A homotopy theory for enrichment in simplicial modules

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ANSYS错误提示及其含义

1 在Ansys中出现“Shape testing revealed that 450 of the 1500 new or modified elements violate shape warning limits.”，是什么原因造成的呢？单元网格质量不够好，尽量用规则化网格，或者再较为细密一点。

2 在Ansys中，用Area Fillet对两空间曲面进行倒角时出现以下错误：Area 6 offset could not fully converge to offset distance 10. Maximum error between the two surfaces is 1% of offset distance.请问这是什么错误？怎么解决？其中一个是圆柱接管表面，一个是碟形封头表面。

ansys的布尔操作能力比较弱。

如果一定要在ansys里面做的话，那么你试试看先对线进行倒角，然后由倒角后的线形成倒角的面。

建议最好用UG、PRO/E这类软件生成实体模型然后导入到ansys。

3 在Ansys中，出现错误“There are 21 small equation solver pivot terms。

”，是否是在建立接触contact时出现的错误？不是建立接触对的错误，一般是单元形状质量太差（例如有接近零度的锐角或者接近180度的钝角）造成small equation solver pivot terms4 在Ansys中，出现警告“SOLID45 wedges are recommended only in regions of relatively low stress gradients.”，是什么意思？"这只是一个警告，它告诉你：推荐SOLID45单元只用在应力梯度较低的区域。

它只是告诉你注意这个问题，如果应力梯度较高，则可能计算结果不可信。

"5 ansys向adams导的过程中，出现如下问题“There is not enough memory for the Sparse Matrix Solver to proceed.Please shut down other applications that may be running or increase the virtual memory on your system and return ANSYS.Memory currently allocated for the Sparse Matrix Solver=50MB.Memory currently required for the Sparse Matrix Solver to continue=25MB”，是什么原因造成的？不清楚你ansys导入adams过程中怎么还需要使用Sparse Matrix Solver（稀疏矩阵求解器）。

FLUENT软件操作界面中英文对照

FLUENT 软件操作界面中英文对照File 文件Read 读取文件：scheme 方案journal 日志 profile 外形 Write 保存文件 Import :进入另一个运算程序 Interpolate :窜改，插入Hardcopy :复制，Batch opti ons —组选项Save layout 保存设计 Grid 网格| F 屆 Grid Define Solve i-Read► Write►Import ► Export-, Interpolate —Hardcopy...Batch Options.^Save LayoutRun...RSF...ExitDefine Models模型：solver解算器Pressure based 基于压力Den sity based基于密度末解用丁 pressure based,雀改用Censity based 岀观不苻合秦相36的摄不，请甸pres 羊WE base d Al density based 慎仆则迢.41刊卜 IS 况？北外.血OC ・EHU ^ cotipled ＞（/^ ptiaw ccupl ed simple 可这孔 & pimple ijEpiso 逞划.--■轟=Sortg 冲-丸布时制* ^jfe-5-6 1844DOden ba&ed 亘工F 吋.压聲血ptessurt ba$«i 适可于那可乐册斤"dens Ply based 把丸“河悴掏t ：至殳嗟之一.刑牛可爪門d 的创R 监當歎1一，般如人锻暮叫人丄有用轴怀，火薩墟迭牛才H5）-0 程：yje8D8-丸布门 1叫；20GS-3-7 10-2ODQ歆；I ：谢PrflSsdrfl-Bawd Soker ^Fluflnt 它星英于压力快的束解孤便丹的圮压力修止畀法"求释旳控制片悝足标联式的，擅K 家鮮不町压縮舐Mb 对于町压砂也可旦索麟；Fk«nl 6-3 tl 前的I 板』冷孵臥 B fjS4^r«^at«d &Olvar fli Cfruptod Solvtr,的实也fltje Pr#4iurft-ea5<KJ ScFvAi 约两种虽幵方ib应理拈Fluent 氐J 眇坝犢型小的.它拈垒于曲喪红旳求聲塞，最辑的出H .也桿艮先■那式的.丰誓■載密式冇Roz AU$hk>谏方法的初和&让Fhwrt 耳有比我甘的求IT 可压胃（说劫謹力，們耳帕榕式淮冇琴加IF 科限闾辉，倒比还丰A 完荐匚Coupted 的算送t 对子慨站何Hh 地们足便用円匕口讷油皿购加£未处為性上世魅端计棒低逢刈趾.擁说的DansJty-Bjiwd Solver F ft St SiMPLEC, P 怡DE 些选以的.闪为進些ffliQl ；力修止鼻袪・不金在这种类崖的我押■中氏现泊I 建址祢匹足整用Fw»ur 护P.sed Solver 堺决昧的利底*implicit 隐式， explicit 显示Space 空间：2D , axisymmetric （转动轴）， axisymmetric swirl （漩涡转动轴）；Time 时间 :steady 定常，unsteady 非定常 Velocity formulatio n 制定速度： absolute 绝对的；relative 相对的Gradient option 梯度选择：以单元作基础；以节点作基础；以单元作梯度的最小正方形。

W (P, t)=

ON THE CHARNEY-DAVIS AND NEGGERS-STANLEYCONJECTURESVICTOR REINER AND VOLKMAR WELKERAbstract.For a graded naturally labelled poset P,it is shownthat the P-Eulerian polynomialW(P,t):= w∈L(P)t des(w)counting linear extensions of P by their number of descents hassymmetric and unimodal coeﬃcient sequence,verifying the moti-vating consequence of the Neggers-Stanley conjecture on real zeroesfor W(P,t)in these cases.The result is deduced from McMullen’sg-Theorem,by exhibiting a simplicial polytopal sphere whose h-polynomial is W(P,t).Whenever this simplicial sphere turns out to beﬂag,that is,its minimal non-faces all have cardinality two,it is shown that theNeggers-Stanley Conjecture would imply the Charney-Davis Con-jecture for this sphere.In particular,it is shown that the sphereisﬂag whenever the poset P has width at most2.In this case,the sphere is shown to have a stronger geometric property(locallyconvexity),which then implies the Charney-Davis Conjecture inthis case via a result from[30].It is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras,and someevidence is presented.1.IntroductionThis paper has several goals.Theﬁrst is to show that,in the context of the Neggers-Stanley Conjecture1.2,for every graded poset P there is lurking in the background a polytopal simplicial sphere,which we will denote∆eq(P).This sphere is relevant for two purposes:2VICTOR REINER AND VOLKMAR WELKER⊲The P-Eulerian polynomial(deﬁned below)coincides with theh-polynomial of∆eq(P).As a consequence,its coeﬃcients sat-isfy McMullen’s conditions for the h-vector of a simplicial poly-tope,and are in particular symmetric and unimodal.Therebywe verify the motivating consequence of the Neggers-StanleyConjecture for naturally labeled graded posets(see discussionafter the statement of Conjecture1.2).⊲Whenever the simplicial sphere∆eq(P)isﬂag,the Neggers-Stanley Conjecture1.2for P implies the Charney-Davis Con-jecture for the sphere∆eq(P).Furthermore,when P has widthat most2,it is shown in Theorem3.23that∆eq(P)satisﬁes astronger geometric condition thanﬂag-ness known as local con-vexity,which implies the Charney-Davis Conjecture in this caseby a result from[30].The latter portion of the paper(Section4onward)is aimed toward the thesis that both the Charney-Davis and Neggers-Stanley Conjec-tures,along with some other combinatorial conjectures and results, should be considered in the context of the following question. Question1.1.For which Koszul algebras is the Hilbert function a Polya frequency sequence?To give a more precise discussion,we start by recalling the Neggers-Stanley Conjecture.For any partial order P on[n]:={1,2,...,n}, let L(P)denote its set of linear extensions,that is the set of w= (w1,...,w n)∈S n for which i<P j implies w−1(i)<w−1(j).The P-Eulerian polynomialW(P,t):= w∈L(P)t des(w)is the generating function for the linear extensions L(P)counted ac-cording to cardinality of their descent sets:Des(w):={i∈[n−1]:w i>w i+1}des(w):=#Des(w)Conjecture1.2(Neggers-Stanley).For any labelled poset P on[n] the polynomial W(P,t)has only real(non-positive)zeroes.We are mainly interested in the case where P is naturally labelled, that is i<P j implies i<j.Some history and context for the conjecture follows.For naturally labelled posets Conjecture1.2was made originally by Neggers[32], and generalized to the above statement by Stanley in1986.When P is an antichain of n elements,W(P,t)is the Eulerian polynomial whoseCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES3 real-rootedness was shown by Harper[24]and served as an initial mo-tivation for the conjecture.For the case when P is a naturally labelled disjoint union of chains the result is due to Simion[37].This result was extended to arbitrary labellings by Brenti[7],who also veriﬁed the conjecture for Ferrers posets and Gaussian posets[7].An impor-tant combinatorial implication of the real-rootedness of a polynomial with non-negative coeﬃcients is the unimodality of the coeﬃcients(i.e. for the sequence of coeﬃcients a0,...,a r there is an index j such that a0≤···≤a j≥···≥a r).Gasharov[19]veriﬁed the unimodality consequence of the conjecture for naturally labelled graded posets with at most3ranks.Corollary3.15veriﬁes this(and something stronger) more generally for all naturally labelled graded posets.Next,we recall the Charney-Davis Conjecture.Given an abstract simplicial complex∆triangulating a(d−1)-dimensional(homology) sphere,one can collate the face numbers f i,which count the number of i-dimensional faces,into its f-vector and f-polynomialf(∆):=(f−1,f0,f1,...,f d−1)f(∆,t):=d i=0f i−1t i.The h-polynomial and h-vector are easily seen to encode the same in-formation:(1.1)h(∆):=(h0,h1,...,h d)whereh(∆,t)=di=0h i t i satisﬁes t d h(∆,t−1)= t d f(∆,t−1) t→t−1.The h-polynomial turns out to be a more convenient and natural encoding in several ways,closely related to commutative algebra,toric geometry,and shellability.For example,the fact that homology spheres are Cohen-Macaulay implies non-negativity of the h i,and the Dehn-Sommerville equations for simplicial spheres assert that h i=h d−i for 0≤i≤d(see[46,§II.6]).Note that the latter implies that the h-polynomial is symmetric,h(∆,t)=t d h(∆,t−1),and that h(∆,−1)=0 whenever d is odd.The Charney-Davis Conjecture[11,Conjecture D]concerns the quan-tity h(∆,−1)in the case where d is even and∆is a simplicial homology (d−1)-sphere which happens to be aﬂag complex,that is the minimal subsets of vertices which do not span a simplex all have cardinality4VICTOR REINER AND VOLKMAR WELKERtwo.For polytopal simplicial spheres ∆,this quantity is known [30]to coincide with the signature or index of the associated toric variety X ∆.Conjecture 1.3(Charney-Davis,Conjecture D [11]).When ∆is a ﬂag simplicial homology (d −1)-sphere and d is even,then(−1)d2h (−1)≥0.Proof.Since h (t )has degree d we have h d =0and by symmetry h 0=0.Thus h (t )has d zeroes which must then all be strictly negative since h i ≥0for 0≤i ≤d .Factor h (t )=h d di =1(t −r i )according to its(real)zeroes r i .Symmetry of h (t )implies that r is a zero if and only if 1r is less than −1.Thus for a zero r ,either r =−1is a zero,in which case h (−1)=0and we are done,or else exactly half of the factors in the product h (−1)=h d d i =1(−1−r i )are negative,implying that the product has sign (−1)d 2W (P,−1),for some cases of posets where the Neggers-StanleyConjecture is known,are explored in [36].In Section 3.2it is shown that the sphere ∆eq (P )is the boundary complex of a simplicial convex polytope.Therefore by McMullen’s g -Theorem characterizing the number of faces of such polytopes [40],the coeﬃcients (h 0,h 1,...,h #P −r )are symmetric and unimodal.Convexity has further relevance.In [30]it was shown via the Hirze-bruch signature formula that the Charney-Davis Conjecture holds forCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES5 a simplicial polytope under a certain geometric hypothesis(local con-vexity)stronger than beingﬂag.We show in Section3.2that this hypothesis holds for∆eq(P)whenever P has width(i.e.size of the largest antichain)at most2,thereby providing more evidence for the Neggers-Stanley Conjecture.In Sections4and5we gather evidence for the thesis that both of these conjectures can be fruitfully viewed within the context of Koszul algebras.In particular,we point out ways in which Hilbert series of Koszul algebras interact well with the theory of Polya frequency series and polynomials with real zeroes.After this paper was circulated,C.Athanasiadis[1]has shown that the unimodular triangulation of the order polytope from Section3.1 is a member of a class of triangulations of polytopes that decompose into a join of a simplex and a polytopal sphere.Most notably he has exhibited such a triangulation for the Birkhoﬀpolytope.2.Review:P-Partitions and Order PolytopesIn this section we review some of the theory of P-partitions,distribu-tive lattices and order polytopes;see[25,27,26,39,41]for proofs and more details.Also see[18,§1.2]for deﬁnitions and basic facts about polyhedral cones and fans.Given a naturally labelled poset P on[n]ordered by≤P,the vector space of functions f=(f(1),...,f(n)):P→R will be identiﬁed with R n.One says that f is a P-partition if f(i)≥0for all i and f(i)≥f(j) for all i<P j.Denote by A(P)the cone of all P-partitions in R n.The convex polytopeO(P)=A(P)∩[0,1]nis called the order polytope of P.An order ideal I in P is a subset of P such that i∈I and j<P i implies j∈I.It is known that O(P)is the convex hull of the characteristic vectorsχI∈{0,1}n as I runs through all order ideals I in P.A useful alternative way to view O(P)is provided by the fact that it is isometric to the hyperplane slice at x0=1of the cone A(P0)⊂R n+1,where P0is the naturally labelled poset on[0,n]:={0,1,...,n} obtained from P by adjoining a new minimum element0.We call the cone A(P0)the homogenization of the cone A(P).We recall a few basic deﬁnitions some of which were already men-tioned in the introduction.The set of permutations w=(w1,...,w n)∈S n which extend P to a linear order is called its Jordan-H¨o lder set L(P):= w=(w1,...,w n)∈S n:i<P j implies w−1(i)<w−1(j) .6VICTOR REINER AND VOLKMAR WELKERThe descent set and descent number of w are deﬁned byDes(w):={i∈[n−1]:w i>w i+1}des(w):=#Des(w).Deﬁne a cone for each w∈S nA(w):={f∈R n:f(w i)≥f(w i+1)for i∈[n−1],f(w i)>f(w i+1)if i∈Des(w)}It is not hard to see that the closure of A(w)(deﬁned by removing the strict inequalities above),is a unimodular(simplicial)cone,that is its extreme rays are spanned by a set of vectors forming a lattice basis for Z n.Similarly,the closure of A(w)∩[0,1]n is a unimodular simplex. Now we are in position to formulate the basic fact from the theory of P-partitions which will be crucial for subsequent arguments. Proposition2.1.(i)The cone of P-partitions decomposes into a disjoint union asfollows:A(P)=⊔w∈L(P)A(w)The closures of the cones A(w)for w∈L(P)give a unimodulartriangulation of A(P).(ii)The unimodular triangulation of A(P)described in(i)restricts to a unimodular triangulation of the order polytopeO(P)=⊔w∈L(P)A(w)∩[0,1]n.We call the triangulations of A(P)(into simplicial cones)and O(P) (into simplices)from Proposition2.1their canonical triangulations. Note that via homogenization the canonical triangulation of O(P)is easily seen to be the restriction of the canonical triangulation of the homogenized cone A(P0)to the hyperplane x0=1.This makes sense since there is an obvious bijection between the linear extensions L(P0) and L(P).The combinatorics of these triangulations is closely related to the distributive lattice J(P)of all order ideals I in P ordered by inclusion. The order complex∆J(P)is the abstract simplicial complex having a vertex for each ideal I in P and a simplex for each chain I1⊂...⊂I t of nested ideals.Given a set of vectors V⊂R n,deﬁne their positive span to be the(relatively open)conepos(V):= v∈V c v·v:c v∈R,c v>0 .CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES7Proposition2.2.(i)Every non-zero P-partition f∈A P can be uniquely expressedin the formf=t i=1c iχI iwhere the c i are positive reals,and I1⊂···⊂I t is a chain ofideals in P.In other words,A(P)= ideals I1⊂···⊂I t⊂P pos {χI t}t i=1 .(ii)The canonical triangulation of the order polytope O(P)is iso-morphic(as an abstract simplicial complex)to∆J(P),via anisomorphism sending an ideal I to its characteristic vectorχI.(iii)The lexicographic order of permutations in L(P)gives rise to a shelling order on∆J(P).(iv)In this shelling,for each w in L(P),the minimal face of its cor-responding simplex in∆J(P)which is not contained in a lexico-graphically earlier simplex is spanned by the ideals{w1,w2,...,w i}where i∈Des(w).Using basic facts about shellings(see[4]),part(iv)of the preceding proposition implies that one can re-interpret the polynomial W(P,t): (2.1)W(P,t):= w∈L(P)t des(w)=h(∆J(P),t)This connection with J(P)also allows one to re-interpret these re-sults in terms of Ehrhart polynomials.Recall that for a convex poly-tope Q in R n having vertices in Z n,the number of lattice points con-tained in an integer dilation dQ grows as a polynomial in the dilation factor d∈N.This polynomial in d is called the Ehrhart polynomial: Ehrhart(O(Q),d):=# d O(P)∩N n .Whenever Q has a unimodular triangulation abstractly isomorphic toa simplicial complex∆,there is the following relationship:(2.2) d≥0Ehrhart(O(Q),d)t d=h(∆,t)8VICTOR REINER AND VOLKMAR WELKERthe equatorial triangulation.This triangulation has several pleasant properties,proven in this and the next subsection,which may be sum-marized as follows:⊲It is a unimodular triangulation.(See Proposition3.6)⊲It is isomorphic,as an abstract simplicial complex,to the joinof an r-simplex with a simplicial(#P−r−1)-sphere,which wewill denote∆eq(P),and call the equatorial sphere.(See Corollary3.8)⊲h(∆eq(P),t)=h(∆J(P),t)=W(P,t).(See Corollary3.8)⊲The equatorial sphere∆eq(P)is polytopal,and hence shellableand a PL-sphere.(See Theorem3.14)⊲When P has width at most2,the equatorial sphere∆eq(P)is realized by a locally convex simplicial fan.Hence is aﬂagsubcomplex of∆J(P),and aﬂag sphere for which the Charney-Davis Conjecture holds.(See Theorem3.23)Example3.1.Let P be the graded naturally labelled poset on[4] with r=2ranks shown in Figure1(a).Let J(P)be its associated (distributive)lattice of order ideals(see Figure1(b)).The4-dimensional order polytope O(P),and its canonical triangula-tion by∆J(P),may be“visualized”as follows.Start with the convex pentagonπwhich is the convex hull of{χ1,χ2,χ12,χ13,χ123,χ124},and triangulateπas shown in Figure1(c).The canonical triangulation is obtained by taking the simplicial join of this triangulation ofπwith the edge{χ∅,χ1234}.The equatorial triangulation(see Proposition3.6)is obtained start-ing from the alternate triangulation ofπdepicted in Figure1(d)and taking the simplicial join with the edge{χ∅,χ1234}.Equivalently,it is obtained from the equatorial1-sphere∆eq(P)depicted in Figure1(e) and taking the simplicial join with the triangle{χ∅,χ12,χ1234}.Fix a naturally labelled poset P on[n],and assume that it is graded, with r rank sets P1,...,P r.The following are the key deﬁnitions.Deﬁnition3.2.A P-partition f will be called rank-constant if it is constant along ranks,i.e.f(p)=f(q)whenever p,q∈P j for some j.A P-partition f will be called equatorial if min p∈P f(p)=0and for every j∈[2,r]there exists a covering relation between ranks j−1,jCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES9Figure 1.(a)A graded poset P .(b)The distributivelattice of order ideals J (P ).(c)Part of the canonicaltriangulation ∆J (P )of its order polytope O (P ).(d)The analogous part of the equatorial triangulation.(e)The equatorial 1-sphere ∆eq (P ).in P along which f is constant,i.e.there exist p j −1<P p j withp j −1∈P j −1,p j ∈P j and f (p j −1)=f (p j ).An order ideal I in P will be called rank-constant (resp.equatorial )if its characteristic vector χI is rank-constant (resp.equatorial).More generally,a collection of ideals {I 1,...,I t }forming a chain I 1⊂...⊂I t will be called rank-constant (resp.equatorial )if the sum χI 1+...+χI t (or equivalently,any vector in the cone pos({χI j }t j =1)is rank-constant (resp.equatorial).Note that the only rank-constant ideals are the ones in the chain∅=I rc 0⊂I rc 1⊂···⊂I rc r =P10VICTOR REINER AND VOLKMAR WELKERwhere I rc j:=⊔i≤j P i.Also note that the only P-partition which is both rank-constant and equatorial is the zero P-partition f(p)=0.Thus the only rank-constant and equatorial order ideal is I rc0=∅. Proposition3.3.Every non-zero P-partition f can be uniquely ex-pressed asf=f rc+f eq,where f rc,f eq are rank-constant and equatorial P-partitions,respec-tively.Proof.To show existence,for j∈[r−1]deﬁne non-negative constantsc j:=min{f(p j−1)−f(p j):p j−1∈P j−1,p j∈P j,p j−1<P p j}c r:=min{f(p r):p r∈P r},and setf rc:=r j=1c jχI rc jf eq:=f−f rc.Obviously f rc is a rank-constant P-partition.It is a straightforward veriﬁcation,left to the reader,that f eq is a P-partition,and that it is equatorial by construction.For uniqueness,assume f=g rc+g eq is an additive decomposi-tion of f into a rank-constant and an equatorial P-partition.It is again straightforward to show that the equatoriality of g eq and rank-constancy of g rc forces g rc= r j=1c jχI rc j,where c j is deﬁned as above in terms of f.We wish to deduce our equatorial triangulation of A(P)from Propo-sition3.3,and for this we need to understand both rank-constant and equatorial chains of ideals better.Equatoriality and rank-constancy of a chain of ideals I1⊂...⊂I t are intimately related with properties of its jumpsJ i:=I i−I i−1for i=1,...,t+1(where by convention I0:=∅,I t+1=P).It is easy to see that the rank-constant P-partitions form an r-dimensional simplicial subcone within the n-dimensional cone A(P), and that this subcone is the non-negative span of the vectors{χI rcj}r j=1. Proposition3.4.The rank-constant subcone of A(P)is interior,that is,it does not lie in the boundary subcomplex of the cone A(P). Proof.In a triangulation of a polyhedral cone,a subcone lies on the boundary if and only if it is contained in a codimension one subcone that lies on the boundary.For codimension one subcones,lying in theCHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES11 boundary is equivalent to being contained in a unique top dimensional subcone.Specializing to the case of the canonical triangulation of the cone A(P)from Proposition2.1,one sees that this means a chain of ideals I1⊂···⊂I t corresponds to a subcone on the boundary if and only if one of at least one of its jumps J i contains a pair of elements which are comparable in P.But for I rc1⊂···⊂I rc r,since the jumps J i=I rc i−I rc i−1=P i are antichains,this property fails to hold. Proposition3.5.A chain of non-empty ideals I1⊂...⊂I t,is equa-torial if and only if its jumps J i have the following property:For every j∈[2,r],there exist p j−1<P p j with p j−1∈P j−1,p j∈P j and a value i∈[t+1],such that p j−1,p j∈J i.The chain I1⊂...⊂I t is maximal with respect to the equatorial property if and only if its jumps J i for i∈[t+1]satisfy the following two conditions:(i)The J i are all maximal(saturated)chains in P,possibly single-tons.(ii)The non-singleton J i can be re-ordered J i1,J i2,...,J isso thatmin Ji1has rank1,max Ji shas rank r,and max J ik,min J ik+1have the same rank in P for k∈[s−1].Consequently,t=n−r for any maximal equatorial chain of non-empty ideals.Proof.Since the jumps J i are the domains on which the associated P-partitionχI1+...+χItis constant,theﬁrst assertion is direct fromDeﬁnition3.2.It is then easy to see that a chain of non-empty ideals having proper-ties(i),(ii)will be equatorial,and maximal with respect to reﬁnement.Conversely,suppose one is given a maximal equatorial chain of non-empty ideals.If there exists an incomparable pair p,p′in one of its jumps J i,it is straightforward to check that one can reﬁne the chainfurther while preserving the equatorial property,e.g.by adding in theideal I i−1∪{q∈J i:q≤p}.Thus each jump J i must be a maximalchain,proving(i).Furthermore,the pairs of adjacent ranks{j−1,j} spanned by two diﬀerent jumps J i,J i′must be disjoint,else one could reﬁne the chain equatorially by“breaking”J i between two such ranks{j−1,j}which they share.The jumps J i must then disjointly coverall possible adjacent rank pairs{j−1,j}r j=2,so they can be re-ordered as in(ii). Proposition3.6.The collection of all conespos {χI:I∈R∪E} ,12VICTOR REINER AND VOLKMAR WELKERwhere R(resp.E)is a chain of non-empty rank-constant(resp.equa-torial)ideals in P,gives a unimodular triangulation of the cone of P-partitions A(P).Proof.First we check that these polytopal cones indeed decompose A(P).Given f∈A,write f=f rc+f eq as in Proposition3.3.Then use these easy facts:⊲f rc lies in the cone of rank-constant P-partitions,which is thesimplicial cone positively spanned by the(non-empty)rank-constant ideals{I rc j}r j=1,⊲When f eq is expressed in the unique way as a positive combina-tion of characteristic vectors of a chain of ideals,as in Propo-sition2.2part(i),this chain of ideals must be equatorial sincef eq is.It remains to check that all such cones are unimodular.Thus it suﬃces to show that whenever R∪E is maximal under inclusion,then #R∪E=n and the Z-span of the set{χI:I∈R∪E}additively generates inside R n is the full integer lattice Z n.To see#R∪E= n,ﬁrst note that when R∪E is maximal,one has R={I rc j}r j=1, and then#E=n−r follows from Proposition3.5.To show they additively generate Z n,we show by induction on the rank r of P that the subgroup they generate contains each standard basis vector e p for p∈P.The base case r=1has P an antichain,hence all ideals I P are equatorial,so the cones in question coincide with the cones in the canonical triangulation,which are unimodular by Proposition2.1.In the inductive step,note that this subgroup generated by{χI:I∈R∪E}has the alternate description as the subgroup generated bythe characteristic vectorsχPj of all of the ranks of P along with thecharacteristic vectorsχJi of all of the jumps between the equatorialideals in E.Proposition3.5shows that there will be exactly one element q of the top rank r in P which does not occur in a singleton jump J i.Namely,q=max J is after the re-labelling as in Proposition3.6.Hencefor every p∈P r−{q},one has e p in the subgroup,but then one alsohas e q in the subgroup,since the subgroup containsχPr .Now applyinduction to the graded poset P−P r of rank r−1,replacing the ideals in R∪E by their intersections with P−P r and removing multiple copies of the same ideal created by the intersection process. The triangulation of A(P)given in Proposition3.6induces a uni-modular triangulation of O(P),which we will call the equatorial trian-gulation of O(P).CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES13 Deﬁnition3.7.The equatorial complex∆eq(P)is deﬁned to be the subcomplex of the order complex∆J(P)whose faces are indexed by the equatorial chains of non-empty ideals.For the formulation of the next corollary we need the concept of simplicial join.For two simplicial complexes∆1,∆2which are deﬁned over disjoint vertex sets,the simplicial join∆1∗∆2is the simplicial complex{σ1∪σ2:σi∈∆i,i=1,2}.Note that we always assume that the empty face∅is a face of a simplicial complex.Corollary3.8.The equatorial triangulation of the order polytope O(P) is abstractly isomorphic to the simplicial joinσr∗∆eq(P),whereσr is the interior r-simplex spanned by the chain of rank-constant ideals {I rc j}r j=0.As a consequence of its unimodularity,one hash(∆eq(P),t)=h(∆J(P),t)=W(P,t).Proof.Theﬁrst assertion follows directly from Proposition3.6,noting thatσr is interior due to Proposition3.4.For the second,note that bothσr∗∆eq(P)and∆J(P)index unimodular triangulations of the order polytope,so(2.2)impliesh(σr∗∆eq(P),t)=h(∆J(P),t).On the other hand,the deﬁning equation(1.1)of the h-polynomial shows thatf(∆1∗∆2,t)=f(∆1,t)∗f(∆2,t)h(∆1∗∆2,t)=h(∆1,t)∗h(∆2,t)h(σr,t)=1,and hence h(σr∗∆,t)=h(∆,t). Remark3.9.Corollary3.8has the following consequence:for a graded poset P, the set of linear extensions L(P)is equinumerous with the set L eq(P) of all maximal equatorial chains of ideals in P,as both coincide with [W(P,t)]t=1.This begs for a bijectionφ:L(P)→L eq(P).The authors thank Dennis White[53]for supplying one which is elegant,using the idea of jeu-de-taquin on linear extensions of P,thought of as P-shaped tableaux that use each entry1,2,...,n exactly once.Given such a linear extension w,replace the highest label n(at top rank r)by a jeu-de-taquin hole,and slide it past other entries down to rank1,du-plicating the last entry that it slid past in the hole’s resting position at rank1.Then repeat this with the entry n−1,sliding it down to rank2,and similarly with the entries n−2,n−3,...,n−r+1.The14VICTOR REINER AND VOLKMAR WELKERresult is a P-shaped tableaux that can be interpreted as an equato-rial P-partition,compatible with a unique maximal equatorial chain of idealsφ(w).It is not hard to check that this map w→φ(w)is a bijection.3.2.Geometric and Convexity Properties of∆eq(P).In this sec-tion,we use convexity and the concrete geometric realization of∆eq(P) to learn more about it.Deﬁnition3.10.The rank-constant subspace V rc⊂R n is the R-linear}r j=1.span of the set{χI rcjLet Q be a convex polytope,and V a linear subspace,both inside R n.Then there is a well deﬁned quotient polytopeQ/V:={q+V:q∈Q}⊂R n/V.Ifπ:R n→R n−dim V is any linear surjection with kernel V(such as an orthogonal projection onto V⊥),then the polytope Q/V can be identiﬁed with the imageπ(Q).Also note that if V is a rational subspace of R n with respect to the integer lattice Z n⊂R n,the quotient lattice Z n/(V∩Z n)is well-deﬁned,and a full rank sublattice in R n/V. Proposition3.11.The collection of quotient conesC E=pos {χI:I∈E} +V rc ,as E runs through all equatorial chains of non-empty ideals in P,forms a complete simplicial fan in R n/V rc.(i)This simplicial fan is unimodular with respect to the quotientlattice Z n/(V rc∩Z n).(ii)The simplices(C E∩O(P))+V rc form a unimodular triangula-tion of the quotient polytope O eq(P):=O(P)/V rc.(iii)This triangulation of O(P)/V rc is isomorphic,as an abstract simplicial complex,to the cone0∗∆eq(P)with base∆eq(P)andapex at the interior point0=V rc.Consequently,∆eq(P)triangulates the(n−r−1)-dimensional bound-ary sphere∂O eq(P).Proof.Apply the following general statement,Proposition3.12,about polytopes(and the analogous statement about fans)withQ=O(P),∆=the equatorial triangulation,∆′=∆eq(P),V=V rc.CHARNEY-DAVIS AND NEGGERS-STANLEY CONJECTURES15Proposition3.12.Let Q be an n-dimensional convex polytope in R n. Assume Q has a triangulation abstractly isomorphic to a simplicial complex∆of the form∆∼=σr∗∆′,whereσr is an r-simplex not lying on the boundary of Q.Let V be the r-dimensional linear subspace parallel to the aﬃne span of the vertices ofσr.Then the quotient(n−r)-dimensional polytope Q/V⊂R n/V in-herits a triangulation abstractly isomorphic toσ0∗∆′,whereσ0is an interior point of Q/V⊂R n/V.Furthermore,when V is rational with respect to Z n⊂R n and if the triangulation of Q is unimodular with respect to Z n,then the triangu-lation of Q/V rc is unimodular with respect to Z n/(V rc∩Z n).The proof of Proposition3.12is straightforward.We leave it as an exercise.Proposition3.11,shows that∆eq(P)corresponds to a complete uni-modular fan.This fact suﬃces to infer both that it is spherical,andthat it corresponds to a smooth,complete toric variety X∆eq(P)(see[18,§2.1]).Our next goal will be to show that∆eq(P)corresponds to a polytopal fan,as this has multiple consequences;see Corollary3.15 below.We prove polytopality of∆eq(P)by choosing for each equatorial ideal I of P a point on its ray pos(χI+V rc)so that the convex hull of all such points is a simplicial polytope having∆eq(P)as its boundary complex.Here we employ the following strategy.We start with the (usually)non-simplicial polytope O eq(P)and pull each of its vertices in a certain order to produce a simplicial polytope with boundary complex ∆eq(P).Recall[31,§2.5]that if Q is a convex polytope,one pulls the vertex v in Q to produce a new polytope pull v(Q)by taking the convex hull after moving v slightly outward past the supporting hyperplanes of all facets that contain v,but past no other facet-supporting hyperplanes of Q.Assuming that Q contains the origin in its interior,this can clearly be achieved by replacing v with(1+ǫ)v whereǫ>0is suﬃciently small.We will require the following proposition describing the1-skeleton resulting from pulling all the vertices of a polytope: Proposition3.13.Let Q be the polytope resulting from pulling all of the vertices of a polytope Q in some order v1,v2,...,and let v i denote the corresponding vertices in Q.。

Kernels and regularization on graphs

Kernels and Regularization on GraphsAlexander J.Smola1and Risi Kondor21Machine Learning Group,RSISEAustralian National UniversityCanberra,ACT0200,AustraliaAlex.Smola@.au2Department of Computer ScienceColumbia University1214Amsterdam Avenue,M.C.0401New York,NY10027,USArisi@Abstract.We introduce a family of kernels on graphs based on thenotion of regularization operators.This generalizes in a natural way thenotion of regularization and Greens functions,as commonly used forreal valued functions,to graphs.It turns out that diﬀusion kernels canbe found as a special case of our reasoning.We show that the class ofpositive,monotonically decreasing functions on the unit interval leads tokernels and corresponding regularization operators.1IntroductionThere has recently been a surge of interest in learning algorithms that operate on input spaces X other than R n,speciﬁcally,discrete input spaces,such as strings, graphs,trees,automata etc..Since kernel-based algorithms,such as Support Vector Machines,Gaussian Processes,Kernel PCA,etc.capture the structure of X via the kernel K:X×X→R,as long as we can deﬁne an appropriate kernel on our discrete input space,these algorithms can be imported wholesale, together with their error analysis,theoretical guarantees and empirical success.One of the most general representations of discrete metric spaces are graphs. Even if all we know about our input space are local pairwise similarities between points x i,x j∈X,distances(e.g shortest path length)on the graph induced by these similarities can give a useful,more global,sense of similarity between objects.In their work on Diﬀusion Kernels,Kondor and Laﬀerty[2002]gave a speciﬁc construction for a kernel capturing this structure.Belkin and Niyogi [2002]proposed an essentially equivalent construction in the context of approx-imating data lying on surfaces in a high dimensional embedding space,and in the context of leveraging information from unlabeled data.In this paper we put these earlier results into the more principled framework of Regularization Theory.We propose a family of regularization operators(equiv-alently,kernels)on graphs that include Diﬀusion Kernels as a special case,and show that this family encompasses all possible regularization operators invariant under permutations of the vertices in a particular sense.2Alexander Smola and Risi KondorOutline of the Paper:Section2introduces the concept of the graph Laplacian and relates it to the Laplace operator on real valued functions.Next we deﬁne an extended class of regularization operators and show why they have to be es-sentially a function of the Laplacian.An analogy to real valued Greens functions is established in Section3.3,and eﬃcient methods for computing such functions are presented in Section4.We conclude with a discussion.2Laplace OperatorsAn undirected unweighted graph G consists of a set of vertices V numbered1to n,and a set of edges E(i.e.,pairs(i,j)where i,j∈V and(i,j)∈E⇔(j,i)∈E). We will sometimes write i∼j to denote that i and j are neighbors,i.e.(i,j)∈E. The adjacency matrix of G is an n×n real matrix W,with W ij=1if i∼j,and 0otherwise(by construction,W is symmetric and its diagonal entries are zero). These deﬁnitions and most of the following theory can trivially be extended toweighted graphs by allowing W ij∈[0,∞).Let D be an n×n diagonal matrix with D ii=jW ij.The Laplacian of Gis deﬁned as L:=D−W and the Normalized Laplacian is˜L:=D−12LD−12= I−D−12W D−12.The following two theorems are well known results from spectral graph theory[Chung-Graham,1997]:Theorem1(Spectrum of˜L).˜L is a symmetric,positive semideﬁnite matrix, and its eigenvaluesλ1,λ2,...,λn satisfy0≤λi≤2.Furthermore,the number of eigenvalues equal to zero equals to the number of disjoint components in G.The bound on the spectrum follows directly from Gerschgorin’s Theorem.Theorem2(L and˜L for Regular Graphs).Now let G be a regular graph of degree d,that is,a graph in which every vertex has exactly d neighbors.ThenL=d I−W and˜L=I−1d W=1dL.Finally,W,L,˜L share the same eigenvectors{v i},where v i=λ−1iW v i=(d−λi)−1L v i=(1−d−1λi)−1˜L v i for all i.L and˜L can be regarded as linear operators on functions f:V→R,or,equiv-alently,on vectors f=(f1,f2,...,f n) .We could equally well have deﬁned Lbyf,L f =f L f=−12i∼j(f i−f j)2for all f∈R n,(1)which readily generalizes to graphs with a countably inﬁnite number of vertices.The Laplacian derives its name from its analogy with the familiar Laplacianoperator∆=∂2∂x21+∂2∂x22+...+∂2∂x2mon continuous spaces.Regarding(1)asinducing a semi-norm f L= f,L f on R n,the analogous expression for∆deﬁned on a compact spaceΩisf ∆= f,∆f =Ωf(∆f)dω=Ω(∇f)·(∇f)dω.(2)Both(1)and(2)quantify how much f and f vary locally,or how“smooth”they are over their respective domains.Kernels and Regularization on Graphs3 More explicitly,whenΩ=R m,up to a constant,−L is exactly theﬁnite diﬀerence discretization of∆on a regular lattice:∆f(x)=mi=1∂2∂x2if≈mi=1∂∂x if(x+12e i)−∂∂x if(x−12e i)δ≈mi=1f(x+e i)+f(x−e i)−2f(x)δ2=1δ2mi=1(f x1,...,x i+1,...,x m+f x1,...,x i−1,...,x m−2f x1,...,x m)=−1δ2[L f]x1,...,x m,where e1,e2,...,e m is an orthogonal basis for R m normalized to e i =δ, the vertices of the lattice are at x=x1e1+...+x m e m with integer valuedcoordinates x i∈N,and f x1,x2,...,x m=f(x).Moreover,both the continuous and the dis-crete Laplacians are canonical operators on their respective domains,in the sense that they are invariant under certain natural transformations of the underlying space,and in this they are essentially unique.Regular grid in two dimensionsThe Laplace operator∆is the unique self-adjoint linear second order diﬀer-ential operator invariant under transformations of the coordinate system under the action of the special orthogonal group SO m,i.e.invariant under rotations. This well known result can be seen by using Schur’s lemma and the fact that SO m is irreducible on R m.We now show a similar result for L.Here the permutation group plays a similar role to SO m.We need some additional deﬁnitions:denote by S n the group of permutations on{1,2,...,n}withπ∈S n being a speciﬁc permutation taking i∈{1,2,...n}toπ(i).The so-called deﬁning representation of S n consists of n×n matricesΠπ,such that[Ππ]i,π(i)=1and all other entries ofΠπare zero. Theorem3(Permutation Invariant Linear Functions on Graphs).Let L be an n×n symmetric real matrix,linearly related to the n×n adjacency matrix W,i.e.L=T[W]for some linear operator L in a way invariant to permutations of vertices in the sense thatΠ πT[W]Ππ=TΠ πWΠπ(3)for anyπ∈S n.Then L is related to W by a linear combination of the follow-ing three operations:identity;row/column sums;overall sum;row/column sum restricted to the diagonal of L;overall sum restricted to the diagonal of W. Proof LetL i1i2=T[W]i1i2:=ni3=1ni4=1T i1i2i3i4W i3i4(4)with T∈R n4.Eq.(3)then implies Tπ(i1)π(i2)π(i3)π(i4)=T i1i2i3i4for anyπ∈S n.4Alexander Smola and Risi KondorThe indices of T can be partitioned by the equality relation on their values,e.g.(2,5,2,7)is of the partition type [13|2|4],since i 1=i 3,but i 2=i 1,i 4=i 1and i 2=i 4.The key observation is that under the action of the permutation group,elements of T with a given index partition structure are taken to elements with the same index partition structure,e.g.if i 1=i 3then π(i 1)=π(i 3)and if i 1=i 3,then π(i 1)=π(i 3).Furthermore,an element with a given index index partition structure can be mapped to any other element of T with the same index partition structure by a suitable choice of π.Hence,a necessary and suﬃcient condition for (4)is that all elements of T of a given index partition structure be equal.Therefore,T must be a linear combination of the following tensors (i.e.multilinear forms):A i 1i 2i 3i 4=1B [1,2]i 1i 2i 3i 4=δi 1i 2B [1,3]i 1i 2i 3i 4=δi 1i 3B [1,4]i 1i 2i 3i 4=δi 1i 4B [2,3]i 1i 2i 3i 4=δi 2i 3B [2,4]i 1i 2i 3i 4=δi 2i 4B [3,4]i 1i 2i 3i 4=δi 3i 4C [1,2,3]i 1i 2i 3i 4=δi 1i 2δi 2i 3C [2,3,4]i 1i 2i 3i 4=δi 2i 3δi 3i 4C [3,4,1]i 1i 2i 3i 4=δi 3i 4δi 4i 1C [4,1,2]i 1i 2i 3i 4=δi 4i 1δi 1i 2D [1,2][3,4]i 1i 2i 3i 4=δi 1i 2δi 3i 4D [1,3][2,4]i 1i 2i 3i 4=δi 1i 3δi 2i 4D [1,4][2,3]i 1i 2i 3i 4=δi 1i 4δi 2i 3E [1,2,3,4]i 1i 2i 3i 4=δi 1i 2δi 1i 3δi 1i 4.The tensor A puts the overall sum in each element of L ,while B [1,2]returns the the same restricted to the diagonal of L .Since W has vanishing diagonal,B [3,4],C [2,3,4],C [3,4,1],D [1,2][3,4]and E [1,2,3,4]produce zero.Without loss of generality we can therefore ignore them.By symmetry of W ,the pairs (B [1,3],B [1,4]),(B [2,3],B [2,4]),(C [1,2,3],C [4,1,2])have the same eﬀect on W ,hence we can set the coeﬃcient of the second member of each to zero.Furthermore,to enforce symmetry on L ,the coeﬃcient of B [1,3]and B [2,3]must be the same (without loss of generality 1)and this will give the row/column sum matrix ( k W ik )+( k W kl ).Similarly,C [1,2,3]and C [4,1,2]must have the same coeﬃcient and this will give the row/column sum restricted to the diagonal:δij [( k W ik )+( k W kl )].Finally,by symmetry of W ,D [1,3][2,4]and D [1,4][2,3]are both equivalent to the identity map.The various row/column sum and overall sum operations are uninteresting from a graph theory point of view,since they do not heed to the topology of the graph.Imposing the conditions that each row and column in L must sum to zero,we recover the graph Laplacian.Hence,up to a constant factor and trivial additive components,the graph Laplacian (or the normalized graph Laplacian if we wish to rescale by the number of edges per vertex)is the only “invariant”diﬀerential operator for given W (or its normalized counterpart ˜W ).Unless stated otherwise,all results below hold for both L and ˜L (albeit with a diﬀerent spectrum)and we will,in the following,focus on ˜Ldue to the fact that its spectrum is contained in [0,2].Kernels and Regularization on Graphs5 3RegularizationThe fact that L induces a semi-norm on f which penalizes the changes between adjacent vertices,as described in(1),indicates that it may serve as a tool to design regularization operators.3.1Regularization via the Laplace OperatorWe begin with a brief overview of translation invariant regularization operators on continuous spaces and show how they can be interpreted as powers of∆.This will allow us to repeat the development almost verbatim with˜L(or L)instead.Some of the most successful regularization functionals on R n,leading to kernels such as the Gaussian RBF,can be written as[Smola et al.,1998]f,P f :=|˜f(ω)|2r( ω 2)dω= f,r(∆)f .(5)Here f∈L2(R n),˜f(ω)denotes the Fourier transform of f,r( ω 2)is a function penalizing frequency components|˜f(ω)|of f,typically increasing in ω 2,and ﬁnally,r(∆)is the extension of r to operators simply by applying r to the spectrum of∆[Dunford and Schwartz,1958]f,r(∆)f =if,ψi r(λi) ψi,fwhere{(ψi,λi)}is the eigensystem of∆.The last equality in(5)holds because applications of∆become multiplications by ω 2in Fourier space.Kernels are obtained by solving the self-consistency condition[Smola et al.,1998]k(x,·),P k(x ,·) =k(x,x ).(6) One can show that k(x,x )=κ(x−x ),whereκis equal to the inverse Fourier transform of r−1( ω 2).Several r functions have been known to yield good results.The two most popular are given below:r( ω 2)k(x,x )r(∆)Gaussian RBF expσ22ω 2exp−12σ2x−x 2∞i=0σ2ii!∆iLaplacian RBF1+σ2 ω 2exp−1σx−x1+σ2∆In summary,regularization according to(5)is carried out by penalizing˜f(ω) by a function of the Laplace operator.For many results in regularization theory one requires r( ω 2)→∞for ω 2→∞.3.2Regularization via the Graph LaplacianIn complete analogy to(5),we deﬁne a class of regularization functionals on graphs asf,P f := f,r(˜L)f .(7)6Alexander Smola and Risi KondorFig.1.Regularization function r (λ).From left to right:regularized Laplacian (σ2=1),diﬀusion process (σ2=1),one-step random walk (a =2),4-step random walk (a =2),inverse cosine.Here r (˜L )is understood as applying the scalar valued function r (λ)to the eigen-values of ˜L ,that is,r (˜L ):=m i =1r (λi )v i v i ,(8)where {(λi ,v i )}constitute the eigensystem of ˜L .The normalized graph Lapla-cian ˜Lis preferable to L ,since ˜L ’s spectrum is contained in [0,2].The obvious goal is to gain insight into what functions are appropriate choices for r .–From (1)we infer that v i with large λi correspond to rather uneven functions on the graph G .Consequently,they should be penalized more strongly than v i with small λi .Hence r (λ)should be monotonically increasing in λ.–Requiring that r (˜L) 0imposes the constraint r (λ)≥0for all λ∈[0,2].–Finally,we can limit ourselves to r (λ)expressible as power series,since the latter are dense in the space of C 0functions on bounded domains.In Section 3.5we will present additional motivation for the choice of r (λ)in the context of spectral graph theory and segmentation.As we shall see,the following functions are of particular interest:r (λ)=1+σ2λ(Regularized Laplacian)(9)r (λ)=exp σ2/2λ(Diﬀusion Process)(10)r (λ)=(aI −λ)−1with a ≥2(One-Step Random Walk)(11)r (λ)=(aI −λ)−p with a ≥2(p -Step Random Walk)(12)r (λ)=(cos λπ/4)−1(Inverse Cosine)(13)Figure 1shows the regularization behavior for the functions (9)-(13).3.3KernelsThe introduction of a regularization matrix P =r (˜L)allows us to deﬁne a Hilbert space H on R m via f,f H := f ,P f .We now show that H is a reproducing kernel Hilbert space.Kernels and Regularization on Graphs 7Theorem 4.Denote by P ∈R m ×m a (positive semideﬁnite)regularization ma-trix and denote by H the image of R m under P .Then H with dot product f,f H := f ,P f is a Reproducing Kernel Hilbert Space and its kernel is k (i,j )= P −1ij ,where P −1denotes the pseudo-inverse if P is not invertible.Proof Since P is a positive semideﬁnite matrix,we clearly have a Hilbert space on P R m .To show the reproducing property we need to prove thatf (i )= f,k (i,·) H .(14)Note that k (i,j )can take on at most m 2diﬀerent values (since i,j ∈[1:m ]).In matrix notation (14)means that for all f ∈Hf (i )=f P K i,:for all i ⇐⇒f =f P K.(15)The latter holds if K =P −1and f ∈P R m ,which proves the claim.In other words,K is the Greens function of P ,just as in the continuous case.The notion of Greens functions on graphs was only recently introduced by Chung-Graham and Yau [2000]for L .The above theorem extended this idea to arbitrary regularization operators ˆr (˜L).Corollary 1.Denote by P =r (˜L )a regularization matrix,then the correspond-ing kernel is given by K =r −1(˜L ),where we take the pseudo-inverse wherever necessary.More speciﬁcally,if {(v i ,λi )}constitute the eigensystem of ˜L,we have K =mi =1r −1(λi )v i v i where we deﬁne 0−1≡0.(16)3.4Examples of KernelsBy virtue of Corollary 1we only need to take (9)-(13)and plug the deﬁnition of r (λ)into (16)to obtain formulae for computing K .This yields the following kernel matrices:K =(I +σ2˜L)−1(Regularized Laplacian)(17)K =exp(−σ2/2˜L)(Diﬀusion Process)(18)K =(aI −˜L)p with a ≥2(p -Step Random Walk)(19)K =cos ˜Lπ/4(Inverse Cosine)(20)Equation (18)corresponds to the diﬀusion kernel proposed by Kondor and Laf-ferty [2002],for which K (x,x )can be visualized as the quantity of some sub-stance that would accumulate at vertex x after a given amount of time if we injected the substance at vertex x and let it diﬀuse through the graph along the edges.Note that this involves matrix exponentiation deﬁned via the limit K =exp(B )=lim n →∞(I +B/n )n as opposed to component-wise exponentiation K i,j =exp(B i,j ).8Alexander Smola and Risi KondorFig.2.Theﬁrst8eigenvectors of the normalized graph Laplacian corresponding to the graph drawn above.Each line attached to a vertex is proportional to the value of the corresponding eigenvector at the vertex.Positive values(red)point up and negative values(blue)point down.Note that the assignment of values becomes less and less uniform with increasing eigenvalue(i.e.from left to right).For(17)it is typically more eﬃcient to deal with the inverse of K,as it avoids the costly inversion of the sparse matrix˜L.Such situations arise,e.g.,in Gaussian Process estimation,where K is the covariance matrix of a stochastic process[Williams,1999].Regarding(19),recall that(aI−˜L)p=((a−1)I+˜W)p is up to scaling terms equiv-alent to a p-step random walk on the graphwith random restarts(see Section A for de-tails).In this sense it is similar to the dif-fusion kernel.However,the fact that K in-volves only aﬁnite number of products ofmatrices makes it much more attractive forpractical purposes.In particular,entries inK ij can be computed cheaply using the factthat˜L is a sparse matrix.A nearest neighbor graph.Finally,the inverse cosine kernel treats lower complexity functions almost equally,with a signiﬁcant reduction in the upper end of the spectrum.Figure2 shows the leading eigenvectors of the graph drawn above and Figure3provide examples of some of the kernels discussed above.3.5Clustering and Spectral Graph TheoryWe could also have derived r(˜L)directly from spectral graph theory:the eigen-vectors of the graph Laplacian correspond to functions partitioning the graph into clusters,see e.g.,[Chung-Graham,1997,Shi and Malik,1997]and the ref-erences therein.In general,small eigenvalues have associated eigenvectors which vary little between adjacent vertices.Finding the smallest eigenvectors of˜L can be seen as a real-valued relaxation of the min-cut problem.3For instance,the smallest eigenvalue of˜L is0,its corresponding eigenvector is D121n with1n:=(1,...,1)∈R n.The second smallest eigenvalue/eigenvector pair,also often referred to as the Fiedler-vector,can be used to split the graph 3Only recently,algorithms based on the celebrated semideﬁnite relaxation of the min-cut problem by Goemans and Williamson[1995]have seen wider use[Torr,2003]in segmentation and clustering by use of spectral bundle methods.Kernels and Regularization on Graphs9Fig.3.Top:regularized graph Laplacian;Middle:diﬀusion kernel with σ=5,Bottom:4-step random walk kernel.Each ﬁgure displays K ij for ﬁxed i .The value K ij at vertex i is denoted by a bold line.Note that only adjacent vertices to i bear signiﬁcant value.into two distinct parts [Weiss,1999,Shi and Malik,1997],and further eigenvec-tors with larger eigenvalues have been used for more ﬁnely-grained partitions of the graph.See Figure 2for an example.Such a decomposition into functions of increasing complexity has very de-sirable properties:if we want to perform estimation on the graph,we will wish to bias the estimate towards functions which vary little over large homogeneous portions 4.Consequently,we have the following interpretation of f,f H .As-sume that f = i βi v i ,where {(v i ,λi )}is the eigensystem of ˜L.Then we can rewrite f,f H to yield f ,r (˜L )f = i βi v i , j r (λj )v j v j l βl v l = iβ2i r (λi ).(21)This means that the components of f which vary a lot over coherent clusters in the graph are penalized more strongly,whereas the portions of f ,which are essentially constant over clusters,are preferred.This is exactly what we want.3.6Approximate ComputationOften it is not necessary to know all values of the kernel (e.g.,if we only observe instances from a subset of all positions on the graph).There it would be wasteful to compute the full matrix r (L )−1explicitly,since such operations typically scale with O (n 3).Furthermore,for large n it is not desirable to compute K via (16),that is,by computing the eigensystem of ˜Land assembling K directly.4If we cannot assume a connection between the structure of the graph and the values of the function to be estimated on it,the entire concept of designing kernels on graphs obviously becomes meaningless.10Alexander Smola and Risi KondorInstead,we would like to take advantage of the fact that ˜L is sparse,and con-sequently any operation ˜Lαhas cost at most linear in the number of nonzero ele-ments of ˜L ,hence the cost is bounded by O (|E |+n ).Moreover,if d is the largest degree of the graph,then computing L p e i costs at most |E | p −1i =1(min(d +1,n ))ioperations:at each step the number of non-zeros in the rhs decreases by at most a factor of d +1.This means that as long as we can approximate K =r −1(˜L )by a low order polynomial,say ρ(˜L ):= N i =0βi ˜L i ,signiﬁcant savings are possible.Note that we need not necessarily require a uniformly good approximation and put the main emphasis on the approximation for small λ.However,we need to ensure that ρ(˜L)is positive semideﬁnite.Diﬀusion Kernel:The fact that the series r −1(x )=exp(−βx )= ∞m =0(−β)m x m m !has alternating signs shows that the approximation error at r −1(x )is boundedby (2β)N +1(N +1)!,if we use N terms in the expansion (from Theorem 1we know that ˜L≤2).For instance,for β=1,10terms are suﬃcient to obtain an error of the order of 10−4.Variational Approximation:In general,if we want to approximate r −1(λ)on[0,2],we need to solve the L ∞([0,2])approximation problemminimize β, subject to N i =0βi λi −r −1(λ) ≤ ∀λ∈[0,2](22)Clearly,(22)is equivalent to minimizing sup ˜L ρ(˜L )−r−1(˜L ) ,since the matrix norm is determined by the largest eigenvalues,and we can ﬁnd ˜Lsuch that the discrepancy between ρ(λ)and r −1(λ)is attained.Variational problems of this form have been studied in the literature,and their solution may provide much better approximations to r −1(λ)than a truncated power series expansion.4Products of GraphsAs we have already pointed out,it is very expensive to compute K for arbitrary ˆr and ˜L.For special types of graphs and regularization,however,signiﬁcant computational savings can be made.4.1Factor GraphsThe work of this section is a direct extension of results by Ellis [2002]and Chung-Graham and Yau [2000],who study factor graphs to compute inverses of the graph Laplacian.Deﬁnition 1(Factor Graphs).Denote by (V,E )and (V ,E )the vertices V and edges E of two graphs,then the factor graph (V f ,E f ):=(V,E )⊗(V ,E )is deﬁned as the graph where (i,i )∈V f if i ∈V and i ∈V ;and ((i,i ),(j,j ))∈E f if and only if either (i,j )∈E and i =j or (i ,j )∈E and i =j .Kernels and Regularization on Graphs 11For instance,the factor graph of two rings is a torus.The nice property of factor graphs is that we can compute the eigenvalues of the Laplacian on products very easily (see e.g.,Chung-Graham and Yau [2000]):Theorem 5(Eigenvalues of Factor Graphs).The eigenvalues and eigen-vectors of the normalized Laplacian for the factor graph between a regular graph of degree d with eigenvalues {λj }and a regular graph of degree d with eigenvalues {λ l }are of the form:λfact j,l =d d +d λj +d d +d λ l(23)and the eigenvectors satisfy e j,l(i,i )=e j i e l i ,where e j is an eigenvector of ˜L and e l is an eigenvector of ˜L.This allows us to apply Corollary 1to obtain an expansion of K asK =(r (L ))−1=j,l r −1(λjl )e j,l e j,l .(24)While providing an explicit recipe for the computation of K ij without the need to compute the full matrix K ,this still requires O (n 2)operations per entry,which may be more costly than what we want (here n is the number of vertices of the factor graph).Two methods for computing (24)become evident at this point:if r has a special structure,we may exploit this to decompose K into the products and sums of terms depending on one of the two graphs alone and pre-compute these expressions beforehand.Secondly,if one of the two terms in the expansion can be computed for a rather general class of values of r (x ),we can pre-compute this expansion and only carry out the remainder corresponding to (24)explicitly.4.2Product Decomposition of r (x )Central to our reasoning is the observation that for certain r (x ),the term 1r (a +b )can be expressed in terms of a product and sum of terms depending on a and b only.We assume that 1r (a +b )=M m =1ρn (a )˜ρn (b ).(25)In the following we will show that in such situations the kernels on factor graphs can be computed as an analogous combination of products and sums of kernel functions on the terms constituting the ingredients of the factor graph.Before we do so,we brieﬂy check that many r (x )indeed satisfy this property.exp(−β(a +b ))=exp(−βa )exp(−βb )(26)(A −(a +b ))= A 2−a + A 2−b (27)(A −(a +b ))p =p n =0p n A 2−a n A 2−b p −n (28)cos (a +b )π4=cos aπ4cos bπ4−sin aπ4sin bπ4(29)12Alexander Smola and Risi KondorIn a nutshell,we will exploit the fact that for products of graphs the eigenvalues of the joint graph Laplacian can be written as the sum of the eigenvalues of the Laplacians of the constituent graphs.This way we can perform computations on ρn and˜ρn separately without the need to take the other part of the the product of graphs into account.Deﬁnek m(i,j):=l ρldλld+de l i e l j and˜k m(i ,j ):=l˜ρldλld+d˜e l i ˜e l j .(30)Then we have the following composition theorem:Theorem6.Denote by(V,E)and(V ,E )connected regular graphs of degrees d with m vertices(and d ,m respectively)and normalized graph Laplacians ˜L,˜L .Furthermore denote by r(x)a rational function with matrix-valued exten-sionˆr(X).In this case the kernel K corresponding to the regularization operator ˆr(L)on the product graph of(V,E)and(V ,E )is given byk((i,i ),(j,j ))=Mm=1k m(i,j)˜k m(i ,j )(31)Proof Plug the expansion of1r(a+b)as given by(25)into(24)and collect terms.From(26)we immediately obtain the corollary(see Kondor and Laﬀerty[2002]) that for diﬀusion processes on factor graphs the kernel on the factor graph is given by the product of kernels on the constituents,that is k((i,i ),(j,j ))= k(i,j)k (i ,j ).The kernels k m and˜k m can be computed either by using an analytic solution of the underlying factors of the graph or alternatively they can be computed numerically.If the total number of kernels k n is small in comparison to the number of possible coordinates this is still computationally beneﬁcial.4.3Composition TheoremsIf no expansion as in(31)can be found,we may still be able to compute ker-nels by extending a reasoning from[Ellis,2002].More speciﬁcally,the following composition theorem allows us to accelerate the computation in many cases, whenever we can parameterize(ˆr(L+αI))−1in an eﬃcient way.For this pur-pose we introduce two auxiliary functionsKα(i,j):=ˆrdd+dL+αdd+dI−1=lrdλl+αdd+d−1e l(i)e l(j)G α(i,j):=(L +αI)−1=l1λl+αe l(i)e l(j).(32)In some cases Kα(i,j)may be computed in closed form,thus obviating the need to perform expensive matrix inversion,e.g.,in the case where the underlying graph is a chain[Ellis,2002]and Kα=Gα.Kernels and Regularization on Graphs 13Theorem 7.Under the assumptions of Theorem 6we haveK ((j,j ),(l,l ))=12πi C K α(j,l )G −α(j ,l )dα= v K λv (j,l )e v j e v l (33)where C ⊂C is a contour of the C containing the poles of (V ,E )including 0.For practical purposes,the third term of (33)is more amenable to computation.Proof From (24)we haveK ((j,j ),(l,l ))= u,v r dλu +d λv d +d −1e u j e u l e v j e v l (34)=12πi C u r dλu +d αd +d −1e u j e u l v 1λv −αe v j e v l dαHere the second equalityfollows from the fact that the contour integral over a pole p yields C f (α)p −αdα=2πif (p ),and the claim is veriﬁed by checking thedeﬁnitions of K αand G α.The last equality can be seen from (34)by splitting up the summation over u and v .5ConclusionsWe have shown that the canonical family of kernels on graphs are of the form of power series in the graph Laplacian.Equivalently,such kernels can be char-acterized by a real valued function of the eigenvalues of the Laplacian.Special cases include diﬀusion kernels,the regularized Laplacian kernel and p -step ran-dom walk kernels.We have developed the regularization theory of learning on graphs using such kernels and explored methods for eﬃciently computing and approximating the kernel matrix.Acknowledgments This work was supported by a grant of the ARC.The authors thank Eleazar Eskin,Patrick Haﬀner,Andrew Ng,Bob Williamson and S.V.N.Vishwanathan for helpful comments and suggestions.A Link AnalysisRather surprisingly,our approach to regularizing functions on graphs bears re-semblance to algorithms for scoring web pages such as PageRank [Page et al.,1998],HITS [Kleinberg,1999],and randomized HITS [Zheng et al.,2001].More speciﬁcally,the random walks on graphs used in all three algorithms and the stationary distributions arising from them are closely connected with the eigen-system of L and ˜Lrespectively.We begin with an analysis of PageRank.Given a set of web pages and links between them we construct a directed graph in such a way that pages correspond。

代数英语

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CAT准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

Elsevier期刊被SCI收录最新一期题录信息

Elsevier期刊被SCI收录最新一期题录信息本内容包括：主题、各主题有代表性刊名、该刊最新一期目录信息（包括卷期、论文题名、页码及作者等）Automation & Control Systems（自动化及控制系统）SYSTEMS & CONTROL LETTERSVolume 59, Issue 5, Pages 265-322 (May 2010)1.Editorial BoardPage IFC2.Gain-scheduled open-loop system design for LPV systems using polynomially parameter-dependent Lyapunov functionsPages 265-276Masayuki Sato3.Delay-adaptive feedback for linear feedforward systemsPages 277-283Nikolaos Bekiaris-Liberis, Miroslav Krstic4.Implicit Euler numerical scheme and chattering-free implementation of sliding mode systemsPages 284-293Vincent Acary, Bernard Brogliato5.Decentralized dynamic nonlinear controllers to minimize transmit power in cellular networks, Part IPages 294-298Vishwesh V. Kulkarni, Mayuresh V. Kothare, Michael G. Safonov6. 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Rodaitė-Riševičienė24.Carbon nanotube-enhanced cell electropermeabilisationPages 136-141Vittoria Raffa, Gianni Ciofani, Orazio Vittorio, Virginia Pensabene, Alfred Cuschieri25.Dependence of catalytic activity and long-term stability of enzyme hydrogel films on curing timePages 142-146Joshua Lehr, Bryce E. Williamson, Frédéric Barrière, Alison J. Downard26.Enzymatic flow injection method for rapid determination of choline in urine with electrochemiluminescence detectionPages 147-151Jiye Jin, Masahiro Muroga, Fumiki Takahashi, Toshio NakamuraChemistry Applied（化学应用）CARBOHYDRATE POLYMERSVolume 81, Issue 4, Pages 751-970 (23 July 2010)1.Editorial BoardPage CO22.Adsorption separation of Ni(II) ions by dialdehyde o-phenylenediamine starch from aqueous solutionPages 751-757Ping Zhao, Jian Jiang, Feng-wei Zhang, Wen-feng Zhao, Jun-tao Liu, Rong Li3.Rheological and morphological characterization of the culture broth during exopolysaccharide production by Enterobacter sp.Pages 758-764Vítor D. 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Amr, Dierk Knittel, Eckhard Schollmeyer6.Synthesis and thermo-physical properties of chitosan/poly(dl-lactide-co-glycolide) composites prepared by thermally induced phase separationPages 775-783Santos Adriana Martel-Estrada, Carlos Alberto Martínez-Pérez, José Guadalupe Chacón-Nava, Perla Elvia García-Casillas, Imelda Olivas-Armendarizparison of the immunological activities of arabinoxylans from wheat bran with alkali and xylanase-aided extractionPages 784-789Sumei Zhou, Xiuzhen Liu, Yan Guo, Qiang Wang, Daiyin Peng, Li Cao8.Nano-in-micro alginate based hybrid particlesPages 790-798Abhijeet Joshi, R. Keerthiprasad, Rahul Dev Jayant, Rohit Srivastava9.The effects of reaction conditions on block copolymerization of chitosan and poly(ethylene glycol)Pages 799-804F. Ganji, M.J. Abdekhodaie10.Thermal behaviour and interactions of cassava starch filled with glycerolplasticized polyvinyl alcohol blendsPages 805-810W.A.W.A. Rahman, Lee Tin Sin, A.R. Rahmat, A.A. Samad11.Banana fibers and microfibrils as lignocellulosic reinforcements in polymer compositesPages 811-819Maha M. Ibrahim, Alain Dufresne, Waleed K. El-Zawawy, Foster A. Agblevor12.Variability of biomass chemical composition and rapid analysis using FT-NIR techniquesPages 820-829Lu Liu, X. Philip Ye, Alvin R. Womac, Shahab Sokhansanj13.TEMPO oxidation of gelatinized potato starch results in acid resistant blocks of glucuronic acid moietiesPages 830-838Ruud ter Haar, Johan W. Timmermans, Ted M. Slaghek, Francisca E.M. Van Dongen, HenkA. Schols, Harry Gruppen14.Development of films based on quinoa (Chenopodium quinoa, Willdenow) starch Pages 839-848Patricia C. Araujo-Farro, G. Podadera, Paulo J.A. Sobral, Florencia C. Menegalli 15.Polysaccharide determination in protein/polysaccharide mixtures for phase-diagram constructionPages 849-854Jacob K. Agbenorhevi, Vassilis Kontogiorgos16.Calorimetric and light scattering study of interactions and macromolecular properties of native and hydrophobically modified hyaluronanPages 855-863Martin Chytil, Sabina Strand, Bjørn E. Christensen, Miloslav Pekař17.Spray drying of nopal mucilage (Opuntia ficus-indica): Effects on powder properties and characterizationPages 864-870F.M. León-Martínez, L.L. Méndez-Lagunas, J. Rodríguez-Ramírez18.Preparation and evaluation of nanoparticles of gum cordia, an anionic polysaccharide for ophthalmic deliveryPages 871-877Monika Yadav, Munish Ahuja19.Functional modification of agarose: A facile synthesis of a fluorescent agarose–guanine derivativePages 878-884Mihir D. Oza, Ramavatar Meena, Kamalesh Prasad, P. Paul, A.K. Siddhanta20.Characterization of maize amylose-extender (ae) mutant starches. Part III: Structures and properties of the Naegeli dextrinsPages 885-891Hongxin Jiang, Sathaporn Srichuwong, Mark Campbell, Jay-lin Jane21.Multistage deacetylation of chitin: Kinetics studyPages 892-896N. Yaghobi, F. Hormozi22.Sulfated modification, characterization and structure–antioxidantrelationships of Artemisia sphaerocephala polysaccharidesPages 897-905Junlong Wang, Hongyun Guo, Ji Zhang, Xiaofang Wang, Baotang Zhao, Jian Yao, Yunpu Wang23.Magnetic chitosan/iron (II, III) oxide nanoparticles prepared by spray-drying Pages 906-910Hsin-Yi Huang, Yeong-Tarng Shieh, Chao-Ming Shih, Yawo-Kuo Twu24.Effect of adding a small amount of high molecular weight polyacrylamide on properties of oxidized cassava starchPages 911-918Yan Liu, Xu-chao Lv, Xiao Hu, Zhi-hua Shan, Pu-xin Zhu25.Preparation of nanofibrillar carbon from chitin nanofibersPages 919-924M. Nogi, F. Kurosaki, H. Yano, M. Takano26.Preparation and characterization of cellulose acetate–Fe2O3 composite nanofibrous materialsPages 925-930Costas Tsioptsias, Kyriaki G. Sakellariou, Ioannis Tsivintzelis, Lambrini Papadopoulou, Costas Panayiotou27.Synthesis, characteristic and antibacterial activity of N,N,N-trimethyl chitosan and its carboxymethyl derivativesPages 931-936Tao Xu, Meihua Xin, Mingchun Li, Huili Huang, Shengquan Zhou28.Fast compositional analysis of ramie using near-infrared spectroscopyPages 937-941Wei Jiang, Guangting Han, Yuanming Zhang, Mengmeng Wang29.Structure characterization of polysaccharide isolated from the fruiting bodies of Tricholoma matsutakePages 942-947Xiang Ding, Su Feng, Mei Cao, Mao-tao Li, Jie Tang, Chun-xiao Guo, Jie Zhang, Qun Sun, Zhi-rong Yang, Jian Zhao30.New insights into viscosity abnormality of sodium alginate aqueous solution Pages 948-952Dan Zhong, Xin Huang, Hu Yang, Rongshi Cheng31.Structural characterization and anti-inflammatory activity of two water-soluble polysaccharides from Bellamya purificataPages 953-960Hong Zhang, Lin Ye, Kuiwu WangComputer Science, Artificial Intelligence（计算机科学，人工智能）ARTIFICIAL INTELLIGENCEVolume 174, Issue 11, Pages 639-766 (July 2010)1.Editorial BoardPage IFC2.Partial observability and learnabilityPages 639-669Loizos Michael3.Monte Carlo tree search in KriegspielPages 670-684Paolo Ciancarini, Gian Piero Favini4.Learning conditional preference networksPages 685-703Frédéric Koriche, Bruno Zanuttini5.Planning to see: A hierarchical approach to planning visual actions on a robot using POMDPsPages 704-725Mohan Sridharan, Jeremy Wyatt, Richard Dearden6.Analysis of a probabilistic model of redundancy in unsupervised information extractionPages 726-748Doug Downey, Oren Etzioni, Stephen Soderland7. Designing competitions between teams of individualsPages 749-766Pingzhong Tang, Yoav Shoham, Fangzhen LinEnergy & Fuels（能源和燃料）APPLIED ENERGYVolume 87, Issue 8, Pages 2427-2768 (August 2010)1.IFCPage IFC2. Energy balance analysis of wind-based pumped hydro storage systems in remote island electrical networksPages 2427-2437J.K. Kaldellis, M. Kapsali, K.A. Kavadias3.Energy auditing and energy conservation potential for glass worksPages 2438-2446Yingjian Li, Jiezhi Li, Qi Qiu, Yafei Xu4.Energy demand and comparison of current defrosting technologies of frozen raw materials in defrosting tunnelsPages 2447-2454Marek Bezovsky, Michal Stricik, Maria Prascakova5.Guidelines for clockspeed acceleration in the US natural gas transmission industry Pages 2455-2466Ruud Weijermars6.Multi-objective self-adaptive algorithm for highly constrained problems: Novel method and applicationsPages 2467-2478Abdelaziz Hammache, Marzouk Benali, François Aubé7.Stochastic interest rates in the analysis of energy investments: Implications on economic performance and sustainabilityPages 2479-2490Athanasios Tolis, Aggelos Doukelis, Ilias Tatsiopoulos8.Effects of the PWM carrier signals synchronization on the DC-link current in back-to-back convertersPages 2491-2499L.G. González, G. Garcerá, E. Figueres, R. González9.Efficiency improvement of the DSSCs by building the carbon black as bridge in photoelectrodePages 2500-2505Chen-Ching Ting, Wei-Shi Chao10.Integer programming with random-boundary intervals for planning municipal power systemsPages 2506-2516M.F. Cao, G.H. Huang, Q.G. Lin11. Modeling the relationship between the oil price and global food prices Pages 2517-2525Sheng-Tung Chen, Hsiao-I Kuo, Chi-Chung Chen12.Marginal production in the Gulf of Mexico – II. Model resultsPages 2526-2534Mark J. Kaiser, Yunke Yu13.Marginal production in the Gulf of Mexico – I. Historical statistics & model frameworkPages 2535-2550Mark J. Kaiser14.Assessment of forest biomass for use as energy. GIS-based analysis of geographical availability and locations of wood-fired power plants in PortugalPages 2551-2560H. Viana, Warren B. Cohen, D. Lopes, J. Aranha15.Alkaline catalyzed biodiesel production from moringa oleifera oil with optimized production parametersPages 2561-2565G. Kafuku, M. Mbarawae of two-component Weibull mixtures in the analysis of wind speed in the Eastern MediterraneanPages 2566-2573S.A. Akdağ, H.S. Bagiorgas, G. Mihalakakou17.Evaluation of wind energy investment interest and electricity generation cost analysis for TurkeyPages 2574-2580Seyit Ahmet Akdağ, Önder Güler18.The role of demand-side management in the grid integration of wind power Pages 2581-2588Pedro S. Moura, Aníbal T. de Almeida19.Synthesis of biodiesel from waste vegetable oil with large amounts of free fatty acids using a carbon-based solid acid catalystPages 2589-2596Qing Shu, Jixian Gao, Zeeshan Nawaz, Yuhui Liao, Dezheng Wang, Jinfu Wang20.A study on the overall efficiency of direct methanol fuel cell by methanol crossover currentPages 2597-2604Sang Hern Seo, Chang Sik Lee21.Study of heat transfer between an over-bed oil burner flame and a fluidized bed during start-up: Determination of the flame to bed convection coefficientPages 2605-2614Vijay Jain, Dominic Groulx, Prabir Basu22.Predictive tools for the estimation of downcomer velocity and vapor capacity factor in fractionatorsPages 2615-2620Alireza Bahadori, Hari B. Vuthaluru23. Monitoring strategies for a combined cycle electric power generatorPages 2621-2627Joshua Finn, John Wagner, Hany Bassily24. Combustion and heat transfer at meso-scale with thermal recuperationPages 2628-2639V. Vijayan, A.K. Gupta25.Part-load characteristics of direct injection spark ignition engine using exhaust gas trapPages 2640-2646Yun-long Bai, Zhi Wang, Jian-xin Wang26.Gas–liquid absorption reaction between (NH4)2SO3 solution and SO2 for ammonia-based wet flue gas desulfurizationPages 2647-2651Xiang Gao, Honglei Ding, Zhen Du, Zuliang Wu, Mengxiang Fang, Zhongyang Luo, Kefa Cen27. Direct contact PCM–water cold storagePages 2652-2659Viktoria Martin, Bo He, Fredrik Setterwall28. Fatty acid eutectic/polymethyl methacrylate composite as form-stable phase change material for thermal energy storagePages 2660-2665Lijiu Wang, Duo Meng29.Thermal characteristics of shape-stabilized phase change material wallboard with periodical outside temperature wavesPages 2666-2672Guobing Zhou, Yongping Yang, Xin Wang, Jinming Cheng30. Study on a compact silica gel–water adsorption chiller without vacuum valves: Design and experimental studyPages 2673-2681C.J. Chen, R.Z. Wang, Z.Z. Xia, J.K. Kiplagat, Z.S. Lu31. 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Experimental flow field characteristics of OFA for large-angle counter flow of fuel-rich jet combustion technologyPages 2737-2745Weidong Fan, Zhengchun Lin, Youyi Li, Mingchuan ZhangEngineering, Electrical & Electronic（电机及电子工程）MICROELECTRONIC ENGINEERINGVolume 87, Issue 9, Pages 1655-1808 (November 2010)1. Inside Front Cover - Editorial BoardPage IFC2 PrefacePage 1655Joel Barnett3.In0.53Ga0.47As(1 0 0) native oxide removal by liquid and gas phase HF/H2O chemistriesPages 1656-1660F.L. Lie, W. Rachmady, A.J. Muscat4.A study of the interaction of gallium arsenide with wet chemical formulations using thermodynamic calculations and spectroscopic ellipsometryPages 1661-1664J. Price, J. Barnett, S. Raghavan, M. Keswani, R. Govindarajan5.The removal of nanoparticles from sub-micron trenches using megasonicsPages 1665-1668Pegah Karimi, Taehoon Kim, Juan Aceros, Jingoo Park, Ahmed A. Busnaina6.In-line control of Si loss after post ion implantation stripPages 1669-1673D. Shamiryan, D. Radisic, W. Boullart7.Removal of post-etch 193 nm photoresist in porous low-k dielectric patterning using UV irradiation and ozonated waterPages 1674-1679E. Kesters, Q.T. Le, M. Lux, L. Prager, G. Vereecke8.Repair of plasma-damaged p-SiOCH dielectric films in supercritical CO2Pages 1680-1684Jae Mok Jung, Hong Seok Kwon, Won-Ki Lee, Byung-Chun Choi, Hyun Gyu Kim, Kwon Taek Lim9.Development of compatible wet-clean stripper for integration of CoWP metal cap in Cu/low-k interconnectsPages 1685-1688Aiping Wu, Eugene Baryschpolec, Madhukar Rao, Matthias Schaller, Christin Bartsch, Susanne Leppack, Andreas Ott10.Dissolution and electrochemical impedance spectroscopy studies of thin copper oxide films on copper in semi-aqueous fluoride solutionsPages 1689-1695N. Venkataraman, S. Raghavan11.The sacrificial oxide etching of poly-Si cantilevers having high aspect ratios using supercritical CO2Pages 1696-1700Ha Soo Hwang, Jae Hyun Bae, Jae Mok Jung, Kwon Taek Lim12.Monitoring wafer cleanliness and metal contamination via VPD ICP-MS: Case studies for next generation requirementsPages 1701-1705Meredith Beebe, Scott Anderson13.Fabrication of a two-step Ni stamp for blind via hole application on PWB Pages 1707-1710In-Soo Park, Jin-Soo Kim, Seong-Hun Na, Seung-Kyu Lim, Young-Soo Oh, Su-Jeong Suh 14.Arrays of metallic nanocones fabricated by UV-nanoimprint lithographyPages 1711-1715Juha M. Kontio, Janne Simonen, Juha Tommila, Markus Pessa15.Synthesis, characterization of CeO2@SiO2 nanoparticles and their oxide CMP behaviorPages 1716-1720Xiaobing Zhao, Renwei Long, Yang Chen, Zhigang Chen16.Development of a triangular-plate MEMS tunable capacitor with linear capacitance–voltage responsePages 1721-1727M. Shavezipur, S.M. Hashemi, P. Nieva, A. 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Anodic bonding of glass–ceramics to stainless steel coated with intermediate SiO2 layerPages 1741-1746Dehua Xiong, Jinshu Cheng, Hong Li, Wei Deng, Kai Ye20.Preparation of silica/ceria nano composite abrasive and its CMP behavior on hard disk substratePages 1747-1750Hong Lei, Fengling Chu, Baoqi Xiao, Xifu Tu, Hua Xu, Haineng Qiu21.Investigation on the controllable growth of monodisperse silica colloid abrasives for the chemical mechanical polishing applicationPages 1751-1755XiaoKai Hu, Zhitang Song, Haibo Wang, Weili Liu, Zefang Zhang22.Fabrication and electrical characteristics of ultrathin (HfO2)x(SiO2)1−x films by surface sol–gel method and reaction-anneal treatmentPages 1756-1759You-Pin Gong, Ai-Dong Li, Chao Zhao, Yi-Dong Xia, Di Wu23.Frequency properties of on-die power distribution network in VLSI circuits Pages 1760-1763Pavel Livshits, Yefim Fefer, Anton Rozen, Yoram Shapira24.Analytical modelling for the current–voltage characteristics of undoped or lightly-doped symmetric double-gate MOSFETsPages 1764-1768A. 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BuchaillotPreview PDF (303 K) | Related Articles30.Automated optical method for ultrasonic bond pull force estimationPages 1796-1804Henri Seppänen, Robert Schäfer, Ivan Kassamakov, Peter Hauptmann, Edward Hæggström31.Oxygen incorporation in TiN for metal gate work function tuning with a replacement gate integration approachPages 1805-1807Zilan Li, Tom Schram, Thomas Witters, Joshua Tseng, Stefan De Gendt, Kristin De MeyerEngineering, Industrial（工业工程）APPLIED ERGONOMICSVolume 41, Issue 5, Pages 643-718 (September 2010)1.Editorial BoardPage IFC2.Editorial for special issue of applied ergonomics on patient safetyPages 643-644Pascale Carayon, Peter Buckle3.Systems mapping workshops and their role in understanding medication errors in healthcarePages 645-656P. Buckle, P.J. Clarkson, R. Coleman, J. Bound, J. Ward, J. 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Tunnelling and Underground Space Technology

Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Stress redistribution during tunneling should be three-dimensional (3D), with the exception of a two-dimensional (2D) plain strain condition. However, ideal assumptions of a circular-shaped tunnel and the 2D condition are typically invoked for the initial analytic solution. This stress distribution solution for a circular tunnel has been reported throughout the last century. Kirsch (1898) solved analytically the distribution of stress and displacement in an unsupported circular tunnel. His solution relied upon elasticity theory, using the plane stress condition with different K0 values. Bray (1967) proposed a theoretical model to permit analysis of the extent of failure, the plastic zone, based on Mohr–Coulomb failure criterion. Ladanyi (1974) discussed stress distribution around a circular opening in a hydrostatic stress ﬁeld, and within annular failed rock generated in the excavation periphery, using Mohr–Coulomb elasto-plastic theory.

自紧身管说明书

1 引言1．1 自紧身管的发展自紧技术的产生和发展是随着现代战争的需要和化工产品的需求而迅速发展起来的，并且正广泛的应用于国防工业、化工工业和其它民用工业中。

压力容器和身管自紧是通过一定的工艺手段在产品加工的过程中对其内壁预加载荷，使容器壁达到一定的塑性变形引起有益的预应力(残余应力)，这个残余应力能部分地抵消压力容器和身管在服役下的工作应力，并延迟内壁表面疲劳裂纹的扩展，从而提高压力容器和身管弹性强度和疲劳寿命。

由于自紧技术所存在的这些优势，用自紧技术来提高厚壁圆筒弹性强度的方法已被广泛采用，这项技术为国家的发展起到了举足轻重的作用[1]。

众所周知，现代战争是高新技术条件下的局部战争，它对各类武器装备提出了愈来愈高的性能要求。

像坦克、火炮、轻武器之类的常规兵器，军方对其威力、寿命均提出了更高的要求，为达到此目的，单纯依靠和采用性能更好的合金钢做身管材料的传统做法，已不是一条理想的技术途径。

目前，国内外普遍广为采用的方法是对身管进行超高压自紧，以期提高枪、炮身管的承载能力和寿命[2]。

自紧炮身的身管在制造时要对其内膛施以高压，使内壁部分产生塑性变形。

在内压去除以后，由于管壁各层塑性变形不一致，在各层之间形成相互作用力，使内层产生压应力而外层产生拉应力，就好像无数多层的筒紧身管一样。

在发射时，由于内壁产生与发射时符号相反的预应力，因此，发射时身管壁应力趋于均匀一致，提高了身管的强度，我们将内膛高压处理过的身管称为自紧身管。

早在19世纪末20世纪初，人们就已经将自紧原理作为提高炮身强度的一种手段,那时大多采用液压自紧的方法,身管钢材的ςs约为274.4~343Mpa，第二次世界大战中,一些资本主义国家在炮身制造中较为广泛地采用自紧技术,自紧工艺依然为液压法,此时材料的ςs提高到441～548.8MPa。

二次大战末期和战后,由于冶金技术的发展,炮钢的强度不断提高,又因自紧工艺比较复杂,所以单筒身管得到了广泛采用,而自紧工艺没有明显的进展。

国家基金学科代码

学科代码学科名称A数理科学B化学科学C生命科学D地球科学E工程与材料科学F信息科学G管理科学A01数学A0101基础数学A010101数论A01010101解析数论A01010102代数数论A01010103丢番图分析A01010104超越数论A01010105模型式与模函数论A01010106数论的应用A010102代数学A01010201群论A01010202群表示论A01010203李群A01010204李代数A01010205代数群A01010206典型群A01010207同调代数A01010208代数K理论A01010209Kac-Moody代数A01010210环论A01010211代数(可除代数)A01010212体A01010213编码理论与方法A01010214序结构研究A010103几何学A01010301整体微分几何A01010302代数几何A01010303流形上的分析A01010304黎曼流形与洛仑兹流形A01010305齐性空间与对称空间A01010306调和映照及其在理论物理中的应用A01010307子流形理论A01010308杨--米尔斯场与纤维丛理论A01010309辛流形A010104拓扑学A01010401微分拓扑A01010402代数拓扑A01010403低维流形A01010404同伦论A01010405奇点与突变理论A01010406点集拓扑A010105函数论A01010501多复变函数论A01010502复流形A01010503复动力系统A01010504单复变函数论A01010505Rn中的调和分析的实方法A01010506非紧半单李群的调和分析A01010507函数逼近论A010106泛函分析A01010601非线性泛函分析A01010602算子理论A01010603算子代数A01010604泛函方程A01010605空间理论A01010606广义函数A010107常微分方程A01010701泛函微分方程A01010702特征与谱理论及其反问题A01010703定性理论A01010704稳定性理论、分支理论A01010705混沌理论A01010706奇摄动理论A01010707复域中的微分方程A01010708动力系统A010108偏微分方程A01010801连续介质物理与力学、及反应扩散等应用领域中A01010802几何与数学物理中的偏微分方程A01010803微局部分析与一般偏微分算子理论A01010804非线性椭圆（和抛物）方程研究中的新方法和新A01010805混合型及其它带奇性的方程A01010806非线性波、非线性发展方程和无穷维动力系统A010109数学物理A01010901规范场论A01010902引力场论的经典理论与量子理论A01010903孤立子理论A01010904统计力学A01010905连续介质力学等方面的数学问题A010110概率论A01011001马氏过程A01011002随机过程A01011003随机分析A01011004随机场A01011005鞅论A01011006极限理论A01011007概率论在调和分析、几何及微分方程等方面的应A01011008在物理、生物、化学管理中的概率论问题A01011009平稳过程A010111数理逻辑与数学基础A01011101递归论A01011102模型论A01011103证明论A01011104公理集合证A01011105数理逻辑在人工智能及计算机科学中的应用A0102应用数学A010201数理统计A01020101抽样调查与抽样方法A01020102试验设计A01020103时间序列分析及其算法研究A01020104多元分析及其算法研究A01020105数据分析及其图形处理A01020106非参数统计方法A01020107应用统计中的基础性工作A01020108统计线性模型A01020109参数估计方法A01020110随机过程的统计理论及方法A01020111蒙特卡洛方法 (统计模拟方法) A010202运筹学A01020201线性与非线性规划A01020202整数规划A01020203动态规划A01020204组合最优化A01020205随机服务系统A01020206对策论A01020207不动点算法A01020208随机最优化A01020209多目标规划A01020210不可微最优化A01020211可靠性理论A010203控制论A01020301有限维非线性系统A01020302分布参数系统的控制理论A01020303随机系统的控制理论A01020304最优控制理论与算法A01020305参数辨识与适应控制A01020306线性系统理论的代数与几何方法A01020307控制的计算方法A01020308微分对策理论A01020309稳健控制A010204若干交叉学科A01020401信息论及应用A01020402经济数学A01020403生物数学A01020404不确定性的数学理论A01020405分形论及应用A010205计算机的数学基础A01020501可解性与可计算性A01020502机器证明A01020503计算复杂性A01020504VLSI的数学基础A01020505计算机网络与并行计算A010206组合数学A01020601组合计数A01020602组合设计A01020603图论A01020604线性计算几何A01020605组合概率方法A0103计算数学与科学工程计算A010301偏微分方程数值计算A01030101初边值问题数值解法及应用A01030102非线性微分方程及其数值解法A01030103边值问题数值解法及其应用A01030104有限元、边界元数值方法A01030105变分不等式的数值方法A01030106辛几何差分方法A01030107数理方程反问题的数值解法A010302常微分方程数值解法及其应用A01030201二点边值问题A01030202STIFF 问题研究A01030203奇异性问题A01030204代数微分方程A010303数值代数A01030301大型稀疏矩阵求解A01030302代数特征值问题及其反问题A01030303非线性代数方程A01030304一般线性代数方程组求解A01030305快速算法A010304函数逼近A01030401多元样条A01030402多元逼近A01030403曲面拟合A01030404有理逼近A01030405散乱数据插值A010305计算几何A01030501曲面造型A01030502曲面光滑拼接A01030503曲面设计A01030504体素拼接A01030505几何问题的计算机实现A010306新型算法A01030601并行算法A01030602多重网格技术A01030603自适应方法A01030604区间分析法及其应用A02力学A0201一般力学A020101分析力学A020102动力系统的分岔、混沌A020103运动稳定性与控制A020104非线性振动与控制A020105多体动力学A020106转子动力学A020107弹道力学和飞行力学A020108理性力学A020109力学中的反问题A020110力学发展史学A0202固体力学A020201弹性力学与塑性力学A020202疲劳与断裂力学A020203损伤、破坏机理和微结构演化A020204本构关系A020205复合材料力学A020206新型材料的力学问题A020207极端条件下的材料和结构A020208微机电系统中的固体力学问题A020209岩体力学和土力学A020210冲击动力学A020211结构力学A020212结构振动与噪声A020213结构优化和可靠性分析A020214制造工艺力学A020215实验固体力学A020216计算固体力学A020217流固耦合作用A0203流体力学A020301流动的稳定性A020302湍流A020303水动力学A020304空气动力学A020305分层流A020306非平衡流A020307渗流A020308多相流A020309非牛顿流A020310内流A020311化工流体力学A020312工业空气动力学A020313微重力流体力学A020314微机电系统中的流体力学问题A020315流动噪声与控制A020316稀薄气体力学A020317实验流体力学A020318计算流体力学A0204交叉与边缘领域的力学A020401物理力学A020402爆炸力学A020403环境流体力学A020404生物力学A020405电磁流体力学和等离子体动力学A03天文学A0301宇宙学A0302星系和类星体A0303恒星物理与星际物质A0304太阳和太阳系A0305射电天文A0306空间天文A0307理论天体物理A0308天体测量和天文地球动力学A0309天体力学和人造卫星动力学A0310时间、频率A0311天文仪器A0312天文学史A0313其它A04物理学(Ⅰ)A0401凝聚态物性Ｉ：结构、力学和热学性质A040101液体和固体结构；晶体、非晶、准晶的物质结构A040102凝聚态物质的力学和声学性质A040103晶格动力学和晶体统计学A040104状态方程、相平衡和相变A040105凝聚态物质的热学性质A040106凝聚态物质的输运性质A040107量子流体和固体;液态氦和固态氦A040108表面和界面;薄膜和晶须；人工微结构（结构和 A0402凝聚态物性Ⅱ：电子结构、电学、磁学和光学性A040201电子态A040202凝聚态物质中的电子输运A040203表面.界面.薄膜和低维系统的电子结构及电学性A040204超导电性A040205磁学性质A040206凝聚态物质的磁共振和弛豫;穆斯堡尔效应A040207介电性质A040208光学性质、凝聚态物质的波谱学、物质与粒子的A040209液体和固体的电子发射和离子发射;碰撞现象A040210与凝聚态物理有关的交叉学科A0403原子和分子物理A040301原子和分子理论A040302原子光谱及原子与光子相互作用A040303分子光谱及分子与光子相互作用A040304原子和分子碰撞过程及相互作用A040305研究原子和分子性质的实验设备和技术A040306特殊原子和分子的研究A040307与原子、分子有关的其它物理问题和交叉学科A0404光学A040401光在均匀介质中的传播A040402光在非均匀介质中的传播A040403像的形成和分析A040404全息照相A040405量子光学A040406微波激射A040407激光发射过程A040408激光系统和激光与物质相互作用A040409非线性光学A040410光学材料中物理问题及固体发光A040411光源和光学标准A040412光学透镜和反射镜系统A040413光学器件的原理A040414与光学有关的其它物理问题和交叉学科A0405声学A040501普通线性声学A040502非线性声学和强声学A040503航空声学和大气声学A040504水声A040505超声、量子声学和声的物理效应A040506次声A040507噪声、噪声效应及其控制A040508建筑声学A040509声的信号处理A040510声全息照相A040511语言声学A040512乐声A040513声的测量及专用仪器A040514声的转换原理A040515与声学有关的其它物理问题和交叉学科A05物理学(Ⅱ)A0501基础物理学A050101物理教育学及物理学史A050102物理学中的数学问题A050103经典物理学和量子理论A050104相对论与引力A050105热力学与统计物理学 (含混沌)A050106测量科学、一般实验技术和测试系统A0502粒子物理学和场论A050201粒子基本特性及粒子物理一般问题A050202场论中的基本问题和新方法A050203对称性及对称破缺A050204量子色动力学、强相互作用和强子物理A050205电－弱相互作用及其唯象学A050206非标准模型及其唯象学A050207新粒子A050208粒子的延展体理论A050209宇宙射线和超高能现象A050210粒子物理与宇宙学A0503核物理A050301原子核特性A050302原子核结构模型的理论研究A050303原子核统计理论研究A050304原子核高激发态、高自旋态和超形变A050305带奇异数系统、奇异核和超核A050306核内非核子自由度A050307核力与少体系统A050308强子、轻子与核相互作用A050309核物质理论及核多体方法A050310核衰变、核裂变、核聚变A050311低能核反应与散射A050312重离子核物理A050313中高能核物理A050314核天体物理A050315核数据分析和计算机模拟A0504核技术及其应用A050401离子束与物质相作用和辐照损伤A050402核分析技术 ( RBS、PIXE、NRA )A050403穆斯堡尔谱学及其应用A050404正电子湮灭技术及其应用A050405中子衍射及其应用A050406扰动角关联及其应用A050407核磁共振及其应用A050408中子活化和同位素示踪技术A050409离子束材料改性A050410核技术在地学中的应用A050411核技术在医学中的应用A050412核技术在农业中的应用A050413核技术在工业中的应用A050414核科学和其它学科的交叉A0505粒子物理与核物理实验设备A050501加速器原理和关键技术A050502离子源和电子枪A050503预加速装置和加速器部件A050504束流输运和性能测量A050505真空和超高真空技术A050506反应堆A050507辐射探测方法A050508探测技术和谱仪A050509辐射剂量及其防护A050510核电子学A0506等离子体物理A050601等离子体中的基本过程与特性A050602等离子体的加热、约束和辐射A050603等离子体动力学与电磁流体力学A050604等离子体中的混沌、孤立波、湍流等非线性现象A050605等离子体的模拟、数值方法和软件A050607等离子体诊断技术A050608等离子体与固体相互作用A050609激光束、粒子束、微波与等离子体A050610低气压低温等离子体的应用A050611热平衡低温等离子体的应用A050612非中性等离子体A050613强耦合等离子体A050614空间等离子体B01无机化学B0101无机合成和制备化学B010101合成技术B010102合成化学B010103特殊聚集态制备B0102丰产元素化学B010201稀土化学B010202钨化学B010203钼化学B010204锡化学B010205锑化学B010206钛化学B010207钒化学B010208稀有碱金属化学B010209稀散元素化学B0103配位化学B010301固体配位化学B010302溶液配位化学B010303金属有机化学B010304原子簇化学B010305功能配合物化学B0104生物无机化学B010401金属酶化学及其化学模拟B010402金属蛋白化学及其化学模拟B010403生物体内微量元素的状态及功能、受体底物相互B010404金属离子与生物膜的作用及其机理B010405金属离子与核酸化学B0105固体无机化学B010501缺陷化学B010502固体反应B010503固体表面化学B010504无机固体材料化学B0106分离化学B010601萃取化学B010602无机色层B010603无机膜分离B0107物理无机化学B010701无机化合物结构与性质B010702理论无机化学B010703无机反应机制及反应动力学B010704熔盐化学及相平衡B0108同位素化学B010801同位素分离B010802同位素分析B010803同位素应用B0109放射化学B010901核燃料化学B010902超铀元素化学B010903裂片元素化学B010904放射性核素及其标记化合物的制备和应用B010905放射分析化学B010906放射性废物处理和综合利用B0110核化学B011001低能核化学B011002高能核化学B011003裂变化学B011004重离子核化学B011005核天体化学B02有机化学B0201有机合成B020101有机合成反应B020102新化合物和复杂化合物的设计与合成B020103高选择性有机合成试剂B020104不对称合成B0202金属有机及元素有机化学B020201有机磷化学B020202有机硅化学B020203有机硼化学B020204有机氟化学B020205金属有机化合物的合成及其应用B0203天然有机化学B020301甾体及萜类化学B020302糖类黄酮类化学B020303中草药有效成份B020304具有重要应用价值的天然产物的研究B0204物理有机化学B020401活泼中间体化学B020402化学动态学B020403有机光化学B020404立体化学B020405有机分子结构与活性关系B020406具有光、电、磁特性的化合物研究B020407计算有机化学B0205药物化学B020501新药物分子设计和合成B020502药物构效关系B0206生物有机化学B020601多肽化学B020602核酸化学B020603仿生及模拟酶B020604天然酶的化学修饰及应用B020605生物合成及生物转化B0207有机分析B020701新化合物和复杂化合物的结构研究B020702有机分析、分离新方法新技术研究B020703有机化合物结构波谱学B0208应用有机化学B020801除草剂B020802植物生长促进剂B020803害虫引诱剂、昆虫信息素B020804高效、低毒、低抗性农药B020805食品化学B020806香料化学B020807染料化学B03物理化学B0301结构化学B030101体相静态结构B030102表面结构B030103溶液结构B030104动态结构B030105谱学B030106结构化学方法和理论B0302量子化学B030201基础量子化学B030202应用量子化学B0303催化B030301多相催化B030302均相催化B030303人工酶催化B030304光催化B0304化学动力学B030401宏观反应动力学B030402分子动态学B030403反应途径和过渡态B030404快速反应动力学B030405结晶过程动力学B0305胶体与界面化学B030501表面活性剂B030502分散体系B030503流变性能B030504界面吸附现象B030505超细粉和颗粒B0306电化学B030601电极过程及其动力学B030602腐蚀电化学B030603熔盐电化学B030604光电化学B030605半导体电化学B030606生物电化学B030607表面电化学B030608电化学技术B030609电催化B0307光化学B030701激光闪光光解B030702激发态化学B030703电子转移光化学、光敏化B030704光合作用B030705大气光化学B0308热化学B030801热力学参数B030802相平衡B030803电解质溶液化学B030804非电解质溶液化学B030805生物热化学B030806量热学B0309高能化学B030901辐射化学B030902等离子体化学B030903激光化学B0310计算化学B031001化学信息的运筹B031002计算模拟B031003计算控制B031004计算方法的最优化B04高分子科学B0401高分子合成B040101催化剂、聚合反应及聚合方法B040102高分子设计和合成B040103新单体及单体的新合成方法B040104聚合反应动力学B040105高分子光化学、辐射化学、等离子体化学B040106微生物参与的聚合反应、酶催化聚合反应B0402高分子反应B040201高分子老化、降解、交联B040202高分子接枝、嵌段改性B040203高分子功能化改性B040204粒子注入、辐射、激光等方法对高分子的改性B0403功能高分子B040301吸附、分离、离子交换、螯合功能的高分子B040302用于有机合成、医疗、分析等领域的高分子试剂B040303医用高分子、高分子药物B040304液晶态高分子B040305有机固体电子材料、磁性高分子B040306储能、换能、敏感材料及高分子催化剂B040307高分子功能膜B040308微电子材料、分子组装材料及器件B0404天然高分子B0405高分子物理及高分子物理化学B040501高分子溶液性质和溶液热力学B040502高分子链结构B040503高分子流变学B040504高聚物聚集态结构B040505高分子结构与性能关系B040506高聚物测试及表征方法B040507高分子材料的传质理论、强度理论、破坏机理B040508高分子多相体系B0406高分子理论化学B040601高分子聚合、交联、聚集态统计理论B040602数学、计算机方法在高分子凝聚态、分子动态学B0407聚合物工程及材料B040701聚合工程反应动力学及聚合反应控制B040702聚合物成型理论及成型方法B040703塑料、纤维、橡胶及成型研究B040704涂料、粘合剂及高分子肋剂B040705可生物降解薄膜B040706高分子润滑材料B040707其它领域中应用的高分子材料B040708高分子资源的再生和综合利用B05分析化学B0501色谱分析B050101气相色谱B050102液相色谱B050103薄层色谱B050104离子色谱B050105超临界液体色谱B050106毛细管电泳B0502电化学分析B050201伏安法B050202极谱法B050203化学修饰电极B050204库伦分析B050205光谱电化学分析B050206电化学传感器B0503光谱分析B050301原子发射光谱（包括ICP）B050302原子吸收光谱B050303原子荧光光谱B050304X射线荧光光谱B050305分子发射光谱（包括荧光光谱、磷光光谱和化学B050306紫外和可见光谱B050307光声光谱B050308红外光谱B050309拉曼光谱B0504波谱分析B050401顺磁B050402核磁B0505质谱分析B050501有机质谱B050502无机质谱B0506化学分析B050601萃取剂、显色剂、特殊功能试剂B050602色谱柱固定相、分离膜B0507热分析B0508放射分析B050801活化分析B050802质子荧光B0509生化分析及生物传感B0510联用技术B0511采样、分离和富集方法B0512化学计量学B051201分析方法与计算机技术B051202分析讯号与数据解析B0513表面、微区、形态分析B051301表面分析B051302微区分析B051303形态分析B06化学工程及工业化学B0601化工热力学和基础数据B060101状态方程与溶液理论B060102相平衡B060103热化学B060104化学平衡B060105热力学理论模型和分子系统的计算机模拟B060106热力学数据和数据库B0602传递过程B060201化工流体力学和传递性质B060202传热过程及设备B060203传质过程B060204流变学B060205颗粒学及浆料化学B0603分离过程及设备B060301蒸馏B060302蒸发与结晶B060303干燥B060304吸收B060305萃取B060306吸附与离子交换B060307机械分离过程B060308膜分离B060309其他分离技术B0604化学反应工程B060401化学(催化)反应动力学B060402反应器原理及传递特性B060403反应器的模型化和优化B060404流态化技术和多相流反应工程B060405固定床反应工程B060406聚合反应工程B060407电化学反应工程B060408生化反应工程B060409催化剂工程B0605化工系统工程B060501化学过程的控制与模拟B060502化工系统的优化B060503化工过程动态学B0606无机化工B060601常规无机化工B060602工业电化学(电解、电镀、化学腐蚀与防腐)B060603精细无机(无机颜料、吸附剂及表面活性剂等) B060604核化工与放射化工B0607有机化工B060701工业有机化工B060702精细有机化工（染料、涂料、感光剂、粘合剂与B0608生物化工与食品化工B060801生化反应动力学及反应器B060802发酵物的提取和纯化B060803生化过程的化工模拟及人工器官B060804酶化工B060805天然产物和农副产品的化学改性及深度加工B060806生物医药工程B0609能源化工B060901煤化工B060902石油化工B060903燃料电池B060904其它能源化工B0610化工冶金B061001矿产资源的利用研究B061002化学选矿与浸出B061003湿法冶金物理化学B061004等离子体冶金B061005化学涂层B0611环境化工B061101环境治理中的物理化学原理B061102三废治理技术中的化工基础B061103环境友好的化工过程B061104可持续发展环境化工的新概念B07环境化学B0701环境分析化学B070101环境中微量生命元素及其化合物的分离、分析技B070102环境中微量有机污染物的分离、分析技术B0702环境污染化学B070201大气污染化学B070202水污染化学B070203土壤污染化学B070204固体废弃物及放射性核素污染化学B0703污染控制化学B070301化学控制、防治新工艺、新技术及其基础性研究B070302无害化工艺(原料、能源和资源的综合利用)B0704污染生态化学B0705理论环境化学B0706全球性环境化学问题C01基础生物学C0101微生物学C010101微生物分类学C01010101细菌分类C01010102放线菌分类C01010103真菌分类C010102微生物生理及生物化学C010103微生物遗传育种C010104微生物方法学C010105微生物资源与生态C010106应用微生物学基础C01010601工业微生物C01010602农业、土壤微生物C010107病毒学C01010701动物病毒C01010702植物病毒C01010703微生物病毒C010108医学与兽医微生物学C01010801病毒C01010802立克次氏体(含衣原体)C01010803病原细菌(含支原体与螺旋体)C01010804病原真菌C0102植物学C010201植物结构学C01020101植物形态解剖学C01020102植物形态发生C01020103植物胚胎学C010202植物系统学与分类学C01020201植物系统发育与演化C01020202种子植物分类C01020203孢子植物分类C01020204植物区系与地理学C010203植物生理学C01020301光合作用及固氮C01020302呼吸作用、采后生理及次生物质代谢C01020303矿质营养及有机物质运输C01020304水分生理及抗性生理C01020305植物激素、生长发育及生殖生理C010204植物资源学C01020401植物资源评价C01020402植物引种驯化C01020403植物种质保存C01020404资源植物化学C0103动物学C010301动物形态学C010302动物胚胎学C010303动物分类学C010304动物生理学C010305动物行为学C010306动物进化和动物遗传学C010307动物地理学C010309保护生物学C010310实验动物学C0104生物化学和分子生物学C010401生物分子的结构与功能、合成机理及调节过程C01040101蛋白质与肽C01040102核酸C01040103酶C01040104多糖及糖复合物C01040105激素C01040106天然产物化学C010402生物膜的结构与功能C010403无机生物化学C0105生物物理学与生物医学工程学C010501理论生物物理C01050101量子生物学C01050102生物信息论和生物控制论C01050103生物功能的计算机模拟、生物数学C01050104生命现象的生物物理理论阐述C010502环境生物物理C01050201电离辐射生物物理C01050202光生物物理C01050203电磁辐射生物物理C01050204声生物物理C01050205其它环境因素对生物的作用C01050206自由基生物学C010503生物组织的物理特性C01050301生物光学C01050302生物电磁学C01050303生物声学C01050304生物力学和生物流变学C01050305生物组织的其它物理特性C010504分子生物物理C01050401生物分子结构的运动性C01050402生物分子的相互作用C01050403生物分子中的能量传递与电子传递C010505膜与细胞生物物理C010506感官与神经生物物理C010507生物物理技术C010508生物物理学研究中的新概念和新方法C010509人工器官C010510生物医学信号处理C010511生物医学测量技术C010512生物系统的建模与应用C010513生物医学超声C010514生物医学传感技术C010515生物材料C010516生物医学图象C010517其它生物医学工程学研究C0106神经生物学C010601分子神经生物学C010602细胞神经生物学C010603系统神经生物学C010604高级神经生物学C010605比较神经生物学C010606发育神经生物学C010607感觉系统神经生物学C0107生理学C010701循环生理学C010702血液生理学C010703呼吸生理学C010704消化生理学C010705泌尿生理学C010706内分泌生理学C010707特殊环境生理学C010708生殖生理学C010709年龄生理学C0108心理学C010801心理学的基本过程研究C010802认知心理学C010803生物心理学C010804医学心理学(含精神卫生学)C010805工程心理学C010806发展与教育心理学C010807运动心理学C0109细胞生物学及发育生物学C010901细胞结构与功能C010902细胞增长、分裂与分化C010903模型动植物及实验体系的建立C010904细胞工程(生物技术和细胞培养) 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C03010101环境卫生监测与卫生工程学C03010102环境流行病学C03010103环境毒理学C030102劳动卫生学与职业病学C030103营养与食品卫生学C03010301营养学C03010302食品卫生学C030104儿童与少年卫生学C030105毒理学C03010501分子、遗传毒理学。

On Maximizing the Second Smallest Eigenvalue of a

On Maximizing the Second Smallest Eigenvalue of aState-Dependent Graph LaplacianYoonsoo Kim and Mehran MesbahiAbstract—We consider theset consisting of graphs of ﬁxed order andweighted edges.The vertex set of graphsinwill correspond to point masses and the weight for an edge between two vertices is a functional of the distance b etween them.We pose the prob lem of ﬁnding the b est vertex po-sitional conﬁguration in the presence of an additional proximity constraint,in the sense that,the second smallest eigenvalue of the corresponding graph Laplacian is maximized.In many recent applications of algeb raic graph theory in systems and control,the second smallest eigenvalue of Laplacian has emerged as a critical parameter that inﬂuences the stability and ro-bustness properties of dynamic systems that operate over an information network.Our motivation in the present work is to “assign”this Laplacian eigenvalue when relative positions of various elements dictate the intercon-nection of the underlying weighted graph.In this venue,one would then b e able to “synthesize”information graphs that have desirable system theo-retic properties.Index Terms—Euclidean distance matrix,graph Laplacian,networked dynamic systems,semideﬁnite programming.I.I NTRODUCTIONConsider the set of n mobile elements as vertices of a graph,with the edge set determined by the relative positions between the respective elements.Speciﬁcally,we let G denote the set of graphs of order n with vertex set V =f 1;2;...;n g and edge set E =f e ij ;i =1;2;...;n 01;j =2;...;n;i <j g with the weight functionw :R 32R 3!R +assigning to eachedge e ij ,a function of the distance between the two nodes i and j .Thus,we havew ij :=w (x i ;x j )=f (k x i 0x j k )(1)for some f :R +!R +,with x i 2R 3denoting the position of ele-ment i .In our setup the function f in (1)will be required to exhibit a distinct behavior as it traverses the positive real line.For example,we will require that this function assume a constant value of one when the distance between i and j is less than some threshold and then rapidly drop to zero (or some small value)as the distance between these el-ements increases.Such a requirement parallels the behavior of an in-formation link in a wireless network where the signal power at the re-ceiver side is inversely proportional to the some power of the distance between transmitting and receiving elements [18].Using this frame-work,we now consider the conﬁguration problem3:maxx2(L G (x ))(2)where x :=[x 1;x 2;...;x n ]T 2R 3n is the vector of positions for the distributed system,the matrix L G (x )is a weighted graph Laplacian deﬁned element-wise as[L G (x )]ij:=0w ij ;if i =js =iw is ;if i =j(3)Manu script received April 29,2004;revised March18,2005and Au gu st 1,2005.Recommended by Associate Editor C.D.Charalambous.This work was supported by the National Science Foundation under Grant NSF/CMS-0301753.Y .Kim is withth e Department of Engineering,University of Leicester,Lei-ceister LE17RH,U.K.(e-mail:yk17@).M.Mesbahi is with the Department of Aeronautics and Astronautics,Univer-sity of Washington,Seattle,W A 98195-2400USA (e-mail:mesbahi@).Digital Object Identiﬁer 10.1109/TAC.2005.861710and 2(L G (x ))denotes the second smallest eigenvalue of the state-dependent Laplacian matrix L G (x )withits spectru m ordered as1(L G ) 2(L G ) 111 n (L G ):Furthermore,we restrict the feasible set of (2)by imposing the prox-imity constraintd ij :=k x i 0x j k 2 1;for all i =j (4)preventing the elements from getting arbitrary close to each other intheir desire to maximize 2(L G )in (2).The second smallest eigenvalue of the graph Laplacian L G ,also known as the algebraic connectivity of G [4],[8],[14],has emerged as an important parameter in many systems problems deﬁned over net-works [7],[12],[15],[17],[20].In fact,in several recent works [7],[17],[19],it has been observed that 2(L G )is a measure of stability and robustness of the networked dynamic system.This observation im-plies,for example,that small perturbations in the conﬁguration of the networked system will be attenuated back to its equilibrium state(s)witha rate th at is proportional to 2(L G ).When this important graph parameter is considered in a state-dependent setting as proposed in [15],the characterization of a distributed system states that maximize 2(L G )emerges as a natural optimization problem.In this venue how-ever,there are only a handful of studies in the literature that are re-lated to su cha grapheigenvalu e assignment problem (2).In particu lar we mention the work of Fallat and Kirkland [6]where a graph-the-oretical approachh as been proposed to extremize 2(L G )over the set of trees of ﬁxed diameter.Also related to the present work are those by Chung and Oden [5]pertaining to bounding the gap between the ﬁrst two eigenvalues of graph Laplacians,and Berman and Zhang [2]and Guattery and Miller [11],where,respectively,isoperimetric numbers of weighted graphs and graph embeddings are employed for lower bounding the second smallest Laplacian eigenvalue.We note that maximizing the second smallest eigenvalue of state dependent graph Laplacians over arbitrary graphconstraints is a difﬁcu lt compu tational problem [16].The contribution of this note is to propose an iterative greedy-type algorithm for problem (2)with a guaranteed local conver-gence behavior.Although the convergence of this algorithm is provably local in nature,extensive simulations suggest that it often converges to the global maximum when the initial graph is taken to be a path.The outline of the note is as follows.In Section II-A we delineate on the various possible choices for the edge weights for our state-depen-dent weighted Laplacians.Sections II-B and C are devoted to the main result of the note where an iterative semideﬁnite programming-based approachis proposed for th e solu tion of problem 3(2).A numerical example is then presented in Section III followed by a few concluding remarks.A few words on the notation.The 2-norm of vector x will be denoted by k x k .The spaces of n 2n real matrices and n 2n real symmetric matrices are designated by R n 2n and S n ,respectively;I n will be the n 2n identity matrix.The inequalities between symmetric matrices are interpreted in the sense of Löwner ordering,i.e.,A >B and A B indicate,respectively,the positive deﬁniteness and positive semideﬁ-niteness of the matrix difference A 0B .II.M ETHODAs we mentioned in Section I,the general formulation of the problem 3(2)does not readily hint at being tractable,in the sense of admit-ting an efﬁcient algorithm for its solution.Generally,maximizing the second smallest eigenvalue of a symmetric matrix subject to matrix inequalities,does not yield to a standard linear matrix inequality ap-proach[3]and,su bsequ ently,a solu tion procedu re th at relies solely0018-9286/$20.00©2006IEEEFig.1.Several candidates for the function f in(1)where =1and =2.on an interior point method[1].The previous complication however is alleviated in case of graph Laplacians,where the smallest eigenvalue 1(L G)is always zero withth e associated eigenvector of1composed of unit entries.This observation follows directly from the deﬁnition(3). Nevertheless,due to the nonlinear dependency of entries of L G on the relative distance d ij and the presence of constraints(4),the problem 3(2)assumes the form of a nonconvex optimization.In light of this fact,we will proceed to propose an iterative SDP-based approachfor this problem.However,before we proceed,we make a few remarks on some judicious choices for the function f in(1).The choice of f in(1)is not only guided by particular applications but also by numerical considerations.A few candidate functions are shown in Fig.1.Although there are a host of choices for f,for our analysis and numerical experimentation we have chosen to work with Type-IV functions(the lower right corner in Fig.1),where f assumes the formf(d ij)= ( 0d)=( 0 ); >0(5) given that d ij 1.1We note that f( 1)=1and f( 2)= .Among the advantages of working with functions(5)are their differentiability properties,as well as their ability to capture a situations that is of prac-tical relevance.In many su chsitu ations,th e strengthof an information link is inversely proportional to the relative distance and decays expo-nentially after a given threshold is passed.Furthermore,and possibly more importantly,functions(5)lead to a stable algorithm for our nu-merical experimentation;a representative set of examples is discussed in Section III.1We have also used functions of the form(1=d),where is a positive number and f( )= .Our simulation results in Section III turned out to be exactly the same for these functions as compared with those obtained using functions of the form(5).A.Maximizing 2(L G)Weﬁrst present a linear algebraic resu lt in conju nction withth e gen-eral problem of maximizing the second smallest eigenvalue of graph Laplacians.Proposition2.1:Consider the m-dimensional subspace P R n spanned by the vectors p i2R n,i=1;...;m.Denote P:=[p1;...;p m]2R n2m.Then,for M2S n one hasx T Mx>0for all nonzero x2Pif and only ifP T MP>0:(6)Proof:An arbitrary nonzero element x2P can be written asx= 1p1+ 2p2+111+ m p mfor some 1;...; m2R,not all zeros and,thus,x=P y,where y:=[ 1; 2;...; m]T.Consequently,theﬁrst inequality in(6)is equivalent to(P y)T M(P y)=y T P T MP y>0for all nonzero y2R m,or in other words,having P T MP>0;we note that P T MP2S m.Corollary2.2:For a graphLaplacian L G the constraint2(L G)>0(7) is equivalent toP T L G P>0(8)where P=[p1;p2;...;p n01],and the unit vectors p i2R n are chosen such thatp T i1=0;(i=1;2;...;n01)andp T i p j=0;(i=j):(9) Proof:It is well-known that for G2GL G 0and L G1=0(10)and,thereby,the smallest eigenvalue of L G is always zero and rank L G n01.This implies that(7)is equivalent to havingx T L G x>0;for all nonzero x21?(11) where1?:=f x2R n j1T x=0g:(12)In view of Proposition2.1,the condition(11)is equivalent to having P T L G P>0,with P denoting the matrix of vectors spanning the subspace1?.Without loss of generality,this subspace can be identiﬁed withth e basis u nit vectors satisfying(9).Corollary2.3:The problem3(2)is equivalent to3:maxx(13)s:t:d ij:=k x i0x j k2 1(14)P T L G(x)P I n01(15)where i=1;2;...;n01,j=2;...;n,i<j,and the pairwise or-thogonal unit vectors p0i s forming the columns of P span the subspace 1?(12).Proof:The proof follows from Corollary2.2.One of the consequences of Corollary2.3pertains to the following graphsynth esis problem2:determine graphs satisfying an upper bound on the number of their edges with maximum smallest second Laplacian eigenvalue.Although this problem will not be further considered in this note,we point out that it can be reformulated asmaxG2Gf j Trace L G ;P T L G P I n01gwhere P is deﬁned as in Corollary2.3and is twice the maximum number of edges allowed in the desired graph.In this venue,a compli-cation that needs to be further addressed pertains to the integrality of the entries of the sought matrix L G.B.Discrete and GreedyWe now proceed to view the problem3(2)in an iterative setting, where the goal is shifted towardﬁnding an algorithm that attempts to maximize the second smallest eigenvalue of the graph Laplacian at each step.Toward this aim,weﬁrst differentiate(14)with respect to time as 2f_x i(t)0_x j(t)g T f x i(t)0x j(t)g=_d ij(t)(16) and then employ Euler’sﬁrst discretization method,with1t as the sampling timex(t)!x(k);_x(t)!x(k+1)0x(k)1t2This connection was pointed to us by one of the referees.to rewrite(14)as2f x i(k+1)0x j(k+1)g T f x i(k)0x j(k)g=d ij(k+1)+d ij(k): Similarly,the state dependent Laplacian L G(x)in(15)is discretized by ﬁrst differentiating the terms w ij with respect to time,and then having w ij(k+1)=w ij(k)0 ( 0d(k))=( 0 )f d ij(k+1)0d ij(k)grecall that we are employing functions of the form(5)in(1).The dis-crete version of the state dependent Laplacian,L G(k),assumes the form[L G(k)]ij=0w ij(k);if i=js=iw is(k);if i=j.Putting it all together,we arrive at the iterative step of solving the op-timization problem3k:maxx(k+1)(17)s:t:2f x i(k+1)0x j(k+1)g T f x i(k)0x j(k)g=d ij(k+1)+d ij(k)(18)d ij(k+1) 1(19)P T L G(k+1)P I n01(20)for i=1;2;...;n01,j=2;...;n,i<j,and x(k):= [x1(k);x2(k);...;x n(k)]T2R3n.Thereby,the algorithm is ini-tiated at time k=0withan initial graph(conﬁgu ration)G0,and then for k=0;1;2;...,we proceed to iterativelyﬁnd a graphth at maximizes 2(L G(k+1)).This greedy procedure is then iterated upon until the value of 2(L G(k))can not be improved further.We note that the proposed greedy algorithm converges,as the sequence generated by it is nondecreasing and bounded from above.3C.Further ConsiderationsIn previous section,we proposed an algorithm that converges to a local optimal vertex positional conﬁguration,in terms of maximizing the quantity 2(L G).However,by replacing the nonconvex constraint (14)withits linear approximation(18)–(19),one introdu ces a poten-tial inconsistency between the position and the distance vectors.In this section,we provide two remedies to avoid such potential com-plications.Let usﬁrst recall the notion of Euclidean distance matrix (EDM).Given the position vectors x1;x2;...;x n2R3,the EDM D=[d ij]2R n2n is deﬁned entry-wise as[D]ij=d ij=k x i0x j k2;for i;j=1;2;...;n:The EDM matrices are nicely characterized in terms of linear matrix inequalities[10].Theorem2.4:A matrix D=[d ij]2R n2n is an EDM if and only ifJDJ 0(21)d ii=0;for i=1;2;...;n(22) where J:=I n011T=n.3The second smallest eigenvalue ofL for a graphof order n is bounded from above by n01[9].Fig.2.Trajectory generated by the proposed algorithm for six nodes in R:the conﬁguration evolves from a path(circles)to a truss(squares).Fig.3.Trajectory generated by the proposed algorithm for six nodes in R:the conﬁguration evolves from a path(circles,1;...;6)to an octahedron(squares, 1;...;6).Theorem2.4allows us to guarantee that by adding the two convexconstraints(21)–(22)to problem3k(17)–(20),we always obtainconsistency among the position and distance variables at each iterationstep.Moreover,by updating the values of d ij(k)’s and[L(k)]ij’s in(18)and(20)after calculating the values of x(k),we can furtherreduce the effect of linearization in the proposed procedure.To furtherexpand on this last point,suppose that x1(k);x2(k);...;x n(k), d ij(k)’s and[L(k)]ij,i=1;2;...;n01,j=2;...;n,i<j,h ave been obtained after solving the problem3k(17)–(20).Our proposed modiﬁcation to the original algorithm thus amounts to updating the values of d ij(k)and[L(k)]ij,based on the computed values of x1(k);x2(k);...;x n(k),before initiating the next iteration.III.S IMULATION R ESULTSFor our simulations we used SeDuMi[1]to solve the required semideﬁnite programs.Fig.2depicts the behavior of six mobile elements under the guidance of the proposed algorithm,leading to a planar conﬁguration that locally maximizes 2(L G).The constants , 1,and 2in(5)are chosen to be0.1,1,and1.5,respectively. The algorithm was initialized with a conﬁguration that corresponds to a path.The sequence of conﬁgurations thereafter converges to the truss-shape graph with the 2(L G)of1.6974.For these set of param-eters,the truss-shape graph as suggested by the algorithm is the global maximum over the set of graphs on six vertices that can be conﬁgured in R2.4Using the same simulation scenario,but this time,in search of an optimal positional conﬁguration in R3,the algorithm leads to the trajectories shown in Fig.3.In this case,the graph sequence converges to an octahedron-shape conﬁguration with 2(L G)=4:02. Increasing the number of nodes to eight,the algorithm was initial-ized as the unit cube;the resulting trajectories are shown in Fig.4.4A global maximum may be found in the following exhaustive manner:First, deﬁne a space large enou ghgu aranteed to contain th e optimal conﬁgu ration. Then grid this region and search over the set of all n grid points for the conﬁg-uration that leads to maximum (L).Fig.4.Evolution of the proposed algorithm for eight nodes in R :the conﬁguration evolves from 3-cube (circles)to octahedron (squares).TABLE IC OMPARING THE V ALUES FOR THE T YPE -IV W EIGHTED G RAPH G AS R EALIZED BY THE A LGORITHM AND T HOSE C ORRESPONDING TO THEA SSOCIATED 0–1W EIGHTED G RAPHGIn this ﬁgure,the edges between vertices i and j indicate that d ij 2=1:5.The solid lines in Fig.4represent the ﬁnal conﬁguration with 2(L G )=2:7658.Once again,an exhaustive search procedure indicates that the proposed algorithm does lead to the global optimal conﬁguration (see Table I).We like to remark however that the choice for the function f in (5)and the initial conﬁguration,are critical to the performance of the proposed algorithm.For example,when this func-tion is chosen to be of Type-I in Fig.1and the initial graph as a dis-connected graph,the algorithm terminates right after initialization,as any small perturbation on the initial graph does not lead to an improve-ment in the value of 2(L G ).Choosing a Type-IV function in Fig.1on the other hand,always lead to a connected conﬁguration with a posi-tive 2(L G ),even when the algorithm is initialized via a disconnected graph.IV .C ONCLUDING R EMARKSWe considered the problem of maximizing the second smallest eigenvalu es of a state-dependent graphLaplacian.Th is problem is of importance,for example,when the positions of a set of dynamic elements-operating over an information network-can be chosen for robust system performance.We proposed an iterative algorithm for this problem that employs a semideﬁnite programming solver at each recursive step.Although the algorithm has a local convergence behavior,extensive simulations suggest that it often leads to a globally optimal state conﬁguration.A CKNOWLEDGMENTThe authors gratefully acknowledge suggestions and comments by the anonymous reviewers.R EFERENCES[1]SeDuMi.McMaster Univ..[Online].Available:http://sedumi.mc-master.ca[2] A.Berman and X.-D.Zhang,“Lower bounds for the eigenvalues of Laplacian matrices,”Linear Alg.Appl.,vol.316,pp.13–20,2000.[3]S.Boyd and L.Vandenberghe,Convex Programming .Cambridge,U.K.:Cambridge Univ.Press,2003.[4] F.R.K.Chung,Spectral Graph Theory .Providence,RI:AMS,1997.[5]F.R.K.Chung and K.Oden,“Weighted graph Laplacians and isoperi-metric inequalities,”Paciﬁc J.Math.,vol.192,no.2,pp.257–273,2000.[6]S.Fallat and S.Kirkland,“Extremizing algebraic connectivity subject to graphth eoretic constraints,”Elect.J.Linear Alg.,vol.1,no.3,pp.48–74,1998.[7]J.A.Fax and R.M.Murray,“Information ﬂow and cooperative control of vehicle formations,”IEEE Trans.Autom.Control ,vol.49,no.9,pp.1465–1476,Sep.2004.[8]M.Fiedler,“A property of eigenvectors of nonnegative symmetric ma-trices and its applications in graphth eory,”Czech.Math.J.,vol.100,no.26,pp.619–633,1975.[9] C.Godsil and G.Royle,Algebraic Graph Theory .New York:Springer-Verlag,2001.[10]J.Gower,“Properties of Euclidean and non-Euclidean distance ma-trices,”Linear Alg.Appl.,vol.1,no.67,pp.81–97,1985.[11]S.Guattery and ler,“On the quality of spectral separators,”SIAM J.Matrix Anal.Appl.,vol.19,no.3,pp.701–719,1998.[12] A.Jadbabaie,J.Lin,and A.S.Morse,“Coordination of groups of mobile autonomous agents using nearest neighbor rules,”IEEE Trans.Autom.Control ,vol.48,no.9,pp.988–1001,Sep.2003.[13]Y .Kim and M.Mesbahi,“Quadratically constrained attitude control via semideﬁnite programming,”IEEE Trans.Autom.Control ,vol.49,no.5,pp.731–735,May 2004.[14]R.Merris,“Laplacian matrices of graphs:A survey,”Linear Alg.Appl.,vol.197,no.1,pp.143–176,1994.[15]M.Mesbahi,“On state-dependent dynamic graphs and their control-lability properties,”IEEE Trans.Autom.Control ,vol.50,no.3,pp.387–392,Mar.2005.[16]H.Q.Ngo and D.-Z Du,“Notes on the complexity of switching net-works,”in Advances in Switching Networks ,H.Q.Ngo and D.-Z.Du,Eds.Norwell,MA:Kluwer,2000,pp.307–357.[17]R.Olfati-Saber and R.M.Murray,“Consensus problems in networks of agents withswitch ing topology and time-delays,”IEEE Trans.Autom.Control ,vol.49,no.9,pp.1520–1533,Sep.2004.[18]K.Pahlavan and A.H.Levesque,Wireless Information Networks .New York:Wiley,1995.[19]H.Tanner, A.Jadbabaie,and G.Pappas,“Flocking in ﬁxed and switching networks,”Automatica,submitted for publication.[20]L.Xiao and S.Boyd,“Fast linear iterations for distributed averaging,”Syst.Control Lett.,vol.53,pp.65–78,2004.。

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arXiv:0712.1319v1 [math.CT] 9 Dec 2007AHOMOTOPYTHEORYFORENRICHMENTINSIMPLICIALMODULES

ALEXANDRUE.STANCULESCUAbstract.WeputaQuillenmodelstructureonthecategoryofsmallcate-goriesenrichedinsimplicialk-modulesandnon-negativelygradedchaincom-plexesofk-modules,wherekisacommutativering.ThemodelstructureisobtainedbytransferfromthemodelstructureonsimplicialcategoriesduetoJ.Bergner.

1.Introduction:DK-equivalencesandDK-fibrations1.1.LetCatthecategoryofsmallcategories.Ithasanaturalmodelstructureinwhichacoﬁbrationisafunctormoniconobjects,aweakequivalenceisanequivalenceofcategoriesandaﬁbrationisanisoﬁbration[5].Theﬁbrationweakequivalencesaretheequivalencessurjectiveonobjects.LetVbeamonoidalmodelcategory[8]withunitI.WedenotebyWtheclassofweakequivalencesofV,byFibtheclassofﬁbrationsandbyCoftheclassofcoﬁbrations.ThesmallV-categoriestogetherwiththeV-functorsbetweenthemformacat-egorywrittenVCat.LetMbeaclassofmapsofV.WesaythataV-functorf:A→BislocallyinMifforeachpairx,y∈Aofobjects,themapfx,y:A→BisinM.Wehaveafunctor[

Ho(V)HomHo(V)(I,Set.

Deﬁnition1.1.Letf:A→BbeamorphisminVCat.1.Themorphismfishomotopyessentiallysurjectiveiftheinducedfunctor[f]V:[A]V→[B]Visessentiallysurjective.2.ThemorphismfisaDK−equivalenceifitishomotopyessentiallysurjectiveandlocallyinW.3.ThemorphismfisaDK−ﬁbrationifitsatisﬁesthefollowingtwocondi-tions.(a)fislocallyinFib.(b)Foranyx∈A,andanyisomorphismv:[f]V(x)→y′in[B]V,thereexistsanisomorphismu:x→yin[A]Vsuchthat[f]V(u)=v.Thatis,if[f]Visanisoﬁbration.

OnecaneasilyseethatamorphismfisaDK-equivalenceandaDK-ﬁbrationiﬀfissurjectiveonobjectsandlocallyinW∩Fib.TheclassofmapshavingtheleftliftingpropertywithrespecttotheV-functorssurjectiveonobjectandlocallyinW∩Fibisgeneratedbythemapu:∅→I,whereIistheV-categorywithasingleobject∗andI(∗,∗)=I,togetherwiththemaps¯2i:¯2A→¯2B,2ALEXANDRUE.STANCULESCUwhereiisageneratingcoﬁbrationofV.HeretheV-category¯2Ahasobjects0and1,with¯2A(0,0)=¯2A(1,1)=I,¯2A(0,1)=Aand¯2A(1,0)=∅.1.2.Letkbeacommutativering.WedenotebySModkthecategoryofsimplicialk-modulesandbyCh+(k)thecategoryofnon-negativelygradedchaincomplexesofk-modules.Thepurposeofthisnoteistoprovethefollowingtheorem.

Theorem1.2.LetVbeoneofthecategoriesSModkorCh+(k).ThenVCatadmitsamodelstructureinwhichtheweakequivalencesaretheDK-equivalencesandtheﬁbrationsaretheDK-ﬁbrations.

Toprovethisresultweusethe(similar)modelstructureonsimplicialcategories[2]andQuillen’spathobjectargument([7],Lemma2.3(2)and[1],2.6).AnexplicitdescriptionofacoﬁbrationofVCatcanbegiven[9].1.3.Theproofoftheorem1.2reliesdecisivelyontheconstructionofpathobjectsfordg-categoriesduetoG.Tabuada([11],4.1).Infact,ourattempttounderstandhisconstructionledustotheproofofourresult.1.4.In[12],B.To¨encharacterisedthemapsinthehomotopycategoryofdg-categories,wherethecategoryofdg-categorieshasamodelstructureinwhichtheweakequivalencesaretheDK-equivalencesandtheﬁbrationsaretheDK-ﬁbrations.Onecanshowthathisresults(loc.cit.,Thm.4.2and6.1)holdforVCat,whereVisSModkorCh+(k).Noteaddedinproof.AfterthecompletionofthisworkwelearnedabouttheexistenceofapaperbyG.Tabuada[10],whichtreatsthesamesubjectmatter,andmore,butdiﬀerently.Onecanseethatthemodelstructureproposedintheorem1.2coincideswiththeonein[10],althoughtheclassesofﬁbrationsandcoﬁbrationsarenotexplicitlyidentiﬁedinloc.cit.Ontheotherhand,TabuadashowsthatthemodelstructuresonSModkCatandCh+(k)CatareQuillenequivalent,anissuethatwehaveinitiallyneglected.Onecaneasilygiveaproofofthisfact,adaptedtoourcontext,usingsection2.2belowandthegeneralresultsof[9].

2.CategoriesenrichedinSModkandCh+(k)2.1.ThecategorySModkisaclosedsymmetricmonoidalcategorywithtensorproductdeﬁnedpointwiseandunitck,where(ck)n=kforalln≥0.AmodelstructureonSModkisobtainedbytransferfromthecategorySofsimplicialsets,regardedashavingtheclassicalmodelstructure,viathefree-forgetfuladjunction

k:S⇄SModk:U.Allobjectsareﬁbrantandthemodelstructureissimplicial.Thefunctorkisstrongsymmetricmonoidal(anditpreservestheunit),henceSModkisamonoidalmodelcategory.Theadjunction(k,U)inducesanadjunction

k′:SCat⇄SModkCat:U′.WeclaimthatamapfofSModkCatisaDK-equivalence(resp.DK-ﬁbration)iﬀU′(f)isaweakequivalence(resp.ﬁbration)intheBergnermodelstructureonSCat[2].Clearly,fislocallyinW(resp.Fib)iﬀU′(f)islocallyinW(resp.Fib).Intheinducedadjointpair

Lk:Ho(S)⇄Ho(SModk):RU,thefunctorLkisstrongsymmetricmonoidalandpreservestheunitobject,henceonehasanaturalisomorphismoffunctors

η:[]SU′:SModkCat→Cat

suchthatforallA∈SModkCat,ηAistheidentityonobjects.Therestoftheclaimfollowsfromthisobservation.