 磁体69. ultraviolet 紫外线32. electromagnet 电磁70. X rays X 射线33. magnetic field 磁场71. Gamma rays γ 射线34. electric current 电流72. infrared rays 红外线35. direct current (DC) 直流电73. microwaves 微波36.alternating current (AC) 沟通电74. dispersion 色散37. electric circuit 电路75. transparent 透亮38. electric charge 电荷76. translucent 半透亮托福阅读长难句：水循环圈Perhapsthe fact many of these first studies considered only algae of a size that couldbe collected in a net, a practice that overlooked the smaller phytoplanktonthat we now know grazers are most likely to feed on, led to a de-emphasis ofthe role of grazers in subsequent research.这个句子谓语动词的识别稍有难度。