A generalization of the binomial coefficients
Brunner-Minkowski不等式
BULLETIN(New Series)OF THEAMERICAN MATHEMATICAL SOCIETYVolume39,Number3,Pages355–405S0273-0979(02)00941-2Article electronically published on April8,2002THE BRUNN-MINKOWSKI INEQUALITYR.J.GARDNERAbstract.In1978,Osserman[124]wrote an extensive survey on the isoperi-metric inequality.The Brunn-Minkowski inequality can be proved in a page,yet quickly yields the classical isoperimetric inequality for important classesof subsets of R n,and deserves to be better known.This guide explains therelationship between the Brunn-Minkowski inequality and other inequalitiesin geometry and analysis,and some applications.1.IntroductionAll mathematicians are aware of the classical isoperimetric inequality in the plane:(1)L2≥4πA,where A is the area of a domain enclosed by a curve of length L.Many,including those who read Osserman’s long survey article[124]in this journal,are also aware that versions of(1)hold not only in n-dimensional Euclidean space R n but also in various more general spaces,that these isoperimetric inequalities are intimately related to several important analytic inequalities,and that the resulting labyrinth of inequalities enjoys an extraordinary variety of connections and applications to a number of areas of mathematics and physics.Among the inequalities stated in[124,p.1190]is the Brunn-Minkowski inequal-ity.One form of this states that if K and L are convex bodies(compact convex sets with nonempty interiors)in R n and0<λ<1,then(2)V((1−λ)K+λL)1/n≥(1−λ)V(K)1/n+λV(L)1/n.Here V and+denote volume and vector sum.(These terms will be defined in Sections2and3.)Equality holds precisely when K and L are equal up to translation and dilatation.Osserman emphasizes that this inequality(even in a more general form discussed below)is easy to prove and quickly implies the classical isoperimetric inequality for important classes of sets,not only in the plane but in R n.And yet,outside geometry,relatively few mathematicians seem to be familiar with the Brunn-Minkowski inequality.Fewer still know of the potent extensions of(2),some very recent,and their impact on mathematics and beyond.This article will attempt356R.J.GARDNERto explain the current point of view on these topics,as well as to clarify relationsbetween the main inequalities concerned.Figure1indicates that this is no easy task.In fact,even to claim that oneinequality implies another invites debate.When I challenged a colloquium audienceto propose their candidates for the most powerful inequality of all,a wit offered x2≥0,“since all inequalities are in some sense equivalent to it.”The arrows in Figure1mean that one inequality can be obtained from the other with what I regardas only a modest amount of effort.With this understanding,I feel comfortable in claiming that the inequalities at the top level of this diagram are among the most powerful known in mathematics today.The Brunn-Minkowski inequality was actually inspired by issues around theisoperimetric problem and was for a long time considered to belong to geometry,where its significance is widely recognized.For example,it implies the intuitively clear fact that the function that gives the volumes of parallel hyperplane sections of a convex body is unimodal.The fundamental geometric content of the Brunn-Minkowski inequality makes it a cornerstone of the Brunn-Minkowski theory,a beautiful and powerful apparatus for conquering all sorts of problems involving metric quantities such as volume and surface area.By the mid-twentieth century,however,when Lusternik,Hadwiger and Ohmann,and Henstock and Macbeath had established a satisfactory generalization(10)of(2)and its equality condition to Lebesgue measurable sets,the inequality had begun its move into the realm of analysis.The last twenty years have seen the Brunn-Minkowski inequality consolidate its role as an analytical tool,and a compelling picture(Figure1)has emerged of its relations to other analytical inequalities.In an integral version of the Brunn-Minkowski inequality often called the Pr´e kopa-Leindler inequality(21),a reverse form of H¨o lder’s inequality,the geometry seems to have rgely through the efforts of Brascamp and Lieb,this in-equality can be viewed as a special case of a sharp reverse form(50)of Young’s inequality for convolution norms.A remarkable sharp inequality(60)proved by Barthe,closely related to(50),takes us up to the present time.The modern view-point entails an interaction between analysis and convex geometry so fertile that whole conferences and books are devoted to“analytical convex geometry”or“con-vex geometric analysis”.Sections3,4,5,7,13,14,15,and17are devoted to explaining the inequalities in Figure1and the relations between them.Several applications are discussed at some length.Section6explains why the Brunn-Minkowski inequality can be ap-plied to the Wulffshape of crystals.McCann’s work on gases,in which the Brunn-Minkowski inequality appears,is introduced in Section8,along with a crucial idea called transport of mass that was also used by Barthe in his proof of the Brascamp-Lieb and Barthe inequalities.Section9explains that the Pr´e kopa-Leindler inequal-ity can be used to show that a convolution of log-concave functions is log concave, and an application to diffusion equations is outlined.The Pr´e kopa-Leindler in-equality can also be applied to prove that certain measures are log concave.These results on concavity of functions and measures,and natural generalizations of them that follow from the Borell-Brascamp-Lieb inequality,an extension of the Pr´e kopa-Leindler inequality introduced in Section10,are very useful in probability theory and statistics.Such applications are treated in Section11,along with related con-sequences of Anderson’s theorem on multivariate unimodality,the proof of which employs the Brunn-Minkowski inequality.The entropy power inequality(55)ofTHE BRUNN-MINKOWSKI INEQUALITY357Figure1.Relations between inequalities labeled as in the text information theory has a form similar to that of the Brunn-Minkowski inequality. To some extent this is explained by Lieb’s proof that the entropy power inequality is a special case of a sharp form of Young’s inequality(49).Section14elaborates on this and related matters,such as Fisher information,uncertainty principles,and logarithmic Sobolev inequalities.In Section16,we come full circle with applica-tions to geometry.Keith Ball started these rolling with his elegant application of the Brascamp-Lieb inequality(59)to the volume of central sections of the cube and to a reverse isoperimetric inequality(67).In the same camp as the latter is Milman’s reverse Brunn-Minkowski inequality(68),which features prominently in the local theory of Banach spaces.The whole story extends far beyond Figure1and the previous paragraph.Sec-tion12brings versions of the Brunn-Minkowski inequality in the sphere,hyper-bolic space,Minkowski spacetime,and Gauss space,and a Riemannian version of358R.J.GARDNERthe Borell-Brascamp-Lieb inequality,obtained very recently by Cordero-Erausquin, McCann,and Schmuckenschl¨a ger.Essentially the strongest inequality for compact convex sets in the direction of the Brunn-Minkowski inequality is the Aleksandrov-Fenchel inequality(69).In Section17a remarkable link with algebraic geometry is sketched:Khovanskii and Teissier independently discovered that the Aleksandrov-Fenchel inequality can be deduced from the Hodge index theorem.Thefinal section, Section18,is a“survey within a survey”.Analogues and variants of the Brunn-Minkowski inequality include Borell’s inequality(76)for capacity,employed in the recent solution of the Minkowski problem for capacity;a discrete Brunn-Minkowski inequality(84)due to the author and Gronchi,closely related to a rich area of discrete mathematics,combinatorics,and graph theory concerning discrete isoperi-metric inequalities;and inequalities(86),(87)originating in Busemann’s theorem, motivated by his theory of area in Finsler spaces and used in Minkowski geom-etry and geometric tomography.Around the corner from the Brunn-Minkowski inequality lies a slew of related affine isoperimetric inequalities,such as the Petty projection inequality(81)and Zhang’s affine Sobolev inequality(82),much more powerful than the isoperimetric inequality and the classical Sobolev inequality(16), respectively.Finally,pointers are given to several other applications of the Brunn-Minkowski inequality.The reader might share a sense of mystery and excitement.In a sea of mathe-matics,the Brunn-Minkowski inequality appears like an octopus,tentacles reaching far and wide,its shape and color changing as it roams from one area to the next. It is quite clear that research opportunities abound.For example,what is the relationship between the Aleksandrov-Fenchel inequality and Barthe’s inequality? Do even stronger inequalities await discovery in the region above Figure1?Are there any hidden links between the various inequalities in Section18?Perhaps, as more connections and relations are discovered,an underlying comprehensive theory will surface,one in which the classical Brunn-Minkowski theory represents just one particularly attractive piece of coral in a whole reef.Within geometry, the work of Lutwak and others in developing the dual Brunn-Minkowski and L p-Brunn-Minkowski theories(see Section18)strongly suggests that this might well be the case.An early version of the paper was written to accompany a series of lectures given at the1999Workshop on Measure Theory and Real Analysis in Gorizia,Italy.I am very grateful to Franck Barthe,Apostolos Giannopoulos,Helmut Groemer,Paolo Gronchi,Peter Gruber,Daniel Hug,Elliott Lieb,Robert McCann,Rolf Schneider, B´e la Uhrin,Deane Yang,and Gaoyong Zhang for their extensive comments on previous versions of this paper,as well as to many others who provided information and references.2.Basic notationThe origin,unit sphere,and closed unit ball in n-dimensional Euclidean space R n are denoted by o,S n−1,and B,respectively.The Euclidean scalar product of x and y will be written x·y,and x denotes the Euclidean norm of x.If u∈S n−1, then u⊥is the hyperplane containing o and orthogonal to u.Lebesgue k-dimensional measure V k in R n,k=1,...,n,can be identified with k-dimensional Hausdorffmeasure in R n.Then spherical Lebesgue measure in S n−1 can be identified with V n−1in S n−1.In this paper dx will denote integration withTHE BRUNN-MINKOWSKI INEQUALITY359+Figure2.The vector sum of a square and a diskrespect to V k for the appropriate k,and integration over S n−1with respect to V n−1will be denoted by du.The term measurable applied to a set in R n will alwaysmean V n-measurable unless stated otherwise.If X is a k-dimensional body(equal to the closure of its relative interior)in R n,its volume is V(X)=V k(X).The volume V(B)of the unit ball will also be denoted byκn.3.Geometrical originsThe basic notions needed are the vector sum X+Y={x+y:x∈X,y∈Y}ofX and Y,and dilatate rX={rx:x∈X},r≥0of X,where X and Y are sets in R n.(In geometry,the term Minkowski sum is more frequently used for the vector sum.)The set−X is the reflection of X in the origin o,and X is called originsymmetric if X=−X.As an illustration,consider the vector sum of an origin-symmetric square K of side length l and a disk L=εB of radiusε,also centered at o.The vector sum K+L,depicted in Figure2,is a rounded square composed of a copy of K,four rectangles of area lε,and four quarter-disks of radiusε.The volume V(K+L)of K+L(i.e.,its area;see Section2)is√V(K+L)=V(K)+4lε+V(L)≥V(K)+2V(K)V(L)+V(L),which implies thatV(K+L)1/2≥V(K)1/2+V(L)1/2.Generally,any two convex bodies K and L in R n satisfy the inequality(3)V(K+L)1/n≥V(K)1/n+V(L)1/n.In fact,this is the Brunn-Minkowski inequality(2)in an equivalent form.To see this,just replace K and L in(3)by(1−λ)K andλL,respectively,and use the positive homogeneity(of degree n)of volume in R n,that is,V(rX)=r n V(X)for r≥0.This homogeneity of volume easily yields another useful and equivalent form360R.J.GARDNERof (2),obtained by replacing (1−λ)and λby arbitrary positive real numbers s and t :V (sK +tL )1/n ≥sV (K )1/n +tV (L )1/n .(4)Detailed remarks and references concerning the early history of (2)are provided in Schneider’s excellent book [135,p.314].Briefly,the inequality for n =3was discovered by Brunn around 1887.Minkowski pointed out an error in the proof,which Brunn corrected,and found a different proof of (2)himself.Both Brunn and Minkowski showed that equality holds if and only if K and L are homothetic (i.e.,K and L are equal up to translation and dilatation).If inequalities are silver currency in mathematics,those that come along with precise equality conditions are gold.Equality conditions are treasure boxes con-taining valuable information.For example,everyone knows that equality holds in the isoperimetric inequality (1)if and only if the curve is a circle—that a domain of maximum area among all domains of a fixed perimeter must be a disk .It is no coincidence that (2)appeared soon after the first complete proof of the classical isoperimetric inequality in R n was found.To begin to understand the connection between these two inequalities,look again at Figure 2.ClearlyV (K +εB )=V (K +L )=V (K )+4lε+V (εB )=V (K )+4lε+V (B )ε2,(5)and thereforelim ε→0+V (K +εB )−V (K )ε,(6)and it follows immediately from Minkowski’s theorem that S (K )=nV (K,n −1;B ),where the notation means that K appears (n −1)times and the unit ball B appears once.Up to a constant,surface area is just a special mixed volume.The isoperimetric inequality for convex bodies in R n is the highly nontrivial statement that if K is a convex body in R n ,thenV (K )S (B )1/(n −1),(7)THE BRUNN-MINKOWSKI INEQUALITY361 with equality if and only if K is a ball.The inequality can be derived in a few lines from the Brunn-Minkowski inequality!Indeed,by(6)and(4)with s=1and t=ε,V(K+εB)−V(K)S(K)=limε→0+ε=nV(K)(n−1)/n V(B)1/n,and(7)results from recalling that S(B)=nV(B)and rearranging.Surely this alone is good reason for appreciating the Brunn-Minkowski inequality. (Perceptive readers may have noticed that this argument does not yield the equality condition in(7),but in Section5this will be handled with a little extra work.)Many more reasons lie ahead.There is a standard geometrical interpretation of the Brunn-Minkowski inequal-ity(2)that is at once simple and appealing.Recall that a function f on R n is concave on a convex set C iff((1−λ)x+λy)≥(1−λ)f(x)+λf(y),for all x,y∈C and0<λ<1.If K and L are convex bodies in R n,then(2)is equivalent to the fact that the function f(t)=V((1−t)K+tL)1/n is concave for 0≤t≤1.Now imagine that K and L are the intersections of an(n+1)-dimensional convex body M with the hyperplanes{x1=0}and{x1=1},respectively.Then (1−t)K+tL is precisely the intersection of the convex hull of K and L with the hyperplane{x1=t}and is therefore contained in the intersection of M with this hyperplane.It follows that the function giving the n th root of the volumes of parallel hyperplane sections of an(n+1)-dimensional convex body is concave.A picture illustrating this can be viewed in[66,p.369].A much more general statement than(2)will be proved in the next section, but certain direct proofs of(2)are still of interest.A standard proof,due to Kneser and S¨u ss in1932and given in[135,Section6.1],is still perhaps the simplest approach for the equality conditions for convex bodies.A quite different proof,due to Blaschke in1917,uses Steiner symmetrization.Symmetrization techniques are extremely valuable in obtaining many inequalities—indeed,Steiner introduced the technique to attack the isoperimetric inequality—so Blaschke’s method deserves some explanation.Let K be a convex body in R n and let u∈S n−1.The Steiner symmetral S u K of K in the direction u is the convex body obtained from K by sliding each of its chords parallel to u so that they are bisected by the hyperplane u⊥and taking the union of the resulting chords.Then V(S u K)=V(K),and it is not hard to show that if K and L are convex bodies in R n,then S u(K+L)⊃S u K+S u L and hence(8)V(K+L)≥V(S u K+S u L).See,for example,[52,Chapter5,Section5]or[151,pp.310–314].One can also prove,as in[56,Theorem2.10.31],that there is a sequence of directions u m∈S n−1 such that if K=K0is any convex body and K m=S uK m−1,then K m convergesmto r K B in the Hausdorffmetric as m→∞,where r K is the constant such that V(K)=V(r K B).Defining r L so that V(L)=V(r L B)and applying(8)repeatedly362R.J.GARDNERthrough this sequence of directions,we obtain(9)V(K+L)≥V(r K B+r L B).By the homogeneity of volume,it is easy to see that(9)is equivalent to the Brunn-Minkowski inequality(2).4.The move to analysis I:The general Brunn-Minkowski inequality Much more needs to be said about the role of the Brunn-Minkowski inequality in geometry,but it is time to transplant the inequality from geometry to analy-sis.We shall call the following result the general Brunn-Minkowski inequality in R n.As always,measurable in R n means measurable with respect to n-dimensional Lebesgue measure V n.Theorem4.1.Let0<λ<1and let X and Y be nonempty bounded measurable sets in R n such that(1−λ)X+λY is also measurable.Then(10)V n((1−λ)X+λY)1/n≥(1−λ)V n(X)1/n+λV n(Y)1/n.Again,by the homogeneity of n-dimensional Lebesgue measure(V n(rX)= r n V n(X)for r≥0),there are the equivalent statements that for s,t>0,(11)V n(sX+tY)1/n≥sV n(X)1/n+tV n(Y)1/n,and this inequality with the coefficients s and t omitted.Yet another equivalent statement is that(12)V n((1−λ)X+λY)≥min{V n(X),V n(Y)}holds for0<λ<1and all X and Y that satisfy the assumptions of Theorem4.1. Of course,(10)trivially implies(12).For the converse,suppose without loss of generality that X and Y also satisfy V n(X)V n(Y)=0.Replace X and Y in(12) by V n(X)−1/n X and V n(Y)−1/n Y,respectively,and takeV n(Y)1/nλ=THE BRUNN-MINKOWSKI INEQUALITY 363Proof of Theorem 4.1.Theideaistoprovethe result first for boxes,rectangular parallelepipeds whose sides are parallel to the coordinate hyperplanes.If X and Y are boxes with sides of length x i and y i ,respectively,in the i th coordinate directions,thenV (X )=n i =1x i ,V (Y )=n i =1y i ,and V (X +Y )=n i =1(x i +y i ).Now ni =1x i x i +y i 1/n ≤1x i +y i +1x i +y i =1,by the arithmetic-geometric mean inequality.This gives the Brunn-Minkowski in-equality for boxes.One then uses a trick sometimes called a Hadwiger-Ohmann cut to obtain the inequality for finite unions X and Y of boxes,as follows.By translating X ,if necessary,we can assume that a coordinate hyperplane,{x n =0}say,separates two of the boxes in X .(The reader might find a picture illustrating the planar case useful at this point.)Let X +(or X −)denote the union of the boxes formed by intersecting the boxes in X with {x n ≥0}(or {x n ≤0},respectively).Now translate Y so thatV (X ±)V (Y ),(13)where Y +and Y −are defined analogously to X +and X −.Note that X ++Y +⊂{x n ≥0},X −+Y −⊂{x n ≤0},and that the numbers of boxes in X +∪Y +and X −∪Y −are both smaller than the number of boxes in X ∪Y .By induction on the latter number and (13),we haveV (X +Y )≥V (X ++Y +)+V (X −+Y −)≥V (X +)1/n +V (Y +)1/n n + V (X −)1/n +V (Y −)1/n n =V (X +) 1+V (Y )1/n V (X )1/n n =V (X ) 1+V (Y )1/n364R.J.GARDNERwhen X and Y are compact convex sets,equality holds in(10)or(11)if and only if X and Y are homothetic or lie in parallel hyperplanes;see[135,Theorem6.1.1].Since H¨o lder’s inequality((25)below)in its discrete form implies the arithmetic-geometric mean inequality,there is a sense in which H¨o lder’s inequality implies the Brunn-Minkowski inequality.The dotted arrow in Figure1reflects the controversial nature of this implication.5.Minkowski’s first inequality,the isoperimetric inequality,andthe Sobolev inequalityIn order to derive the isoperimetric inequality with its equality condition,a slight detour via another inequality of Minkowski is needed.This involves a quantity V1(K,L)depending on two convex bodies K and L in R n that can be defined bynV1(K,L)=limε→0+V(K+εL)−V(K)t(1−t)n−1=limt→0+V((1−t)K+tL)−V(K)t=limt→0+V((1−t)K+tL)−V(K)V(K)(n−1)/n.Therefore(15)is equivalent to f (0)≥f(1)−f(0).As was noted in Section3, the Brunn-Minkowski inequality(2)says that f is concave,so Minkowski’sfirst inequality follows.THE BRUNN-MINKOWSKI INEQUALITY365 Suppose that equality holds in(15).Then f (0)=f(1)−f(0).Since f is concave,we havef(t)−f(0)366R.J.GARDNER6.Wulff shape of crystals and surface area measuresA crystal in contact with its melt (or a liquid in contact with its vapor)is modeled by a bounded Borel subset M of R n of finite surface area and fixed volume.If f is a nonnegative function on S n −1representing the surface tension,assumed known by experiment or theory,then the surface energy is given by F (M )=∂Mf (u x )dx,(17)where u x is the outer unit normal to M at x and ∂M denotes the boundary of M .(Measure-theoretic subtleties are ignored in this description;it is assumed that f and M are such that the various ingredients are properly defined.)By the Gibbs-Curie principle,the equilibrium shape of the crystal minimizes this surface energy among all sets of the same volume.This shape is called the Wulffshape .For example,in the case of a soapy liquid drop in air,f is a constant (neglecting external potentials such as gravity)and the Wulffshape is a ball.For crystals,however,f will generally reflect certain preferred directions.In 1901,Wulffgave a construction of the Wulffshape W :W =∩u ∈S n −1{x ∈R n :x ·u ≤f (u )};each set in the intersection is a half-space containing the origin with bounding hyperplane orthogonal to u and containing the point f (u )u at distance f (u )from the origin.The Brunn-Minkowski inequality can be used to prove that,up to translation,W is the unique shape among all with the same volume for which F is minimum;see,for example,[144,Theorem 1.1].This was done first by A.Dinghasin 1943for convex polygons and polyhedra and then by various people in greater generality.In particular,Busemann [37]solved the problem when f is continuous,and Fonseca [60]and Fonseca and M¨u ller [61]extended the results to include setsM of finite perimeter in R n .Good introductions with more details and referencesare provided by Taylor [144]and McCann [116].In fact,McCann [116]also proves more general results that incorporate a convex external potential,by a technique developed in his paper [115]on interacting gases;see Section 8.To understand how the Brunn-Minkowski inequality assists in the determination of Wulffshape,a glimpse into later developments in the Brunn-Minkowski theory is helpful.There are (see [135,Theorem 5.1.6])integral representations for mixed volumes and,in particular,V 1(K,L )=1nS n −1h L (u )dS (K,u )(19)THE BRUNN-MINKOWSKI INEQUALITY367 is more common than(18).Here the measure S(K,·)is afinite Borel measure in S n−1called the surface area measure of K,an invention of A.D.Aleksandrov, W.Fenchel,and B.Jessen from around1937that revolutionized convex geometry by providing the key tool to treat convex bodies that do not necessarily have smooth boundaries.If E is a Borel subset of S n−1,then S(K,E)is the V n−1-measure of the set of points x∈∂K where the outer normal u x∈E.When K is sufficientlysmooth,it turns out that dS(K,u)=f K(u)du,where f K(u)is the reciprocal of the Gauss curvature of K at the point on∂K where the outer unit normal is u.A fundamental result called Minkowski’s existence theorem gives necessary and sufficient conditions for a measureµin S n−1to be the surface area measure of some convex body.Minkowski’sfirst inequality(15)and(19)imply that if S(K,·)=µ, then K minimizes the functionalL→S n−1h L(u)dµunder the condition that V(L)=1,and this fact motivates the proof of Minkowski’s existence theorem.See[66,Theorem A.3.2]and[135,Section7.1],where pointers can also be found to the vast literature surrounding the so-called Minkowski prob-lem,which deals with existence,uniqueness,regularity,and stability of a closed convex hypersurface whose Gauss curvature is prescribed as a function of its outer normals.7.The move to analysis II:The Pr´e kopa-Leindler inequalityThe general Brunn-Minkowski inequality(10)appears to be as complete a gen-eralization of(2)as any reasonable person could wish.Yet even before Hadwiger and Ohmann found their wonderful proof,a completely different proof,published in1953by Henstock and Macbeath[77],pointed the way to a still more general inequality.This is now known as the Pr´e kopa-Leindler inequality.Theorem7.1.Let0<λ<1and let f,g,and h be nonnegative integrable func-tions on R n satisfyingh((1−λ)x+λy)≥f(x)1−λg(y)λ,(20)for all x,y∈R n.ThenR n h(x)dx≥R nf(x)dx1−λR ng(x)dxλ.(21)The Pr´e kopa-Leindler inequality(21),with its strange-looking assumption(20), looks exotic at this juncture.It may be comforting to see how it quickly implies the general Brunn-Minkowski inequality(10).Suppose that X and Y are bounded measurable sets in R n such that(1−λ)X+λY is measurable.Let f=1X,g=1Y,and h=1(1−λ)X+λY,where1E denotes the characteristic function of E.If x,y∈R n,then f(x)1−λg(y)λ>0(and in fact equals 1)if and only if x∈X and y∈Y.The latter implies(1−λ)x+λy∈(1−λ)X+λY, which is true if and only if h((1−λ)x+λy)=1.Therefore(20)holds.We conclude368R.J.GARDNERby Theorem 7.1that V n ((1−λ)X +λY )=R n 1(1−λ)X +λY (x )dx ≥R n 1X (x )dx 1−λ R n 1Y (x )dx λ=V n (X )1−λV n (Y )λ.We have obtained the inequalityV n ((1−λ)X +λY )≥V n (X )1−λV n (Y )λ.(22)To understand how this relates to the general Brunn-Minkowski inequality (10),some basic facts are useful.If 0<λ<1and p =0,we defineM p (a,b,λ)=((1−λ)a p +λb p )1/pif ab =0and M p (a,b,λ)=0if ab =0;we also defineM 0(a,b,λ)=a 1−λb λ,M −∞(a,b,λ)=min {a,b },and M ∞(a,b,λ)=max {a,b }.These quantities and their natural generalizations for more than two numbers are called p th means or p -means .The classic text of Hardy,Littlewood,and P´o lya [76]is still the best general reference.(Note,however,the different convention here when p >0and ab =0.)The arithmetic and geometric means correspond to p =1and p =0,respectively.Jensen’s inequality for means (see [76,Section 2.9])implies that if −∞≤p <q ≤∞,thenM p (a,b,λ)≤M q (a,b,λ),(23)with equality if and only if a =b or ab =0.Now we have already observed that (10)is equivalent to (12),the inequality that results from replacing the (1/n )-mean of V n (X )and V n (Y )by the −∞-mean.In(22)the (1/n )-mean is replaced by the 0-mean,so the equivalence of (10)and (22)follows from (23).If the Pr´e kopa-Leindler inequality (21)reminds the reader of anything,it is prob-ably H¨o lder’s inequality with the inequality reversed .Recall that if f i ∈L p i (R n ),p i ≥1,i =1,...,m are nonnegative functions,where1p m =1,(24)then H¨o lder’s inequality in R n states thatR n m i =1f i (x )dx ≤m i =1 f i p i =m i =1 R n f i (x )p i dx 1/p i .(25)Let 0<λ<1.If m =2,1/p 1=1−λ,1/p 2=λ,and we let f =f p 11and g =f p 22,we getR n f (x )1−λg (x )λdx ≤ R n f (x )dx 1−λ R ng (x )dx λ.THE BRUNN-MINKOWSKI INEQUALITY369 The Pr´e kopa-Leindler inequality can be written in the formR n supmi=1f i(x i):mi=1x iFu(t)−∞f(x)dx=1F=g(v(t))v (t)。
计量经济学专业英汉词典
计量经济学专业英汉词典计量经济学专业英汉词典中文英文调整的R^2 （确定系数）adjusted R^2调整系数adjustment coefficient调整系数矩阵adjustment coefficient matrix赤池信息准则（AIC）Akaike’s information criterion (AIC) 阿尔蒙分布滞后模型Almen distributed lag model阿尔蒙滞后Almon lags备择假设alternative hypothesis方差分析analysis of variance辅助变量ancillary variable近似协方差矩阵approximate covariance matrix近似正态分布approximate normal distribution自回归模型AR model自回归过程AR process自回归条件异方差模型ARCH model自回归移动平均模型ARMA model假定assumption渐近χ2分布asymptotic χ2 distribution渐近协方差矩阵asymptotic covariance matrix渐近分布asymptotic distribution渐近有效性asymptotic efficiency渐近性质asymptotic properties渐近抽样特性asymptotic sampling properties渐近设定asymptotic specification渐近标准误差asymptotic standard error渐近检验asymptotic test渐近检验统计量asymptotic test statistic渐近逼近asymptotically approximation渐近有效估计式asymptotically efficient estimator渐近无偏估计式asymptotically unbiased estimatorADF检验，增项（增广）DF检验Augmented Dickey-Fuller test AEG检验，增项（增广）EG检验Augmented Engle-Granger test自相关方程误差autocorrelated equation error自相关autocorrelation自相关函数autocorrelation function自协方差autocovariance自协方差函数autocovariance function自回归autoregression自回归条件异方差autoregressive conditional heteroscedasticity自回归分布滞后模型autoregressive distributed lag (ADL) model自回归单整移动平均（ARIMA）autoregressive integrated moving average process 过程自回归（AR）摸型autoregressive model自回归移动平均（ARMA）过程autoregressive moving-average process自回归算子autoregressive operator辅助回归auxiliary regression平均值average行为方程behavioral equation贝拉－哈尔克（BJ）统计量Bera-Jarque statistic贝努利分布Bernoulli distribution最佳决策best decision最佳线性无偏估计式（BLUE）best linear unbiased estimator (BLUE)最佳线性无偏预测best linear unbiased prediction最佳无偏估计式best unbiased estimator偏倚bias偏倚向量bias vector有偏估计式biased estimator二元选择模型binary choice model二项分布binomial distribution二元正态随机变量bivariate normal random variable自举法，靴襻法bootstrap procedure博克斯－考克斯变换Box-Cox transformation博克斯－詹金斯方法Box-Jenkins approach布罗施－帕甘检验Breusch-Pagan test布朗运动Brownian motion典型相关canonical correlation因果性causality中心极限定理central limit theorem特征方程characteristic equation特征根characteristic root特征向量characteristic vector卡埃方分布chi-square distribution古典统计学classical statistics柯布－道格拉斯生产函数Cobb-Douglas production function 科克伦－奥克特方法Cochrane-Orcutt procedure“概率极限”概念concept of “plim”条件推断conditional inference条件概率conditional probability条件概率密度函数conditional probability density function 置信区间confidence interval一致性consistency一致估计式consistent estimator一致性检验consistent test消费函数consumption function同期相关contemporaneous correlation同期协方差矩阵contemporaneous covariance matrix同期扰动相关contemporaneous disturbance correlation同期独立随机回归自变量contemporaneous independent stochastic regressor 连续映射理论continuous mapping theorem 连续随机变量continuous random variable连续回归函数continuous regression function常规抽样理论conventional sampling theory依概率收敛converge in probability收敛convergence依分布收敛convergence in distribution相关correlation相关系数correlation coefficient相关矩阵correlation matrix相关图correlogram成本cost协方差covariance协方差矩阵covariance matrix协方差矩阵估计式covariance matrix estimator克拉美规则Cramér rule克拉美－拉奥不等式Cramér-Rao inequality克拉美－拉奥下界Cramér-Rao lower bound临界区域critical region临界值critical value截面数据cross-section data累积分布函数cumulative distribution function 数据data数据生成过程（dgp）date generation process数据标准化date normalization盲始模型dead-start model决策decision making决策规则decision rule决策规则选择decision rule choice决策理论decision theory演绎系统deductive system定义方程definitional equation解释程度degree of explanation自由度degree of freedom密度函数density function相依变量dependent variable设计矩阵design matrix检验方法detection methods方阵的行列式determinant of a square matrix确定系数，可决系数determination coefficient诊断校验diagnostic checking对角矩阵diagonal matrix对称矩阵的对角化diagonalization of a symmetric matrix 差分difference差分方程difference equation离散随机变量discrete random variable离散样本空间discrete sample space离散随机过程discrete stochastic process非均衡误差disequilibrium error不相交集disjoint set分布滞后distributed lag分布滞后模型distributed lag model分布distribution分布函数distribution function分布理论distribution theory扰动协方差矩阵disturbance covariance matrix扰动方差disturbance variance位移项drift虚拟变量dummy variable虚拟变量估计式dummy variable estimatorDW（德宾—沃森）统计量Durbin-Watson statisticDW（德宾—沃森）检验Durbin-Watson test动态模型dynamic model动态乘数dynamic multiplier动态回归dynamic regression动态联立方程dynamic simultaneous equation计量经济学，经济计量学econometrics经济变量economic variables经济学economics经济economy有效性efficiencyEG检验EG test特征值eigen value弹性elasticity椭圆ellipse空集empty set内生变量endogenous variableEG两步估计量Engel-Granger (EG) two-step estimate EG两步法Engel-Granger (EG) two-step method 方程误差equation error 方程识别equation identification均衡equilibrium均衡分析equilibrium analysis均衡条件equilibrium condition均衡乘子equilibrium multiplier均衡关系equilibrium relationship均衡状态equilibrium state遍历性ergodicity误差error误差分量error component误差修正机制error correction mechanism误差修正模型error correction model误差修正项error correction term误差平方和error sum of squares误差向量error vector估计量estimate估计estimation估计式estimator欧氏空间Euclidean space外生前定变量exogenous predetermined variable 外生变量exogenous variable期望算子expectation operator期望值expected value试验experiment被解释变量explained variable解释变量explaining variable解释explanation指数分布exponential distributionF分布 F distributionF统计量 F statisticF检验 F test因子分解准则factorization criterion反馈feedback最终形式final form有限分布滞后模型finite distribution lag model有限非奇异矩阵finite nonsingular matrix有限多项式滞后finite polynomial lag有限抽样特性finite sampling property有限方差finite variance一阶自回归模型first-order autoregressive model 一阶条件first-order condition一阶差分算子first-order difference operator 一阶泰勒级数first-order Taylor series拟合值fitted value固定回归自变量fixed regressor预测区间forecast interval预测区域forecast region预测方差forecast variance预测forecasting频数，频率frequency完全信息估计full information estimation完全信息极大似然法full information maximum likelihood method 函数形式function form函数空间function space泛函中心极限定理functional central limit theorem (FCLT)伽玛分布Gamma distribution伽玛函数Gamma function广义自回归条件异方差模型GARCH高斯白噪声Gaussian white noise高斯－马尔可夫定理Gauss-Markov theorem高斯－牛顿算法Gauss-Newton algorithm一般协方差矩阵general covariance matrix一般均衡general equilibrium一般线性假设general linear hypothesis一般线性统计模型general linear statistical model一般随机回归自变量模型general stochastic regressor model“一般到特殊”方法general to special method广义自回归算子generalized autoregressive operator广义最小二乘法generalized least squares广义最小二乘估计generalized least squares estimation 广义最小二乘估计式generalized least squares estimator 广义最小二乘方法generalized least squares procedure 广义最小二乘残差generalized least squares residual广义最小二乘规则generalized least squares rule几何滞后模型估计geometric lag model estimation总体极小值global minimum拟合优度goodness of fit格兰杰因果性Granger causality格兰杰因果性检验Granger causality test格兰杰非因果性Granger noncausality格兰杰定理Granger representation theorem增长率模型growth rate model豪斯曼设定检验Hausman specification test重（厚）尾heavy tail海赛矩阵Hessian matrix异方差误差heteroscedastic error异方差heteroscedasticity同一性homogeneity同方差误差homoscedastic error同方差homoscedasticity假设hypothesis假设检验hypothesis test同分布随机变量identically distributed random variable 识别identification识别规则identification rules单位矩阵identity matrix压缩矩阵，影响矩阵impact matrix影响乘数矩阵impact multiplier matrix非一致性inconsistency错误约束incorrect restriction独立同一分布independent and identical distribution (IID) 独立分布independent distribution独立事件independent event独立随机变量independent random variable独立随机回归自变量independent stochastic regressor独立变量independent variable间接最小二乘法indirect least squares不等式约束inequality restriction推断inference无限分布滞后infinite distributed lag无限累加算子infinite summation operator无限方差infinite variance有影响的观测值influential observation信息矩阵information matrix内积inner product新息过程innovation sequence投入产出关系input-output relationship工具变量instrumental variable工具变量估计instrumental variable estimation 单整integration截距intercept区间估计interval estimation区间预测interval forecast不变性invariance逆矩阵inverse matrix信息矩阵的逆inverse of information matrix可逆性invertibility可逆移动平均过程invertible moving-average process 投资investment迭代方法iterative procedure大折刀方法jackknife procedure雅可比变换Jacobian of the transformation联合置信区间joint confidence interval联合置信区域joint confidence region联合密度函数joint density function联合扰动向量joint disturbance vector联合假设检验joint hypothesis test联合区间估计joint interval estimation联合零（原）假设joint null hypothesis联合概率分布joint probability distribution联合被确定变量jointly determined variable恰好识别方程just identified equation核kernel凯恩斯消费函数Keynesian consumption function 凯恩斯模型Keynesian model克莱因－戈德伯格消费函数Klein-Goldberger consumption克莱因－鲁滨效用函数Klein-Rubin utility function柯依克变换Koyck transformation克罗内克尔积Kronecker product库恩－塔克条件Kuhn-Tucker condition峰度，峭度kurtosis滞后lag滞后长度lag length滞后算子lag operator滞后权数lag weight滞后变量lagged variable拉格朗日乘数Lagrange multiplier拉格朗日乘子检验Lagrange multiplier test拉普拉斯展开Laplace expansion大样本特性large sample properties全概率定律law of total probability前导模型leading indication model最小绝对离差least absolute deviation最小绝对误差估计式least absolute error estimator 最小平方偏倚least squares bias最小平方准则least squares criterion最小平方估计式least squares estimator最小平方法least squares procedure最小平方残差least squares residual最小平方规则least squares rule最小平方方差估计式least squares variance estimator左逆矩阵left-inverse matrix显著性水平level of significance杠杆率leverage似然函数likelihood function似然原理likelihood principle似然比原理likelihood ratio principle似然比统计量likelihood ratio statistic似然比检验likelihood ratio test线性代数linear algebra线性联系linear association线性相依linear dependency线性相依向量linear dependent vector线性等式约束linear equality restriction 线性方程linear equation线性方程系统linear equation system线性估计式linear estimator线性形式linear form线性参数linear in parameter线性无关向量linear independent vector线性不等式假设linear inequality hypothesis 线性不等式约束linear inequality restriction 线性损失函数linear loss function 线性算子linear operator线性概率模型linear probability model线性规划模型linear programming model线性约束linear restriction线性规则linear rule线性联立方程linear simultaneous equation 线性统计模型linear statistical model线性变换linear transformation线性无偏估计式linear unbiased estimator线性linearity局部极小值local minima罗基斯迪随机变量logistic random variable罗基特（Logit）模型logit model对数似然函数log-likelihood function对数线性函数log-linear function长期效应long-run effect损失loss损失函数loss function下三角矩阵lower triangular matrix矩（M）估计式M estimator移动平均模型MA model宏观经济学macroeconomics边缘分布marginal distribution边缘概率密度函数marginal probability density function 边际消费倾向marginal propensity to consume数理经济学mathematical economics数学期望mathematical expectation矩阵matrix矩阵分解matrix decomposition极大似然估计maximum likelihood estimation极大似然估计式maximum likelihood estimator极大似然法maximum likelihood method均值mean均方误差mean square error均方误差准则mean square error criterion均方误差矩阵mean square error matrix均值向量mean vector测量误差measurement error中位数median矩法method of moments极小极大准则minimax criterion使损失最小minimizing loss使风险最小minimizing risk最小绝对离差估计式minimum absolute deviation estimator 最小方差minimum variance最小方差无偏估计minimum variance unbiased estimation 错误设定misspecification混合估计mixed estimation众数mode模型model模型设定model specification模数module复数的模modulus of a complex number矩moment蒙特卡罗Monte Carlo蒙特卡罗数据Monte Carlo data蒙特卡罗试验Monte Carlo experiment蒙特卡罗模拟Monte Carlo simulation移动平均moving average移动平均（MA）模型moving average (MA) model移动平均过程moving average process移动平均表示法moving average representation移动平均季节过滤算子moving average seasonal filter多重共线性multicollinearity多项选择模型multinomial choice models多项分布multinomial distribution多元回归multiple regression多重解multiple solution多重时间序列分析multiple time-series analysis乘法multiplication乘子，乘数multiplier多元分布multivariate distribution多元函数multivariate function多元正态分布multivariate normal distribution多元正态随机变量multivariate normal random variable 多元随机变量multivariate random variable多元t 分布multivariate t distribution互斥集mutually exclusive set自然共轭先验概率密度函数natural conjugate prior probability density function半负定矩阵negative semidefinite matrix嵌套nest牛顿－拉夫森算法和方法Newton-Raphson algorithm and method非线性函数nonlinear function参数非线性nonlinear in the parameter非线性最小平方法nonlinear least squares非线性最小平方估计nonlinear least squares estimation非线性似然函数nonlinear likelihood function非线性极大似然估计nonlinear maximum likelihood estimation 非线性回归nonlinear regression非线性似不相关回归方程nonlinear seemingly unrelated regression equation非线性nonlinearity非负定矩阵nonnegative definite matrix非嵌套模型nonnested models非正态分布nonnormal distribution非正态误差nonnormal error非正定矩阵nonpositive definite matrix非纯量单位协方差矩阵nonscalar identity covariance matrix 非奇异矩阵nonsingular matrix非平稳nonstationary非平稳过程nonstationary process非随机变量nonstochatic variable正态分布normal distribution正态分布理论normal distribution theory正态误差的检验normal error testing正态线性统计模型normal linear statistical model正态概率密度函数的核normal probability density function 正态随机向量normal random vector正态变量normal variable正态向量normal vector标准化常数normalizing constant正态分布随机变量normally distribution random variable 多余参数nuisance parameter零（原）假设null hypothesis零矩阵null matrix空集，零集null set可观测随机变量observable random variable可观测随机向量observable random vector观测值样本observation sample观测上的等价模型observationally equivalent model阶order阶条件order condition普通最小二乘法ordinary least squares正交矩阵orthogonal matrix正交向量orthogonal vector正交orthogonality标准正交线性统计模型orthonormal linear statistical model 离群值outliers过度识别方程overidentified equation参数parameter参数估计parameter estimation参数方差parameter variance参数检验parametric test帕累托分布Pareto distribution局部调整分布滞后模型partial adjustment distributed lag model 偏（局部）调整模型partial adjustment model偏自相关partial autocorrelation偏自相关系数partial autocorrelation coefficient偏自相关函数partial autocorrelation function偏相关partial correlation偏相关图partial correlogram偏导数partial derivative局部均衡partial equilibrium分块逆规则partitioned inverse rule完全多重共线性perfect multicollinearity长期收入假设permanent income hypothesis分段线性回归piecewise linear regression分段回归函数piecewise regression function点估计量point estimate点估计point estimation点估计式point estimator点估计式性质point estimator properties多项式polynomial多项式滞后polynomial lag多项式矩阵polynomial matrix合并数据pooling data合并模型pooling model合并模型选择pooling model selection合并时间序列pooling time series合并时间序列数据pooling time series data总体population正定矩阵positive definite matrix正定对称矩阵positive definite symmetric matrix 半正定矩阵positive semidefinite matrix后验密度posterior density后验密度函数posterior density function后验分布posterior distribution后验信息posterior information后验均值posterior mean后验优势posterior odds后验优势比posterior odds ratio后验概率posterior probability后验概率密度函数posterior probability density function 后验概率区域posterior probability region假设过程postulation process功效函数power function检验功效power of a test前定变量predetermined variable预测误差prediction error随机分量的预测prediction of random components预测精度prediction precision主分量模型principal components model先验协方差矩阵prior covariance matrix先验分布prior distribution先验均值prior mean先验概率prior probability先验概率密度函数prior probability density function先验概率区域prior probability region概率probability概率密度probability density概率分布probability distribution离散随机变量的概率分布probability distribution for discrete random variable概率分布函数probability distribution function概率测度probability measure概率单位（probit）模型probit model积矩product moment积矩量矩阵product moment matrix积算子product operator生产函数production function生产过程production process比例响应模型proportional response model 伪样本数据pseudo sample data二次型quadratic form二次损失函数quadratic loss function二次矩阵quadratic matrix定量选择模型quantitative choice model 定量因素quantitative factors定量信息quantitative information随机系数模型random coefficient model随机分量预测random component prediction 随机误差random error随机试验random experiment随机变量random variable随机向量random vector随机向量分量random vector component随机游走random walk秩rank秩条件rank condition矩阵的秩rank of a matrix简化型reduced form简化型系数reduced form coefficient简化型扰动reduced form disturbance简化型方程reduced form equation简化型估计式reduced form estimator。
On the Asymptotic Eigenvalue Distribution of Concatenated Vector--Valued Fading Channels
On the Asymptotic Eigenvalue Distribution of Concatenated Vector–Valued Fading ChannelsRalf R.M¨u llerJanuary31,2002AbstractThe linear vector–valued channel with and denoting additive white Gaussian noise and independent random matrices,respectively,is analyzed in the asymptotic regime as the dimensions of the matrices and vectors involved become large.The asymptotic eigenvalue distribution of the channel’s covariance matrix is given in terms of an implicit equation for its Stieltjes transform as well as an explicit expression for its moments.Additionally,almost all eigenvalues are shown to converge towards zero as the number of factors grows over all bounds. This effect cumulates the total energy in a vanishing number of dimensions.The channel model addressed generalizes the model introduced in[1]for communication via large antennas arrays to–fold scattering per propagation path.As a byproduct,the multiplica-tive free convolution is shown to extend to a certain class of asymptotically large non–Gaussian random covariance matrices.Index terms—random matrices,Stieltjes transform,channel models,fading channels,antenna arrays,multiplicative free convolution,S–transform,Catalan numbers1IntroductionConsider a communication channel with transmitting and receiving antennas grouped into a transmitter and a receiver array,respectively.Let there be clusters of scatterers each with,scattering objects.Assume that the vector–valued transmitted signal propagates from the transmitter array to thefirst cluster of scatterers,from thefirst to the second cluster,and so on,until it is received from the cluster by the receiver array.Such a channel model is discussed and physical motivation is given in[2,Sec.3].Indoor propagation between different floors,for instance,may serve as an environment where multifold scattering can be typical,cf.[3, Sec.13.4.1].The communication link outlined above is a linear vector channel that is canonically described bya channel matrix(1)where the matrices,,and denote the subchannels from the transmitter array to thefirst cluster of scatterers,from the cluster of scatterers to the cluster,and from the cluster to the receiving array,respectively.This means that is of size. Assuming distortion by additive white Gaussian noise,the complete channel is given by(2) with and denoting the vectors of transmitted and received signals,respectively.The capacity of channels as in(2)is well–known conditioned on the channel matrix,see e.g.[4].If is the result of some matrix–valued random process,only few results have been reported in literature:Telatar[5]calculates the channel capacity for if is a random matrix with zero–mean,independent complex Gaussian random entries that are known at receiver site,but unknown at the transmitter.Marzetta and Hochwald[6]found capable information rates for a setting equivalent to Telatar’s,but without knowledge of the receiver about the channel matrix.If the entries of the random matrix are not independent identically distributed(i.i.d.),analysis of those channels becomes very difficult,in general.However,the following analytical results are2known in the asymptotic regime as the size of the channel matrix becomes very large and channel state information is available at receiver site only:Tse and Hanly[7]and Verd´u and Shamai[8] independently report results for the asymptotic case with independent entries for.The case where the channel matrix is composed by with entries of i.i.d.random and denoting i.i.d.random diagonal matrices was solved by Hanly and Tse[9].Finally,M¨u ller [1]solved the case for where is a product of two independent i.i.d.random matrices.The present paper will give results for products of independent i.i.d.random matrices,cf.(1),that do not need to have the same dimensions.2Asymptotic Eigenvalue DistributionThe performance of communication via linear vector channels described as in(1)is determined by the eigenvalues of the covariance matrix.In general,not all its eigenvalues are non–zero,as(3)The empirical eigenvalue distributions,for these are the distributions(5) and the S–transformwith.Further assume:(a)be an random matrix with independent identically distributed entries with zeromean and variance,(b)as,(c)be,random,non–negative definite,with an empirical eigenvalue distributionconverging almost surely in distribution to a probability density function on,as, with non–zero mean,(d)and statistically independent,(e).Then,the empirical eigenvalue distribution of converges almost surely,as,to a non–random limit that is uniquely determined by the S–transform(7) Moreover,.The proof is placed in Appendix A.Note that in addition to the results on multiplicative free convolu-tion in[10],Theorem1states almost sure convergence and it is not restricted to Gaussian,diagonal, or unitary random matrices.The asymptotic limits for may serve as good estimates for the eigenvalues in the non–asymptotic case.This has been verified for code–division multiple–access systems in[11,12]and it is likely to generalize to a broader class of communication systems described by large random matrices. In the following,the asymptotic distributions of the eigenvalues are calculated.Assume that all matrices are statistically independent and their entries are zero–mean i.i.d. random variables with variance.Define the ratiosand assume that all tend to infinity,but the ratios remain constant.Consider the random covariance matrices(9)(10)Note that their non–zero eigenvalues are identical.Thus,by Theorem1and induction over their respective eigenvalue distributions converge to a non–random limit almost surely,as, but.The asymptotic distribution of the eigenvalues is conveniently represented in terms of its Stieltjes transform1(12)It will turn out helpful for calculation of to consider the matrix instead of the original matrix in the following.Since the non–zero eigenvalues of both matrices are identical,their empirical distributions differ only by a scaling factor and a point mass at zero.In the Stieltjes domain,this translates into[1](13) It is straightforward from(12)and(6)that(13)reads in terms of and as(14)(15)respectively.Identifying and I,we get from Theorem1and(15)(17) Moreover,using Theorem1and(15)the following Theorem is shown in Appendix B:Theorem2Let be independent random matrices of respective sizes each with independent identically distributed zero–mean entries with respective variance.Define the ratios(19)The ratios are a generalization of the richness introduced in[1]where only was termed richness while was called system load.The theorem yields with(12)and(6)(21)The Stieltjes transform of the eigenvalue density of is determined in(21)by a polynomial equa-tion of order.For,it cannot be resolved with respect to the Stieltjes transform,in general.However,it will be shown later on,cf.Theorem3,how to obtain an infinite power series for .In addition to the statistics of the eigenvalues of,a dual also important character-ization of the channel is possible in terms of the eigenvalues of(22)6The respective Stieltjes transform can be easily derived from(21)applying the rotation formula(13) consecutively for times.After some re–arrangements of the–fold product,this gives(24) Subsequently,the ratios are termed loads since they can be interpreted as the number of logical channels normalized to the number of dimensions of the signal space at stage.This terminology is consistent with that one introduced in[1]for.It follows from the definition of the Stieltjes transform(11)and the Taylor expansion of its kernel that(25)where denotes the moment of the eigenvalue density and is the Z–transform of the sequence of moments.In terms of the loads,it is convenient to write the moments of the eigenvalue distributions of both and.Theorem3Assume that the conditions required for Theorem2are fulfilled.Let and be defined as in(24)and(22),respectively.Then,for,the moments of the empirical eigenvalue distributions of and converge almost surely to the non–random limits(28)7cf.[14,Problem III.211].The moments in(28)are the generalized Catalan numbers,see e.g.[15]for a tutorial on their properties,and are known to appear in many different problems in combinatorics. Explicit expressions for the moments are particularly useful for the design and analysis of polynomial expansion equalizers,cf.[16,17].Note from the definition of the Stieltjes transform(11)that is the harmonic mean of the eigenvalues of.It can be calculated explicitly with(23)and reads(29)As the number of factors in the product increases,the harmonic mean strictly decreases,while the arithmetic means remain constant due to the assumed normalization of variances of the matrix ele-ments.This indicates that the product matrix becomes closer and closer to singularity as increases, even if all factors are fully ranked(i.e.).This convergence to singularity will be examined more precisely and in greater detail in the next section.3Infinite ProductsIt is interesting to consider the limiting eigenvalue distribution as:In the Appendix D,we proofTheorem4Assume that and the series is upper bounded.Then,almost all eigenvalues of and of converge to zero.Note that this means(30) However,since integrals and limits do not necessarily commute,i.e.(31)8in general.Theorem3and Theorem4do not contradict,although they give different results for the moments of the eigenvalue distribution as:(32)(33)The distributionforFigure1:Convergence of cumulative distribution function for increasing number of factors.Curves are generated numerically by multiplying Gaussian random matrices of size.The dashed lines refer to(34)for comparison.4ProspectPreviously,asymptotic eigenvalue distributions were characterized in terms of their Stieltjes trans-forms and moments.As shown in[7,1],Stieltjes transforms can be used to express more intuitive performance measures of communication systems like signal–to–interference–and–noise ratios and channel capacity.For such purposes the reader is referred to the respective papers.The results derived in this paper are asymptotic with respect to the size of the random matrices involved.However,there is strong numerical evidence supporting the conjecture that Theorem4also holds for a large class offinite-dimensional random matrices with even not i.i.d.entries.An illustrat-ing example in this respect are longfinite impulse responsefilters with i.i.d.random coefficients(they10correspond to circulant random matrices):Passing a white random process repeatedly through inde-pendent realizations of suchfilters gives a sinusoidal signal at the output with a random frequency. Obviously,all but one dimension of the signal space have collapsed.AcknowledgmentThe author would like to thank A.Grant,C.Mecklenbr¨a uker,E.Schofield,H.Hofstetter,K.Kopsa, and the anonymous reviewers for helpful comments.AppendixA Proof of Theorem1Under the assumptions(a)to(e)of Theorem1,the empirical eigenvalue distribution of is shown to converge almost surely to a non–random limit distribution in[19,Theorem1.1].Characterizing this limit distribution by its Stieltjes transform(11),wefind[19,Eq.(1.4)]2(36)(37)2Note the different sign of compared to the reference due to the different definition of the Stieltjes transform in(11).11(40)(41) The definition of the S–transform(6)gives(42) and re–arranging terms yields(44)First,(44)is verified for.Note that(17)holds for all.Therefore,(45) which proofs(44)for.Second,assuming(44)holding for the–fold product,(44)is shown to also hold for the–fold product.Note from(10)that(46)12Theorem1gives(51) Hereby,the induction is complete.C Proof of Theorem3Combining(23)and(12)yields(52) Solving for givesParticularly,wefind(55)(56)The only term of(56)which matters in(55)is the one including.Thus,we can restrict the summation indices of(56)to satiesfywhich is equivalent to(57) Since(58) we getFirst,consider the matrix with.Note from(23)that the asymptotic eigenvalue distribution is invariant to any permutation of the ratios.Thus,without loss of generality,we set(60) From(23),we have(62) Note that due to(60)(63)Note from(11)that is always positive for positive arguments.Thus,for any positive,one of the following three statements must be true:1.2.3.for some positive and.Statement1is in contradiction to(61),since a sum of positive terms with one term larger than1 cannot be1.Statement2,in combination with(61)impliesThus,we have[13]Harry Bateman.Table of Integral Transforms,volume2.McGraw–Hill,New York,1954.[14]George P´o lya and Gabor Szeg¨o.Problems and Theorems in Analysis,volume1.Springer–Verlag,Berlin,Germany,1972.[15]Peter Hilton and Jean Pedersen.Catalan numbers,their generalization,and their uses.The MathematicalIntelligencer,13(2):64–75,1991.[16]Ralf R.M¨u ller and Sergio Verd´u.Design and analysis of low–complexity interference mitigation onvector channels.IEEE Journal on Selected Areas in Communications,19(8):1429–1441,August2001.[17]Ralf R.M¨u ller.Polynomial expansion equalizers for communication via large antenna arrays.In Proc.ofEuropean Personal Mobile Communications Conference,Vienna,Austria,February2001.[18]William C.Y.Lee.Mobile Communications Design Fundamentals.John Wiley&Sons,New York,1993.[19]Jack W.Silverstein.Strong convergence of the empirical distribution of eigenvalues of large dimensionalrandom matrices.Journal of Multivariate Analysis,55:331–339,1995.List of Figures1Convergence of cumulative distribution function for increasing number of factors.Curves are generated numerically by multiplying Gaussian random matrices of size.The dashed lines refer to(34)for comparison (10)17。
The structure and function of complex networks
SIAM R EVIEW c 2003Society for Industrial and Applied Mathematics Vol.45,No.2,pp.167–256The Structure and Function ofComplex Networks∗M.E.J.Newman†Abstract.Inspired by empirical studies of networked systems such as the Internet,social networks, and biological networks,researchers have in recent years developed a variety of techniquesand models to help us understand or predict the behavior of these systems.Here wereview developments in thisfield,including such concepts as the small-world effect,degreedistributions,clustering,network correlations,random graph models,models of networkgrowth and preferential attachment,and dynamical processes taking place on networks.Key works,graph theory,complex systems,computer networks,social networks,random graphs,percolation theoryAMS subject classifications.05C75,05C90,94C15PII.S0036144503424804Contents.1Introduction1681.1Types of Networks (171)1.2Other Resources (172)1.3Outline of the Review (174)2Networks in the Real World1742.1Social Networks (174)2.2Information Networks (176)2.3Technological Networks (178)2.4Biological Networks (179)3Properties of Networks1803.1The Small-World Effect (180)3.2Transitivity or Clustering (183)3.3Degree Distributions (185)3.3.1Scale-Free Networks (186)3.3.2Maximum Degree (188)3.4Network Resilience (189)3.5Mixing Patterns (190)∗Received by the editors January20,2003;accepted for publication(in revised form)March17, 2003;published electronically May2,2003.This work was supported in part by the U.S.National Science Foundation under grants DMS-0109086and DMS-0234188and by the James S.McDonnell Foundation and the Santa Fe Institute./journals/sirev/45-2/42480.html†Department of Physics and Center for the Study of Complex Systems,University of Michigan, Ann Arbor,MI48109(mejn@).167168M.E.J.NEWMAN3.6Degree Correlations (192)3.7Community Structure (193)3.8Network Navigation (195)3.9Other Network Properties (196)4Random Graphs1964.1Poisson Random Graphs (197)4.2Generalized Random Graphs (200)4.2.1The Configuration Model (200)4.2.2Example:Power-Law Degree Distribution (202)4.2.3Directed Graphs (203)4.2.4Bipartite Graphs (204)4.2.5Degree Correlations (205)5Exponential Random Graphs and Markov Graphs206 6The Small-World Model2086.1Clustering Coefficient (210)6.2Degree Distribution (210)6.3Average Path Length (211)7Models of Network Growth2127.1Price’s Model (213)7.2The Model of Barab´a si and Albert (215)7.3Generalizations of the Model of Barab´a si and Albert (219)7.4Other Growth Models (221)7.5Vertex Copying Models (223)8Processes T aking Place on Networks2248.1Percolation Theory and Network Resilience (225)8.2Epidemiological Processes (229)8.2.1The SIR Model (229)8.2.2The SIS Model (232)8.3Search on Networks (233)8.3.1Exhaustive Network Search (234)8.3.2Guided Network Search (235)8.3.3Network Navigation (236)8.4Phase Transitions on Networks (238)8.5Other Processes on Networks (239)9Summary and Directions for Future Research2401.Introduction.A network is a set of items,which we will call vertices or some-times nodes,with connections between them,called edges(Figure1.1).Systems taking the form of networks(also called“graphs”in much of the mathematical lit-erature)abound in the world.Examples include the Internet,the World Wide Web, social networks of acquaintance or other connections between individuals,organiza-tional networks and networks of business relations between companies,neural net-works,metabolic networks,food webs,distribution networks such as blood vessels orTHE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS169vertexFig.1.1A small example network with eight vertices and ten edges.postal delivery routes,networks of citations between papers,and many others(Fig-ure1.2).This paper reviews recent(and some not-so-recent)work on the structure and function of networked systems such as these.The study of networks,in the form of mathematical graph theory,is one of the fundamental pillars of discrete mathematics.Euler’s celebrated1735solution of the K¨o nigsberg bridge problem is often cited as thefirst true proof in the theory of net-works,and during the twentieth century graph theory has developed into a substantial body of knowledge.Networks have also been studied extensively in the social sciences.Already in the 1930s,sociologists realized the importance of the patterns of connection between peo-ple to the understanding of the functioning of human society(Figure1.3).Typical net-work studies in sociology involve the circulation of questionnaires,asking respondents to detail their interactions with others.One can then use the responses to reconstruct a network in which vertices represent individuals and edges the interactions between them.Typical social network studies address issues of centrality(which individuals are best connected to others or have most influence)and connectivity(whether and how individuals are connected to one another through the network).Recent years,however,have witnessed a substantial new movement in network research,with the focus shifting away from the analysis of single small graphs and the properties of individual vertices or edges within such graphs to consideration of large-scale statistical properties of graphs.This new approach has been driven largely by the availability of computers and communication networks that allow us to gather and analyze data on a scale far larger than previously possible.Where studies used to look at networks of maybe tens or in extreme cases hundreds of vertices,it is not uncommon now to see networks with millions or even billions of vertices.This change of scale forces upon us a corresponding change in our analytic approach.Many of the questions that might previously have been asked in studies of small networks are simply not useful in much larger networks.A social network analyst might have asked,“Which vertex in this network would prove most crucial to the network’s connectivity if it were removed?”But such a question has little meaning in most networks of a million vertices—no single vertex in such a network will have much effect at all when removed.On the other hand,one could reasonably ask a question like,“What percentage of vertices need to be removed to substantially affect network connectivity in some given way?”and this type of statistical question has real meaning even in a very large network.However,there is another reason why our approach to the study of networks has changed in recent years,a reason whose importance should not be underestimated, although it often is.For networks of tens or hundreds of vertices,it is a relatively straightforward matter to draw a picture of the network with actual points and lines,170M.E.J.NEWMAN(c)(a)Fig.1.2Three examples of the kinds of networks that are the topic of this review.(a)A visualization of the network structure of the Internet at the level of “autonomous systems”—local groups of computers each representing hundreds or thousands of machines.Picture by Hal Burch and Bill Cheswick,courtesy of Lumeta Corporation.(b)A social network,in this case of sexual contacts,redrawn from the HIV data of Potterat et al.[341].(c)A food web of predator-prey interactions between species in a freshwater lake [271].Picture courtesy of Richard Williams.and to answer specific questions about network structure by examining this picture.This has been one of the primary methods of network analysts since the field be-gan (see Figure 1.3).The human eye is an analytic tool of remarkable power,and eyeballing pictures of networks is an excellent way to gain an understanding of theirTHE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS171Fig.1.3An early hand-drawn social network from1934representing friendships between school children.After Moreno[295].Reprinted with permission from ASGPP.structure.With a network of a million or a billion vertices,however,this approach is useless.(See Figure1.2a for an example of a network that lies at the upper limit of what can usefully be drawn on a piece of paper or computer screen.)One simply can-not draw a meaningful picture of a million vertices,even with modern3D computer rendering tools,and therefore direct analysis by eye is hopeless.The recent devel-opment of statistical methods for quantifying large networks is to a large extent an attempt tofind something to play the part played by the eye in the network analysis of the twentieth century.Statistical methods answer the question,“How can I tell what this network looks like,when I can’t actually look at it?”The body of theory that is the primary focus of this review aims to do three things.First,it aims tofind and highlight statistical properties,such as path lengths and degree distributions,that characterize the structure and behavior of networked systems,and to suggest appropriate ways to measure these properties.Second,it aims to create models of networks that can help us to understand the meaning of these properties—how they came to be as they are,and how they interact with one another. Third,it aims to predict what the behavior of networked systems will be on the basis of measured structural properties and the local rules governing individual vertices.How, for example,will network structure affect traffic on the Internet,or the performance of a Web search engine,or the dynamics of social or biological systems?As we will see,the scientific community has,by drawing on ideas from a broad variety of disciplines,made an excellent start on thefirst two of these aims,the characterization and modeling of network structure.Studies of the effects of structure on system behavior on the other hand are still in their infancy.It remains to be seen what the crucial theoretical developments will be in this area.1.1.T ypes of Networks.A set of vertices joined by edges is only the simplest type of network;there are many ways in which networks may be more complex than this(Figure1.4).For instance,there may be more than one different type of vertex in a network,or more than one different type of edge.And vertices or edges may have a variety of properties,numerical or otherwise,associated with them.Taking the example of a social network of people,the vertices may represent men or women, people of different nationalities,locations,ages,incomes,or many other things.Edges may represent friendship,but they could also represent animosity,or professional acquaintance,or geographical proximity.They can carry weights,representing,say,172M.E.J.NEWMANFig.1.4Examples of various types of networks:(a)an undirected network with only a single type of vertex and a single type of edge;(b)a network with a number of discrete vertex and edge types;(c)a network with varying vertex and edge weights;(d)a directed network in which each edge has a direction.how well two people know each other.They can also be directed,pointing in only one direction.Graphs composed of directed edges are themselves called directed graphs or sometimes digraphs,for short.A graph representing telephone calls or email messages between individuals would be directed,since each message only goes in one direction. Directed graphs can be either cyclic,meaning they contain closed loops of edges,or acyclic,meaning they do not.Some networks,such as food webs,are approximately but not perfectly acyclic.One can also have hyperedges—edges that join more than two vertices together. Graphs containing such edges are called hypergraphs.Hyperedges could be used to indicate family ties in a social network for example—n individuals connected to each other by virtue of belonging to the same immediate family could be represented by an n-edge joining them.Graphs may also be naturally partitioned in various ways.We will see a number of examples in this review of bipartite graphs:graphs that contain vertices of two distinct types,with edges running only between unlike types.So-called affiliation networks in which people are joined together by common membership of groups take this form,the two types of vertices representing the people and the groups. Graphs may also evolve over time,with vertices or edges appearing or disappearing, or values defined on those vertices and edges changing.And there are many other levels of sophistication one can add.The study of networks is by no means a complete science yet,and many of the possibilities have yet to be explored in depth,but we will see examples of at least some of the variations described here in the work reviewed in this paper.The jargon of the study of networks is unfortunately confused by differing usages among investigators from differentfields.To avoid(or at least reduce)confusion,we give in Box1a short glossary of terms as they are used in this paper.1.2.Other Resources.A number of other reviews of this area have appeared recently,which the reader may wish to consult.Albert and Barab´a si[13]and Doro-govtsev and Mendes[120]have given extensive pedagogical reviews focusing on the physics literature.Both devote the larger part of their attention to the models ofTHE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS173 Vertex(pl.vertices):The fundamental unit of a network,also called a site (physics),a node(computer science),or an actor(sociology).Edge:The line connecting two vertices.Also called a bond(physics),a link (computer science),or a tie(sociology).Directed/undirected:An edge is directed if it runs in only one direction(such as a one-way road between two points),and undirected if it runs in both directions.Directed edges,which are sometimes called arcs,can be thought of as sporting arrows indicating their orientation.A graph is directed if all of its edges are directed.An undirected graph can be represented by a directed one having two edges between each pair of connected vertices,one in each direction.Degree:The number of edges connected to a vertex.Note that the degree is not necessarily equal to the number of vertices adjacent to a vertex,since there may be more than one edge between any two vertices.In a few recent articles,the degree is referred to as the“connectivity”of a vertex,but we avoid this usage because the word connectivity already has another meaning in graph theory.A directed graph has both an in-degree and an out-degree for each vertex,which are the numbers of incoming and outgoing edges respectively.Component:The component to which a vertex belongs is that set of vertices that can be reached from it by paths running along edges of the graph.In a directed graph a vertex has both an in-component and an out-component,which are the sets of vertices from which the vertex can be reached and which can be reached from it.Geodesic path:A geodesic path is the shortest path through the network from one vertex to another.Note that there may be and often is more than one geodesic path between two vertices.Diameter:The diameter of a network is the length(in number of edges)of the longest geodesic path between any two vertices.A few authors have also used this term to mean the average geodesic distance in a graph,although strictly the two quantities are quite distinct.Box1A short glossary of terms.growing graphs that we describe in section7.Shorter reviews taking other viewpoints have been given by Newman[308]and Hayes[188,189],who both concentrate on the so-called small-world models(see section6),and by Strogatz[386],who includes an interesting discussion of the behavior of dynamical systems on networks.A number of books also make worthwhile reading.Dorogovtsev and Mendes[122] have expanded their above-mentioned review into a book,which again focuses on models of growing graphs.The edited volumes by Bornholdt and Schuster[70]and by Pastor-Satorras and Rubi[329]both contain contributed essays on various topics by leading researchers.Detailed treatments of many of the topics covered in the present work can be found there.The book by Newman,Barab´a si,and Watts[319] is a collection of previously published papers and also contains some review material by the editors.Three popular books on the subject of networks merit a mention.Barab´a si’s Linked[31]gives a personal account of recent developments in the study of net-works,focusing particularly on Barab´a si’s work on scale-free networks.Watts’s Six Degrees[413]gives a sociologist’s view,partly historical,of discoveries old and new. Buchanan’s Nexus[76]gives an entertaining portrait of thefield from the point of view of a science journalist.174M.E.J.NEWMANFarther afield,there are a variety of books on the study of networks in particular fields.Within graph theory the books by Harary[187]and by Bollob´a s[62]are widely cited as are,among social network theorists,the books by Wasserman and Faust[408] and by Scott[362].The book by Ahuja,Magnanti,and Orlin[7]is a useful source for information on network algorithms.1.3.Outline of the Review.The outline of this paper is as follows.In section2 we describe empirical studies of the structure of networks,including social networks, information networks,technological networks,and biological networks.In section3we describe some of the common properties that are observed in many of these networks, how they are measured,and why they are believed to be important for the functioning of networked systems.Sections4to7form the heart of the review.They describe work on the mathematical modeling of networks,including random graph models and their generalizations,exponential random graphs,p∗models and Markov graphs, the small-world model and its variations,and models of growing graphs including preferential attachment models and their many variations.In section8we discuss the progress,such as it is,that has been made on the study of processes taking place on networks,including epidemic processes,network failure,models displaying phase transitions,and dynamical systems like random Boolean networks and cellular automata.In section9we give our conclusions and point to directions for future research.works in the Real World.In this section we look at what is known about the structure of networks of different types.Recent work on the mathematics of networks has been driven largely by observations of the properties of actual networks and attempts to model them,so network data are the obvious starting point for a review such as this.It also makes sense to examine simultaneously data from different kinds of networks.One of the principal thrusts of recent work in this area, inspired particularly by a groundbreaking1998paper by Watts and Strogatz[415], has been the comparative study of networks from different branches of science,with emphasis on properties that are common to many of them and the mathematical developments that mirror those properties.We here divide our summary into four loose categories of networks:social networks,information networks,technological networks,and biological networks.2.1.Social Networks.A social network is a set of people or groups of people with some pattern of contacts or interactions between them[362,408].The pat-terns of friendships between individuals[295,347],business relationships between companies[268,285],and intermarriages between families[326]are all examples of networks that have been studied in the past.1Of the academic disciplines,the social sciences have the longest history of the substantial quantitative study of real-world networks[162,362].Of particular note among the early works on the subject are the following:Moreno’s networks of friendships within small groups[295],of which Figure1.3is an example;the so-called southern women study of Davis,Gardner,and Gardner[103],which focused on the social circles of women in an unnamed city in the American south in1936;the study by Elton Mayo and colleagues of social net-works of factory workers in the late1930s in Chicago[356];the mathematical models of Rapoport[345],who was one of thefirst theorists,perhaps thefirst,to stress 1Occasionally social networks of animals have been investigated also,such as dolphins[96],not to mention networks offictional characters,such as the protagonists of Tolstoy’s Anna Karenina[243] or Marvel Comics superheroes[10].THE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS175 the importance of the degree distribution in networks of all kinds,not just social networks;and the studies of friendship networks of school children by Rapoport and others[347,149].In more recent years,studies of business communities[167,168,268] and of patterns of sexual contacts[45,217,242,265]have attracted particular atten-tion.Another important set of experiments are the famous“small-world”experiments of Milgram[282,392].No actual networks were reconstructed in these experiments, but nonetheless they tell us about network structure.The experiments probed the distribution of path lengths in an acquaintance network by asking participants to pass a letter2to one of theirfirst-name acquaintances in an attempt to get it to an assigned target individual.Most of the letters in the experiment were lost,but about a quarter reached the target and passed on average through the hands of only about six people in doing so.This experiment was the origin of the popular concept of the “six degrees of separation,”although that phrase did not appear in Milgram’s writing, being coined some decades later by Guare[182].A brief but useful early review of Milgram’s work and work stemming from it was given by Garfield[169].Traditional social network studies often suffer from problems of inaccuracy,sub-jectivity,and small sample size.With the exception of a few ingenious indirect studies such as Milgram’s,data collection is usually carried out by querying participants di-rectly using questionnaires or interviews.Such methods are labor-intensive and there-fore limit the size of the network that can be observed.Survey data are,moreover, influenced by subjective biases on the part of respondents;how one respondent defines a friend,for example,could be quite different from how another does.Although much effort is put into eliminating possible sources of inconsistency,it is generally accepted that there are large and essentially uncontrolled errors in most of these studies.A review of the issues has been given by Marsden[270].Because of these problems many researchers have turned to other methods for probing social networks.One source of copious and relatively reliable data is col-laboration networks.These are typically affiliation networks in which participants collaborate in groups of one kind or another,and links between pairs of individuals are established by common group membership.A classic,though rather frivolous, example of such a network is the collaboration network offilm actors,which is thor-oughly documented in the online Internet Movie Database.3In this network,actors collaborate infilms and two actors are considered connected if they have appeared in afilm together.Statistical properties of this network have been analyzed by a number of authors[4,20,322,415].Other examples of networks of this type are networks of company directors,in which two directors are linked if they belong to the same board of directors[104,105,268];networks of coauthorship among aca-demics,in which individuals are linked if they have coauthored one or more pa-pers[36,43,68,107,181,278,291,310,311,312];and coappearance networks,in which individuals are linked by mention in the same context,particularly on Web pages[3,226]or in newspaper articles[99](see Figure1.2b).Another source of reliable data about personal connections between people is communication records of certain kinds.For example,one could construct a network in which each(directed)edge between two people represented a letter or package sent by mail from one to the other.No study of such a network has been published as far as we are aware,but some similar things have.Aiello,Chung,and Lu[8,9]have analyzed2Actually a folder containing several documents.3/176M.E.J.NEWMANa network of telephone calls made over the AT&T long-distance network on a single day.The vertices of this network represent telephone numbers and the directed edges calls from one number to another.Even for just a single day this graph is enormous, having about50million vertices,one of the largest graphs yet studied after the graph of the World Wide Web.Ebel,Mielsch,and Bornholdt[136]have reconstructed the pattern of email communications betweenfive thousand students at Kiel University from logs maintained by email servers.In this network the vertices represent email addresses and directed edges represent a message passing from one address to another. Email networks have also been studied by Newman,Forrest,and Balthrop[320]and by Guimer`a et al.[184],and similar networks have been constructed for an“instant messaging”system by Smith[370],and for an Internet community Web site by Holme, Edling,and Liljeros[195].Dodds,Muhamad,and Watts[110]have carried out an email version of Milgram’s small-world experiment in which participants were asked to forward an email message to one of their friends in an effort to get the message ultimately to some chosen target individual.Response rates for the experiment were quite low,but a few hundred completed chains of messages were recorded,enough to allow various statistical analyses.rmation Networks.Our second network category is what we will call information networks(also sometimes called“knowledge networks”).The classic example of an information network is the network of citations between academic papers[138].Most learned articles cite previous works by others on related topics. These citations form a network in which the vertices are articles and a directed edge from article A to article B indicates that A cites B.The structure of the citation network then reflects the structure of the information stored at its vertices,hence the term“information network,”although certainly there are social aspects to the citation patterns of papers too[419].Citation networks are acyclic(see section1.1)because papers can only cite other papers that have already been written,not those that have yet to be written.Thus all edges in the network point backwards in time,making closed loops impossible,or at least extremely rare(see Figure2.1).As an object of scientific study,citation networks have a great advantage in the copious and accurate data available for them.Quantitative study of publication patterns stretches back at least as far as Alfred Lotka’s groundbreaking1926discovery of the so-called Law of Scientific Productivity,which states that the distribution of the numbers of papers written by individual scientists follows a power law.That is,the number of scientists who have written k papers falls offas k−αfor some constantα. (In fact,this result extends to the arts and humanities as well.)Thefirst serious work on citation patterns was conducted in the1960s as large citation databases became available through the work of Eugene Garfield and other pioneers in thefield of bibliometrics.The network formed by citations was discussed in an early paper by Price[342],in which,among other things,the author points out for thefirst time that both the in-and out-degree distributions of the network follow power laws,a far-reaching discovery which we discuss further in section3.3.Many other studies of citation networks have been performed since then,using the ever better resources available in citation databases.Of particular note are the studies by Seglen[363]and Redner[350].44An interesting development in the study of citation patterns has been the arrival of automatic citation“crawlers”that construct citation networks from online papers.Examples include Cite-seer(/),SPIRES(/spires/hep/),and Cite-base(/).。
英汉对照计量经济学术语
英汉对照计量经济学术语计量经济学术语A校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的⽅差时对添加的解释变量⽤⼀个⾃由度来调整。
对⽴假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。
AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。
渐近置信区间（Asymptotic Confidence Interval）：⼤样本容量下近似成⽴的置信区间。
渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。
渐近性质（Asymptotic Properties）：当样本容量⽆限增长时适⽤的估计量和检验统计量性质。
渐近标准误（Asymptotic Standard Error）：⼤样本下⽣效的标准误。
渐近t 统计量（Asymptotic t Statistic）：⼤样本下近似服从标准正态分布的t 统计量。
渐近⽅差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须⽤以除估计量的平⽅值。
渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的⼀致性估计量，有最⼩渐近⽅差的估计量。
渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。
衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因⽽有衰减偏误的估计量的期望值⼩于参数的绝对值。
⾃回归条件异⽅差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异⽅差性模型，即给定过去信息，误差项的⽅差线性依赖于过去的误差的平⽅。
⼀阶⾃回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：⼀个时间序列模型，其当前值线性依赖于最近的值加上⼀个⽆法预测的扰动。
计量经济学中英文词汇对照
Common variance Common variation Communality variance Comparability Comparison of bathes Comparison value Compartment model Compassion Complement of an event Complete association Complete dissociation Complete statistics Completely randomized design Composite event Composite events Concavity Conditional expectation Conditional likelihood Conditional probability Conditionally linear Confidence interval Confidence limit Confidence lower limit Confidence upper limit Confirmatory Factor Analysis Confirmatory research Confounding factor Conjoint Consistency Consistency check Consistent asymptotically normal estimate Consistent estimate Constrained nonlinear regression Constraint Contaminated distribution Contaminated Gausssian Contaminated normal distribution Contamination Contamination model Contingency table Contour Contribution rate Control
CFX多孔介质模型介绍
本文主要介绍ANSYS CFX 11.0中多孔介质模型的使用方法。
首先详细讲述了Porosity Settings 对话框的填写方法，之后以附录形式给出了多孔介质模型中的定义、术语、方程等供参考。
以下内容为本人根据CFX 帮助及相关资料编写，错漏之处敬请见谅并指正。
Porosity Settings 对话框填写说明Porosity Settings 对话框包括三项：Area Porosity 、Volume Porosity 和Loss Models 。
1. Area Porosity ： 即面积孔隙率，是指流体可以穿过的面积占物理面积的份额，默认为Isotropic （各向同性的），不能修改。
2. Volume Porosity ：即体积孔隙率，是指允许流体流动的体积与物理体积之比。
3. Loss Models ： 即阻力损失模型，可选择Isotropic Loss （各向同性）或Directional Loss （各向异性）。
此外还有多项需要选择或填写：3.1 Loss Velocity Type ：即阻力损失对应的速度类型。
可选择Superficial （表观流速，即按物理面积计算的流速）或True Velocity （真实流速）。
3.2 若选择了Isotropic Loss ，则需要填写Isotropic Loss 对话框，其界面如图1：图1 Isotropic Loss 对话框在Option （选项栏）中，有两种阻力计算方式可以选择：3.2.1 Permeability and Loss Coefficient （渗透率和损失系数）分别填写渗透率和损失系数即可。
渗透率为多孔介质本身的性质，需通过试验测定，计算式为：Q L K A Pμ=Δ 其中，Q 为通过多孔介质的体积流量，μ为动力粘度，L 为流通长度，A 为横截面积，ΔP 为压差。
由该式可知，渗透率具有面积的量纲。
其常用单位为达西（Darcy ），物理意义为：介质允许粘度为1cp 的流体，在压力梯度为1atm/cm 的作用下，通过横截面积为1cm 2的流量为1cm 3/s ，此时，介质的渗透率称为1达西。
Some General Expressions for the Coefficient of the 14th Chern Form
〈e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ e i ∧ … … ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei ∧ ei , c(14)〉
Quattuordecic
1 1 1 3 5 − 23,401,744,351,572,787,200 − 1,428,329,123,020,800 π14 π14
COEFFICIENT OF THE 14th CHERN FORM The coefficient c(14)i =
1 28!
1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11 i12 i13 i14 i15 i16 i17 i18 i19 i20 i21 i22 i23 i24 i25 i26 i27 i28
1st & 2nd OVERALL NUMERICAL FACTORS ip 22 p π p p!
c (p )i c (14) i
1 i2 … i2 p
p 14
2p 28
—
1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18 i 19 i 20 i 21 i 22 i 23 i 24 i 25 i 26 i 27 i 2 8
b b
1 j 2… j2 p δj i 1 i 2 …i 2 p
1
2
2p
1 2
2p
(6)
is the generalized Kronecker delta.
Reduction of bias in maximum likelihood ellipse fitting
Reduction of Bias in Maximum Likelihood Ellipse FittingBogdan Matei Peter MeerElectrical Engineering Department,Rutgers University94Brett Road,Piscataway,NJ08854,USAmatei,meer AbstractAn improved maximum likelihood estimator for ellipsefitting based on the heteroscedastic errors-in-variables(HEIV)regression algorithm is proposed.The techniquesignificantly reduces the bias of the parameter estimatespresent in the Direct Least Squares method,while it is nu-merically more robust than renormalization,and requiresless computations than minimizing the geometric distancewith the Levenberg-Marquardt optimization procedure.The HEIV algorithm also provides closed-form expressionsfor the covariances of the ellipse parameters and correcteddata points.The quality of the different solutions is assessedby defining confidence regions in the input domain,eitheranalytically,or by bootstrap.The latter approach is exclu-sively data driven and it is used whenever the expression ofthe covariance for the estimates is not available.Keywords:errors-in-variables models,heteroscedasticnoise,maximum likelihood estimation,bootstrap,confi-dence regions.1.IntroductionLet,be the true values of the imagepoints assumed to obey the quadratic model(1)where is the center of ellipse and is a positive definitematrix defining the shape of the ellipse.Throughoutthe paper the subscript‘o’denotes the true,unknown valuesof the data.The true values are corrupted by additivenoise(2)where are independent variables and standsfor a general and distribution with mean and covariance.It is assumed that the covariance matrices areknown up to a common multiplicative factor,the noise vari-ance.(11)Figure1.Minimization of the geometric dis-tance from the measurements to the ellipse. The DLS estimates are significantly biased when the avail-able measurements come from short ellipse segments with low curvature.The approach,however,has the advantage of yielding a noniterative solution and automatically impos-ing the ellipse constraint(7).The Maximum Likelihood(ML)estimator under normal noise for,and is(12) subject to(3)and(7).There are unknowns,,, ,but only available measurements.A way of reducing the number of unknowns in ML esti-mation is to restrict the corrected measurements(the es-timators of the true values)to the projection of onto the ellipse,,as shown in Figure1.The criterion(12) then becomes the minimization of the sum of geometric dis-tances from to under the metric induced by.The solution of this nonlinear problem is usually obtained with the Levenberg-Marquardt(L-M)technique.The approach is computationally intensive,since it requires at each iteration stepfinding the roots of a four-degree polynomial for every point[15].To reduce the amount of computation the ge-ometric distances are often approximated by Taylor expan-sion[5,pp.230–244][13]at the price of introducing a small bias in the estimates.2.HEIV Estimator for Ellipse FittingThe improved ML algorithm for solving the linearized model(3)is presented next.The approach extends[10,11] using the results from[1].As was shown in Section1the carriers are corrupted by noise dependent on the measure-ments,i.e.,heteroscedastic noise.Since the errors cor-rupting the true values appear inside the carriers,(3) is called an errors-in-variables(EIV)model[1][5,pp.30–31].An excellent introduction to EIV nonlinear models is [5].The heteroscedastic EIV(HEIV)algorithm introduced below exploits the linear dependency of(3)in,and itera-tively solves a generalized eigenproblem.Thefirst and sec-ond order moments of the carrier noise are also updated during the iterations.The quality of the HEIV estimates can be assessed by the covariance matrices and confidence re-gions provided.2.1.HEIV AlgorithmThe update rules for the HEIV algorithm are summarized below.The iteration index is dropped for simplicity.1)Start with an initial random.Evaluate atusing(11)and let.Alternatively,may be a TLS estimate obtained by taking and .Let.2)Compute(13)Calculate the weighted centroid of the carriers(14) 3)Calculate the scatter matrix(15)and the weighted covariance matrix(16)Note that and are positive semidefinite. 4)Compute the new as the smallest unit eigenvector(the eigenvector corresponding to the smallest eigen-value)of the generalized eigenproblem(17)Compute the new intercept(18) 5)Update the corrected measurements(20) 7)Update the moments of the carrier noise(21)(22) 8)Update(23)9)Go to step2until the smallest eigenvalue.Thefirst-order renormalization technique of Kanatani for linearized models[8,pp.267–290]is a particular case of the HEIV algorithm,obtained by substituting in(16)and(24) The covariance matrices of the carriers are evaluated at the noisy measurements and are not updated during the it-erations.The renormalization iterates as the smallest eigenvector of.We have found that solving the generalized eigenvalue problem(17)at Step4yields faster convergence and less dependence on the initial. The second-order renormalization algorithm[8,pp.290–294]takes the bias of the carriers also into consideration by using second order Taylor expansion,but the analysis be-comes quite involved.2.2.Statistical Properties of HEIV EstimatesUp to afirst order approximation all the HEIV estimates are unbiased.The covariance of the corrected points given is the rank-one matrix(25)The covariance of is obtained from the perturbation theory of the eigenvectors[9,pp.72–99]where(26)(27) Assuming that the HEIV algorithm has converged,the ma-trix has rank four,and therefore the pseudoin-verse must be used in(27).2.3.Imposing the Ancillary ConstraintWhen the constraint(7)is not satisfied a has to be found such that[4](28)The solution is obtained from(29)with defined in(27).Taking the derivative of(29)with respect to and using(27)wefind(30) The DLS estimate corresponds to substituting in(30) and withFigure2.Mapping of the confidence region of the ellipse estimate into the input do-main.where denotes the quantile of the distribution with degrees of freedom.A similar expression is obtained for.3.2.Bootstrap Confidence RegionsThe bootstrap is a well-established technique to compute statistical measures for estimators using computer simula-tions with resampled data[3].To illustrate this procedure, confidence regions for the DLS ellipse estimates are con-structed.The corrected measurements and their covariance ma-trices(25)returned by the HEIV algorithm are employed together with and in the generation of new boot-strap samples,using the boot-strap of residuals[11].For each bootstrap sample,the boot-strap replicates are computed(using DLS algorithm this time)and employed subsequently for computing the covari-ance matrix and confidence region for the DLS es-timate[12].4.ExperimentsFour algorithms:DLS,HEIV,renormalization and the geometric distance minimized with L-M were compared us-ing synthetic and real data.The DLS estimates are generally biased when the available data comes from short ellipse seg-ments with a low curvature,as illustrated in Figure3.The L-M,though theoretically unbiased,may yield biased when its initialization,most often done using DLS,is far from the true value.To assess the accuracy of the solutions yielded by these ellipse estimators,the true values were selected from different segments of the ellipseNormal noise with zero mean and covariance,was added on the true values,shown in Figure4.Figure3.The four ellipse estimators used in the experiments.The218data points are highlighted in the back.Figure4.Synthetic data.Normal noise with zero-mean and covariance was addedfor(left)and for(right)using the synthetic data from Figure4(a).500trials. for(left)and for(right)using the synthetic data from Figure4(b).500trials.The histograms of errors in the estimation of and using500Monte Carlo trials are presented in Figures5and6.The DLS had the smallest spread,however,it was strongly biased.The MATLAB implementation of the L-M algorithm was used in the simulations.Note in Figures5and6how the bias in estimating and was reduced after100and200L-M iterations.For the data from Figure4(a)the HEIV required about four iterations to reach convergence(with tolerance).The HEIV estimate satisfied the ancillary con-straint(6)in all500trials.The renormalization requiredabout six iterations to reach convergence(same tolerance for the eigenvalue)and in3trials the ancillary constraint(6)had to be imposed.The HEIV and the renormalization are unbi-ased,with the latter having a slightly larger spread.The data from Figure4(b)leads to a more difficult es-timation process since it comes from a shorter ellipse seg-ment.The L-M remained biased even after200iterations. The ancillary constraint had to be imposed in72trials for HEIV and in108trials for the renormalization.Both algo-rithms required more iterations to converge(about seven for HEIV and ten for renormalization).Note again the larger spread of the estimates yielded by the renormalization com-pared with HEIV.The confidence regions for the HEIV and DLS esti-mates constructed using the technique from Section3are presented in Figure7.The covariance matrices were computed using(26)for HEIV and bootstrap repli-cates for DLS.To assess the accuracy of,derived ana-lytically for HEIV,200bootstrap replicates were also calcu-lated.Note from Figure7(a)the symmetric distribution of these replicates around(zero-bias)and inside the theoreti-cal confidence region.On the other hand,the DLS bootstrap replicates are not symmetrically distributed,suggesting a bi-ased estimate.5.ConclusionAn improved ML ellipse estimator based on the HEIV al-gorithm is presented.Closed-form expressions for the co-variance of the HEIV estimates are provided.Confidence regions of the ellipse estimates are constructed either ana-litically,or by bootstrap.The mapping in the input space of these confidence regions allows an easier assessment of the reliability of the solution yielded.References[1]Y.Amemiya,“Generalization of the TLS Approach in Errors-Variables Problem”,Recent Advances in Total Least Squares Tech-niques and Errors-In-Variables Modeling,pp.77–86,SIAM,1997.[2]L.Breiman,Statistics:With a View Toward Applications,HoughtonMifflin Company,1973.[3] B.Efron,R.J.Tibshirani,An Introduction to the Bootstrap,Chap-man&Hall,1993.(a)(b)Figure7.Confidence regions for the esti-mates:(a)HEIV-obtained analitically and with bootstrap;(b)DLS-obtained with boot-strap.The153data points used are on the highlighted segment on the right.[4] A.Fitzgibbon,M.Pilu,R.B.Fisher,“Direct least Square Fitting ofEllipses”,IEEE Trans.on P AMI,vol.21,pp.476–480,1999.[5]W.Fuller,Measurement Error Models,John Wiley&Sons,1987.[6]J.Hornegger,C.Tomasi,“Representation Issues in the ML Estima-tion of Camera Motion”,Proc.7th ICCV,Kerkyra-Greece,pp.640–647,1999.[7]K.Kanatani,“Statistical Bias of Conic Fitting and Renormaliza-tion”,IEEE Trans.on P AMI,vol.16,pp.320-326,1994.[8]K.Kanatani,Statistical Optimization for Geometric Computation:Theory and Practice,Elsevier,1996.[9]T.Kato,A Short Introduction to Perturbation Theory for Linear Op-erators,Springer-Verlag,1982.[10]Y.Leedan,P.Meer,“Estimation with bilinear constraints in com-puter vision,”Proc.6th ICCV,Bombay,India,pp.733–738,January 1998.[11] B.Matei,P.Meer,‘’Optimal rigid motion estimation and perfor-mance evaluation with bootstrap”,Proc.of CVPR’99,Fort Collins CO.,pp.339–345,1999.[12] B.Matei,P.Meer,“Bootstrap for Errors-In-Variables”,To appear inLecture Notes in Computer Science,B.Triggs,R.Szeliski,A.Zis-serman(Eds.),Springer,Spring2000.[13]G.Taubin,“An Improved Algorithm for Algebraic Curve and Sur-face Fitting”,IEEE Trans.on P AMI,vol.13,pp.1115–1138,1991.[14]S.Van Huffel,Joos Vandewalle,The Total Least Squares Problem.Computational Aspects and Analysis,SIAM,1991.[15]Z.Zhang,“Parameter Estimation Techniques:A Tutorial with Ap-plication to Conic Fitting”,Image and Vision Computing,vol.15, pp.59–76,1997.。
SOME CONGRUENCES FOR BINOMIAL COEFFICIENTS
2
s −1
= tr (a p )
χ(a ) = ζt } χ(a p ) = ζt } χ(b ) = ζt }
pj pj
j
# {b ∈ F × q | tr (b ) = 1, cpj
Since c j is determined by j (mod t ) and p ≡ r (mod t ), c j = c p j = cr j . Lemma 2.1.4. √ g (χ ν ) ∈ Q ( −t ) √ Proof. The Galois group Gal (Q (ζt )/Q ( −t )) is cyclic of order s generated by τ : Q (ζt ) ζt −→ −→
q −1 t
−→ ω(a )
where ζt is the primitive t -th root of unity. Note that t |(q − 1). Define the Gauss sum as follows. g (χ) := −
a ∈Fq tr (a ) χ(a )ζ p
where tr : Fq −→ F p is the trace map and ζ p is the primitive p -th root of unity. Note that g (χ) ∈ Q (ζt p ) since χ(a ) ∈ Q (ζt ). Definition 2.1.1. [1]
a ∈F× q
χ ν (a ) = 0
p −1
=
(1 − ζ p )
× a∈ q tr (a )=1
χ ν (a ) + · · · + (1 − ζ p
2015+Brualdi-type eigenvalue inclusion sets of tensors
Linear Algebra and its Applications 480(2015)168–175Contents lists available at ScienceDirectLinear Algebra and its Applications/locate/laaBrualdi-type eigenvalue inclusion sets of tensorsChangjiang Bu a ,b ,∗, Yuanpeng Wei a , Lizhu Sun c , Jiang Zhou a ,da College of Science, Harbin Engineering University, Harbin 150001, PR China bCollege of Automation, Harbin Engineering University, Harbin 150001, PR China cSchool of Science, Harbin Institute of Technology, Harbin 150001, PR China dCollege of Computer Science and Technology, Harbin Engineering University, Harbin 150001, PR Chinaa r t i c l e i n f o ab s t r ac tArticle history:Received 16 March 2015Accepted 29 April 2015A vailable online 16 May 2015Submitted by R. Brualdi MSC:15A6915A18Keywords:TensorEigenvalue DigraphBy using digraphs of tensors, we give Brualdi-type eigenvalue inclusion sets of tensors. We also give some applications of our result to nonsingularity and positive definiteness of tensors.©2015 Elsevier Inc. All rights reserved.1. IntroductionFor a positive integer n , let [n ] ={1, 2, ..., n }. An order m dimension n tensor A =(a i 1i 2...i m )1 i j n (j =1,...,m )over the complex field C is a multidimensional array with entries a i 1i 2...i m ∈C . When m =2, A is an n ×n matrix. Let C n ×n ×···×n be the set of order m dimension n tensors over C . For two real tuples (a 1,a 2,...,a n )and*Corresponding author.E-mail address:buchangjiang@ (C.Bu)./10.1016/a.2015.04.0340024-3795/©2015 Elsevier Inc. All rights reserved.C.Bu et al./Linear Algebra and its Applications 480(2015)168–175169(b 1,b 2,...,b n ), (a 1,a 2,...,a n )=(b 1,b 2,...,b n )means that there exists i ∈[n ]such thata i =b i . For a tensor A =(a i 1i 2...i m ), let R i (A ) =(i 2,...,i m )=(i,...,i )|a ii 2···i m |.In 2005, Qi [12]and Lim [10]defined the eigenvalue of tensors, respectively. ForA =(a i 1i 2···i m ) ∈C n ×n ×···×n and x =(x 1,...,x n )T∈C n , A x m −1is a vector in C n whose i -th component is(A xm −1)i =n i 2,...,i m =1a ii 2···i m x i 2···x i m .A number λ ∈C is called an eigenvalue of A , if there exists a nonzero vector x ∈C nsuch thatA x m −1=λx [m −1],(1)where x [m −1]= x m −11,...,x m −1n T. In this case, x is an eigenvector of A correspond-ing to the eigenvalue λ. Let σ(A )denote the set of all eigenvalues of A . There has been extensive attention and interest in spectral theory of tensors and hypergraphs (see [4,8–14,17]).The well-known Geršgorin’s and Brauer’s eigenvalue inclusion sets of matrices were given in [7]and [1], respectively. In [12], Qi gave Geršgorin-type eigenvalue inclusion sets of real symmetric tensors. This result also holds for general tensors [16]. Li et al. [9]gave Brauer-type eigenvalue inclusion sets of tensors.The associated digraph D (A )of a matrix A =(a ij ) ∈C n ×n has vertex set V ={1, 2, ..., n }and arc set E ={(i, j )|a ij =0, i =j }. Let C (A )denote the set of circuits of D (A ). A matrix A is called weakly irreducible if each vertex in D (A )belongs to some circuit of D (A )(see [2]). By using digraphs of matrices, Brualdi gave the following eigenvalue inclusion set.Theorem 1.1. (See [2].) Let A =(a ij )∈C n ×n be a weakly irreducible matrix. Thenσ(A )⊆γ∈C (A )⎧⎨⎩z ∈C :i ∈γ|z −a ii |i ∈γR i (A )⎫⎬⎭.In this paper, we give Brualdi-type eigenvalue inclusion sets of tensors, which extend some results in [2]to tensors. We also give applications of our result to nonsingularity and positive definiteness of tensors.2. PreliminariesLet Γbe a digraph with vertex set V and arc set E . If there exist directed paths from i to j and j to i for any i, j ∈V (i =j ), then Γis called strongly connected . For each170 C.Bu et al./Linear Algebra and its Applications480(2015)168–175vertex i∈V,if there exists a circuit such that i belong to the circuit,thenΓis called weakly connected.For v∈V,letΓ+(v)={u∈V:(v,u)∈E}.A pre-order defined on V satisfies(i)a a(a∈V)(ii)a b and b c implies a c(a,b,c∈V)(iii)a b and b a cannot conclude a=b[2].Lemma2.1.(See[2].)LetΓbe a digraph for which a pre-order is defined on its vertexset.IfΓ+(v)is nonempty for each v inΓ,then there exists circuit v i1,...,v ik,v ik+1=v i1such that v ij+1is a maximal element inΓ+(v ij)for j=1,...,k.The determinant of a tensor A=(a i1···i m )∈C n×···×n is the resultant of A x m−1=0,denoted by det(A).Lemma2.2.(See[8].)For A=(a i1i2···i m)∈C n×n×···×n,det(A)=0if and only if0is an eigenvalue of A.Shao introduced the following product of tensors,which is a generalization of the matrix multiplication.Definition2.3.(See[13].)Let A and B be order m 2and order k 1,dimension n tensors,respectively.The product AB is the following tensor C of order(m−1)(k−1)+1 and dimension n with entries:c iα1...αm−1=i2,...,i m∈[n]a ii2...i mb i2α1···b i mαm−1,where i∈[n],α1,...,αm−1∈[n]k−1.Lemma2.4.(See[15].)Let A be an order m dimension n tensor,and let B be an order k dimension n tensor.Then det(AB)=det(A)(k−1)n−1det(B)(m−1)n.Lemma2.5.(See[13].)Let A and B be two order m dimension n tensors.If there exists an invertible diagonal matrix D such that B=D−(m−1)A D,thenσ(A)=σ(B).For a tensor A=(a i1···i m )∈C n×···×n,we associate with A a digraphΓA as follows.The vertex set ofΓA is V(A)={1,...,n},the arc set ofΓA is E(A)={(i,j)|a ii2···i m =0,j∈{i2,...,i m}={i,...,i}}.A tensor is called nonnegative if all its entries are nonnegative.A tensor A is called weakly irreducible ifΓA is strongly connected[6,11].Lemma2.6.(See[3].)Let A be a weakly irreducible nonnegative tensor.Then A has an eigenvalueλ>0with a positive eigenvector.C.Bu et al./Linear Algebra and its Applications 480(2015)168–1751713. Main resultsFor a tensor A =(a i 1···i m ) ∈C n ×···×n , let C (A )denote the set of circuits of ΓA , andlet G (A ) = ni =1{z ∈C :|z −a ii ···i | R i (A )}. Qi [12]proved that σ(A ) ⊆G (A ). We extend Theorem 1.1to tensors as follows.Theorem 3.1. Let A =(a i 1...i m )∈C n ×···×n be a tensor such that ΓA is weakly connected. Thenσ(A )⊆D = γ∈C (A )⎧⎨⎩z ∈C : i ∈γ|z −a ii ···i | i ∈γR i (A )⎫⎬⎭⊆G (A ).Proof.Let λbe any eigenvalue of A . Since ΓA is weakly connected, λ ∈D if λ =a ii (i)for some i ∈[n ]. Suppose that λ =a ii ···i (i =1, 2, ..., n ). Let x =(x 1, ..., x n )T ∈C n be an eigenvector corresponding to λ, and let Γ0be the subgraph of ΓA induced by those vertices i for which x i =0. By Eq.(1), we get(λ−a ii ···i )x m −1i=(i 2,...,i m )=(i,...,i )a ii 2...i m x i 2...x i m (i =1,...,n ).(2)Since λ =a ii ···i , by the above equation, we know that Γ+0(i )is nonempty for each vertexi in Γ0. Define the pre-order i j on the vertex set of Γ0if and only if |x i | |x j |. Lemma 2.1implies that Γ0has a circuit γ={i 1, ..., i p , i p +1=i 1}such that x i j +1 |x k |for any k ∈Γ+0(i j )(j =1, ..., p ). By Eq.(2), we getλ−a ij i j ...i j x m −1i jR i j (A ) x m −1i j +1 (j =1,...,p ).Hencep j =1λ−a i j ...i jp j =1x m −1i j p j =1R i j (A )p j =1x m −1i j +1 .Since i p +1=i 1, x i j =0(j =1, ..., p ), we havep j =1λ−a i j ...i jpj =1R i j (A ),that isi ∈γ|λ−a i ···i |i ∈γR i (A ).Hence the region D contains all eigenvalues of A .172 C.Bu et al./Linear Algebra and its Applications 480(2015)168–175Next we show that D ⊆G (A ). For any z ∈D , if z /∈G (A ), then |z −a ii ···i | >R i (A )(i =1, ..., n ). In this case,i ∈γ|z −a ii ···i |> i ∈γR i (A )for any γ∈C (A ), a contradictionto z ∈D . Hence z ∈G (A ), i.e., D ⊆G (A ).2We can get the following result from Lemma 2.2and Theorem 3.1.Corollary 3.2. Let A =(a i 1···i m ) ∈C n ×···×n be a tensor such that ΓA is weakly connected.Ifi ∈γ|a ii ···i |> i ∈γR i (A )for each γ∈C (A ), then det(A ) =0.The following result extends Theorem 2.9 in [2]to tensors.Theorem 3.3. Let A =(a i 1···i m ) ∈C n ×···×n be a tensor such that ΓA is strongly connected.Ifi ∈γ|a ii ···i | i ∈γR i (A )for each γ∈C (A )and there exists at least one circuit such that the inequality holds strictly, then det(A ) =0.Proof.Since ΓA is strongly connected andi ∈γ|a ii ···i | i ∈γR i (A )for each γ∈C (A ), we have a ii ···i =0for each i ∈[n ]. Suppose that det(A ) =0, i.e., 0is an eigenvalue of A . Let x =(x 1, ..., x n )T ∈C n be an eigenvector corresponding to 0, and let Γ0be the subgraph of Γ(A )induced by those vertices i for which x i =0. From the proof of Theorem 3.1, we know that Γ0has a circuit γ1={i 1, ..., i p , i p +1=i 1}such thati ∈γ1|a ii ···i | i ∈γ1R i (A )and x i j +1 |x k |for any k ∈Γ+0(i j )(j =1, ..., p ). Since i ∈γ1|a ii ···i | i ∈γ1R i (A ), we havei ∈γ1|a ii ···i |= i ∈γ1R i (A ). By the proof of Theorem 3.1, we have |x k |= x i j +1 for each k ∈Γ+A (i j )(j =1, ..., p ).Sincei ∈γ|a ii ···i |> i ∈γR i (A )for some γ∈C (A ), ΓA has at least a vertex which is not in γ1. Since ΓA is strongly connected, there exists an arc from some vertex i j of γ1to a vertex v which is not in γ1. Since x i j +1 =|x v |, Γ0has a circuit γ2which is different from γ1, and γ2satisfies the conclusion of Lemma 2.1. Similar with the caseof γ1, we also havei ∈γ2|a ii ···i |= i ∈γ2R i (A ), and for each vertex i of γ2, |x j |is constant over all j ∈Γ+A (i ). Continuing like this, we know that for each vertex i in ΓA , |x j |is constant over all j ∈Γ+A (i ). Hence i ∈γ|a ii ···i |= i ∈γR i (A )for each γ∈C (A ), a contradiction to:there exists at least one circuit such that the inequality holds strictly. Hence det(A ) =0.2The tensor A =(a i 1i 2···i m )is called symmetric if a i 1i 2···i m =a σ(i 1)σ(i 2)···σ(i m ), where σis any permutation of the indices. A real symmetric tensor A of order m and dimension n is called positive definite , if x (A x m −1) >0for all real vector x ∈R n . Clearly, the symmetry and m being even are necessary for positive definite tensors. If m is even and all real eigenvalues of real symmetric tensor A are positive, then A is positive definite [12]. We can obtain the following result from Theorems 3.1and 3.3.Corollary 3.4. Let A =(a i 1···i m ) ∈R n ×···×n be an even order real symmetric tensor withnonnegative diagonal entries, and ΓA is strongly connected. Ifi ∈γ|a ii ···i | i ∈γR i (A )C.Bu et al./Linear Algebra and its Applications 480(2015)168–175173for each γ∈C (A )and there exists at least one circuit such that the inequality holds strictly, then A is positive definite.The unit tensor of order m is a diagonal tensor I =(δi 1i 2···i m )such that δi 1i 2···i m =1if i 1=i 2=···=i m , and δi 1i 2···i m =0otherwise. For a real tensor A =(a i 1···i m ) ∈R n ×···×n , let μ(A )denote the set of complex tensors B =(b i 1i 2···i m ) ∈C n ×···×n , where |b i 1i 2···i m |=a i 1i 2···i m for all (i 1, i 2, ..., i m ) =(i 1, i 1, ..., i 1). We extend the result in [5]to tensors as follows.Theorem 3.5. Let A =(a i 1i 2···i m ) ∈R n ×n ×···×n be a weakly irreducible nonnegative tensor, where a ii ···i =0for each i ∈[n ]. Let ρ1, ..., ρn be n positive numbers such thatσ(B ) ⊆ ni =1{z :|z −b ii ···i | ρi }for each B =(b i 1i 2···i m ) ∈μ(A ). Then there exists a positive vector x =(x 1, ..., x n ) such that(A x m −1)ix m −1iρi (i =1,...,n ).Proof.Let D be the diagonal matrix with diagonal entries ρ1, ..., ρn . From Defini-tion 2.3, we have (D −1A )i 1i 2···i m =ρ−1i 1a i 1i 2···i m . Since A is weakly irreducible, D−1A is also weakly irreducible. By Lemma 2.6, D −1A has an eigenvalue λ >0with a positive eigenvector x =(x 1, ..., x n ) . Henceλx m −1i=1i(A x m −1)i (i =1,...,n ).(3)Let C =D −1A −λI , where I is an order m unit tensor. Then 0is an eigenvalue of C . By Lemmas 2.2and 2.4, we know that 0is an eigenvalue of D C =A −λD I ∈μ(A ). By hypothesis, there exists i ∈[n ]such that |(D C )i ···i | =λρi ρi . Hence λ 1. By Eq.(3),we get (A x m −1)ix m −1iρi (i =1, ..., n ).2We generalize Theorem 2.12 in [2]to tensors as follows.Theorem 3.6. Let A =(a i 1i 2···i m ) ∈R n ×n ×···×n be a weakly irreducible nonnegative tensor, where a ii ···i =0for each i ∈[n ]. Let ρ1, ..., ρn be n positive numbers such that σ( A ) ⊆ n i =1{z :|z −˜a ii ···i | ρi }for each A ∈μ(A ). Then σ( A)⊆γ∈C (A )⎧⎨⎩z : i ∈γ|z −˜a ii ···i | i ∈γρi ⎫⎬⎭for each A∈μ(A ).Proof.For A=(˜a i 1i 2···i m ) ∈μ(A )and a diagonal matrix D =diag (x 1, ..., x n )(x i >0, 1 i n ), let T =D −(m −1) AD . By Lemma 2.5, σ(T ) =σ( A ). From Definition 2.3, we get (T )i 1i 2···i m = a i 1i 2···i m x −(m −1)i 1x i 2···x i m . By computation, we have174 C.Bu et al./Linear Algebra and its Applications 480(2015)168–175Fig.1.Eigenvalue inclusion sets.R i (T )=1x m −1i(i 2,···,i m )=(i,...,i )a ii 2···i m x i 2···x i m=(A x m −1)ix m −1i(i =1,...,n ),where x =(x 1, ..., x n ) . By Theorem 3.5, there exists a positive vector x such thatR i (T ) ρi (i =1, ..., n ). Since ΓT =Γ A =ΓA , by Theorem 3.1, we haveσ( A)=σ(T )⊆γ∈C (A )⎧⎨⎩z : i ∈γ|z −˜a ii ···i | i ∈γρi ⎫⎬⎭.24. ExampleWe give an example to compare the region of Theorem 3.1with those of [12, The-orem 6]and [9, Theorem 2.1]. Let A =(a i 1i 2i 3)∈C 3×3×3, where a 122=a 132=1,a 233=a 211=2, a 311=3and the other entries are zero. Then ΓA is weakly connected and ΓA has 3circuits: 1 →2 →3 →1, 1 →2 →1, 1 →3 →1. By Theorem 3.1,we have σ(A ) ⊆D = z ∈C :|z |324 . From [12, Theorem 6], we get σ(A ) ⊆G ={z ∈C :|z | 4}. From [9, Theorem 2.1], we get σ(A ) ⊆K ={z ∈C :(|z | −2) |z | 6}. It is clear that D is tighter than G and K (see Fig.1).AcknowledgementsThe authors would like to thank the anonymous referee for useful comments and suggestions, and thank Prof. R.A. Brualdi for providing Ref.[2]. This work is supported by the National Natural Science Foundation of China (No. 11371109 and No. 11426075),C.Bu et al./Linear Algebra and its Applications480(2015)168–175175the Natural Science Foundation of the Heilongjiang Province(No.QC2014C001)and the Fundamental Research Funds for the Central Universities(No.2014110015).References[1]A.Brauer,The theorems of Ledermann and Ostrowski on positive matrices,Duke Math.J.24(1957)256–274.[2]R.A.Brualdi,Matrices,eigenvalues,and directed graphs,Linear Multilinear Algebra11(1982)143–165.[3]K.C.Chang,L.Qi,T.Zhang,A survey on the spectral theory of nonnegative tensors,Numer.LinearAlgebra Appl.20(2013)891–912.[4]J.Cooper,A.Dutle,Computing hypermatrix spectra with the Poisson product formula,LinearMultilinear Algebra63(2015)956–970.[5]Ky Fan,Note on circular disks containing the eigenvalues of a matrix,Duke Math.J.25(1958)441–445.[6]S.Friedland,S.Gaubert,L.Han,Perron–Frobenius theorem for nonnegative multilinear forms andextensions,Linear Algebra Appl.438(2013)738–749.[7]S.Geršgorin,Über die Abgrenzung der Eigenwerte einer Matrix,Izv.Akad.Nauk SSSR Ser.Fiz.-Mat.6(1931)749–754.[8]S.Hu,Z.Huang,C.Ling,L.Qi,On determinants and eigenvalue theory of tensors,J.SymbolicComput.50(2013)508–531.[9]C.Li,Y.Li,X.Kong,New eigenvalue inclusion sets for tensors,Numer.Linear Algebra Appl.21(2013)39–50.[10]L.H.Lim,Singular values and eigenvalues of tensors:a variational approach,in:Proceedings of the1st IEEE International Workshop on Computational Advances of Multitensor Adaptive Processing, 2005,pp.129–132.[11]K.Pearson,T.Zhang,On spectral hypergraph theory of the adjacency tensor,Graphs Combin.30(2014)1233–1248.[12]L.Qi,Eigenvalues of a real supersymmetric tensor,J.Symbolic Comput.40(2005)1302–1324.[13]J.Y.Shao,A general product of tensors with applications,Linear Algebra Appl.439(2013)2350–2366.[14]J.Y.Shao,L.Qi,S.Hu,Some new trace formulas of tensors with applications in spectral hypergraphtheory,Linear Multilinear Algebra63(2015)971–992.[15]J.Y.Shao,H.Y.Shan,L.Zhang,On some properties of the determinants of tensors,Linear AlgebraAppl.439(2013)3057–3069.[16]Y.Yang,Q.Yang,Further results for Perron–Frobenius theorem for nonnegative tensors,SIAM J.Matrix Anal.Appl.31(2010)2517–2530.[17]J.Zhou,L.Sun,W.Wang,C.Bu,Some spectral properties of uniform hypergraphs,Electron.J.Combin.21(2014),P4.24.。
BinomialCoefficientsInclusionexclusionprinciple.ppt
We can see that a coefficient is the sum of two coefficients in the previous level. This is called the Pascal’s formula and we will prove it soon.
The sum of the binomial coefficients is equal to 2n.
6
Binomial Coefficients
In general we have the following identity:
Corollary: When x=-1, y=1, it implies that
1. Define a set S. 2. Show that |S| = n by counting one way. 3. Show that |S| = m by counting another way. 4. Conclude that n = m.
Double counting
9
Proving Identities
The sum of “odd” binomial coefficients is equal to the sum of “even” binomial coefficients.
7
Proving Identities
One can often prove identities about binomial coefficients by a counting argument. Direct proof:
So in this case, c0 = 1, c1 = 3, c2 = 3, c3 = 1.
BINOMIAL COEFFICIENT IDENTITIES AND HYPERGEOMETRIC SERIES
m 2k
k n
= 2m−2n−1
m−n m−n−1 + n n−1
,
(2)
j
(2m − 2l)! 1 = 2k−m j (m − k − 2j )!(k + j )!(j − l)!4 (m − k − 2l)!(m − l)!(k + m)!
and (3)
j
(−1)j
m−1 j
2m − 2j − 1 k+m−1
k
m 2k
k . n
BINOMIAL COEFFICIENT IDENTITIES AND HYPERGEOMETRIC SERIES 3
First of all we notice that for any term to be non–zero, we require m ≥ 2k, k ≥ n, so in particular m ≥ 2n and then the sum becomes (5) S=
If we make use of (6), (7), (8), (9), (10) and (11), we find −m + 2n −m + 2n + 1 ( )k ( )k m 2 2 S= 1 2n k≥0 k !(n + )k 2
BINOMIAL COEFFICIENT IDENTITIES AND HYPERGEOMETRIC SERIES 5
BINOMIAL COEFFICIENT IDENTITIES AND HYPERGEOMETRIC SERIES
Michael D. Hirschhorn In recent months I have come across many instances in which someone has found what they believe is a new result, in which they evaluate in closed form a sum involving binomial coefficients or factorials. In each case they have managed to do that either by using the recent powerful method of Wilf and Zeilberger (the W–Z method) [6], or by comparing coefficients in some ad hoc algebraic identity. The aim of this note is to describe, using a few examples, a purely algorithmic method for re–casting the sum as a (multiple of a) hypergeometric series in standard notation, so that one can then simply look up standard tables of hypergeometric series to see if the series under investigation is “summable” via known results. I do not claim any originality in this idea. I got it from Richard Askey, who claims [2] that “at least 90% if not 95% of the formulas in Table 3 of Henry Gould’s Tables [3] yield to this approach”. Indeed, to quote Askey further, “For years [before working with George E. Andrews in 1973] I had been trying to point out that the rather confused world of binomial coefficient summations is best understood in the language of hypergeometric series identities. Time and again I would find first–rate mathematicians who had never heard of this insight and who would waste considerable time proving some apparently new binomial coefficient summation which almost always turned out to be a special case of one of a handful of classical hypergeometric identities.” The identities I will use to illustrate the method are the following. The first came to me in a paper I was asked to referee, but is to be found in Wang and Guo [8] (1989). The second was found in 2001 by an Honours
医学统计学 英语
Introduction to Medical StatisticsMedical statistics, also known as biostatistics or health statistics, is a vital field that applies statistical methods to the study of health and medicine. This discipline is crucial for designing research studies, analyzing data, and interpreting results in a meaningful way. It plays an essential role in medical research, clinical trials, epidemiology, and public health.Key Concepts in Medical Statistics:1. Descriptive Statistics:Descriptive statistics summarize and describe the main features of a dataset. In medical statistics, this includes measures such as mean, median, mode, standard deviation, and range. These measures help researchers understand the distribution and central tendencies of their data.2. Inferential Statistics:Inferential statistics are used to make generalizations from a sample to a larger population. This involves hypothesis testing, confidence intervals, and p-values. Techniques such as t-tests, chi-square tests, and ANOVA are commonly used to determine if observed differences are statistically significant.3. Probability:Understanding probability is fundamental to medical statistics. It helps in assessing the likelihood of events occurring, such as the probability of a patient developing a particular condition. Probability distributions, such as the normal distribution, binomial distribution, and Poisson distribution, are often used.4. Regression Analysis:Regression analysis examines the relationship between dependent and independent variables. In medical research, this can be used to explore how various factors, such as age, weight, or treatment type, affect health outcomes. Linear regression, logistic regression, and Cox proportional hazards models are common methods.5. Survival Analysis:Survival analysis focuses on time-to-event data, often used in clinical trials to study the time until a patient experiences an event of interest, such as relapse or death. The Kaplan-Meier estimator and Cox proportional hazards model are key tools in this area.6. Clinical Trials:Clinical trials are research studies performed on patients to evaluate medical, surgical, or behavioral interventions. Medical statistics is essential in designing trials (randomization, blinding), analyzing the results (efficacy, safety), and ensuring that the findings are valid and reliable.7. Epidemiology:Epidemiology is the study of the distribution and determinants of health-related states and events in populations. Medical statistics is used to identify risk factors, track disease outbreaks, and evaluate preventive measures. Measures such as incidence, prevalence, and odds ratios are commonly used.Applications of Medical Statistics:- Drug Development:Medical statistics is crucial in the development and testing of new drugs. It helps in determining the efficacy and safety of new treatments through carefully designed clinical trials.- Public Health:In public health, medical statistics is used to monitor and control diseases, plan and evaluate health services, and inform policy decisions. It helps in understanding the spread of diseases and the effectiveness of interventions.- Medical Research:Researchers rely on statistical methods to analyze data from experiments and observational studies. This includes everything from basic research in laboratories to applied research in clinical settings.- Healthcare Decision Making:Statistical analysis helps healthcare providers make informed decisions based on evidence. This includes diagnostic tests, treatment plans, and resource allocation.Challenges in Medical Statistics:- Data Quality:Ensuring high-quality, accurate, and complete data is essential for reliable statistical analysis.- Ethical Considerations:Handling patient data requires strict adherence to ethical guidelines to protect patient confidentiality and ensure informed consent.- Complexity of Medical Data:Medical data can be complex, with numerous variables and potential confounding factors. Advanced statistical techniques are often needed to address these challenges.In conclusion, medical statistics is a fundamental discipline that supports the entire spectrum of healthcare, from research and development to public health and clinical practice. Its rigorous methods enable the medical community to make data-driven decisions that improve patient outcomes and advance our understanding of health and disease.。
Score tests for zero-inflation in overdispersed count data
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production_theory
We will show in the next chapter that, alternatively, given the family of level sets L(u ) | u R describing a technology, one may uniquely derive the production function for the technology as
ቤተ መጻሕፍቲ ባይዱ
2
It is convenient to define a particular subset of a level set L(u) called the production isoquant. Definition: The production isoquant corresponding to an output rate u > 0 is a subset of the input set L(u) defined by ISOQ L(u ) x | x L(u ), x L(u ) for all [0,1). Figure 1-1 displays a level set for a technology with 2 inputs (factors). The production isoquant is shown as a darkened line.
n L(u) R is the set of input vectors yielding an output rate of at least u.
Note that, given the production function , the level sets may be determined from as
CFA考试一级章节练习题精选0329-58(附详解)
CFA考试一级章节练习题精选0329-58（附详解）1、An analyst does research about bank discount yield and gathers the following informationabout a U.S Treasury bill:The present value of the U.S Treasury bill is closest to:【单选题】A.$ 99 324B.$ 99 432C.$ 99 439正确答案:B答案解析:假设该国库券的价格为P，则[(100 000 - p)/100 000] × 360/330 =0.62%，得出P =99 432。
2、Independent samples drawn from normally distributed populations exhibit the following characteristics:Assuming that the variances of the underlying populations are equal, the pooled estimate of the sample variance is 2,678.05. The t-test statistic appropriate to test the hypothesis that the two population means are equal is closest to:【单选题】A.0.29.B.0.94.C.1.90.正确答案:B答案解析:“Hypothesis Testing,” Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle2012 Modular Level I, Vol. 1, pp. 608–612Study Session 3-11-gIdentify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the population means of two at least approximately normally distributed populations, based on independent random samples with (1) equal or (2) unequal assumed variances.B is correct. The t statistic for the given information (normal populations, variances assumed equal) is calculated as:3、A sample of 25 observations has a mean of 8 and a standard deviation of 15. The standard error ofthe sample mean is closest to:【单选题】A.1.60.B.3.00.C.3.06.正确答案:B答案解析:The standard error of the sample mean, when the sample standard deviation is known, is:=3.00.CFA Level I"Sampling and Estimation," Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E.RunkleSection 3.14、An investor currently has a portfolio valued at $700,000. The investor’s objective is long-term growth, but the investor will need $30,000 by the end of the year to pay her son’s college tuition and another $10,000 by year-end for her annual vacation. The investor is considering three alternative portfolios:Using Roy’s safety-first criterion, which of the alternative portfolios most likely minimizes the probability that the investor’s portfolio will have a value lower than $700,000 at year-end?【单选题】A.Portfolio 1B.Portfolio 2C.Portfolio 3正确答案:C答案解析:“Common Probability Distributions,” Richard A. Defusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA2010 Modular Level I, Vol. 1, pp. 445-446Study Session 3-9-lDefine shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion.The investor requires a minimum return of $40,000/$700,000 or 5.71 percent. Roy’s safety-first model uses the excess of each portfolio’s expected return over the minimum return and divides that excess by the standard deviation for that portfolio. The highest safety-first ratio is associated with Portfolio 3: (14% – 5.71%)/22% = 0.3768.5、A technical analyst observes a head and shoulders pattern in a stock she has been following. She notes the following information:Based on this information, her estimate of the price target is closest to:【单选题】A.$48.00.B.$59.50.C.$89.75.正确答案:A答案解析:“Technical Analysis,” Barry M. Sine, CFA and Robert A. Strong, CFA 2013 Modular Level I, Vol. 1, Reading 12, Section 3.3.1.3Study Session 3-12-dIdentify and interpret common chart patterns.A is correct.Price target = Neckline ? (Head ? Neckline).In this example, PT=65.75-(83.50-65.75 )= 65.75-17.75 = 48.00。
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1 INTRODUCTION
1.1
The Roman Factorial
We begin by presenting a generalization of the factorial n! which makes sense for negative integral values of n as well as nonnegative called the Roman factorial ⌊n⌉! after its inventor Steve Roman. As usual for n a nonnegative integer the factorial is given by the product ⌊n⌉! = n! = 1 × 2 × 3 × · · · × n. However, for n a negative integer ⌊n⌉! = (−1)n+1 (−n − 1)!
−1 −1
6
1 INTRODUCTION
Table 6: Region 4 n\k 6 5 4 3 2 1 0 1 2 3 4 5
1/5 −1/20 1/30 −1/20 1/5
6
1/6 −1/30 1/60 −1/60 1/30 −1/6
7
1/7 −1/42 1/105 −1/140 1/105 −1/42 1/7
1.2 The Roman Coefficients −n + k − 1 n . = (−1)k k k −k − 1 n . = (−1)n+k n−k k Table 5: Region 3 n\k −6 −5 −1 −1 1 −2 5 −4 6 −3 −10 10 −4 −4 −5 −5 1 1 −6 −4
(1)
a for a = 0 1 for a = 0.
Note that equation (1) and the condition ⌊0⌉ = 1 completely characterizes the Roman factorial of integers.
1.2 The Roman Coefficients
5
21 6 1
6 7
7 1 1
Table 4: Region 2 n\k −1 −2 −3 −4 −5 0
1 1 1 1 1
1
−1 −2 −3 −4 −5
2
1 3 6 10 15
3
−1 −4 −10 −20 −35
4
1 5 15 35 70
5
−1 −6 −21 −56 −126
6
1 7 28 84 210
1
Introduction
Despite being so fundamental to combinatorics, several authors have noticed that ne is virtually unlimited in the choice of definition for the factorial—at least as far as umbral calculus is concerned. Indeed, one is presented with a bewildering number of alternatives each with its own notation. We present a new definition of the factorial which generalizes the usual one, and study the binomial coefficients it induces. They are blessed with a variety of combinatorial properties. However, what we are most interested is studying the interrelationship between this factorial and other famous ones.
Region 1 If n ≥ k ≥ 0, then
4
1 INTRODUCTION
Table 3: Region 1 n\k 0 1 7 1 7 6 1 6 5 1 5 4 1 4 3 1 3 2 1 2 1 1 1 0 1 2
21 15 10 6 3 1
3
35 20 10 4 1
4
35 15 5 1
n+k −1
= (−1)
−1
n+k +1
k 1 n+1 n+1
−1
= (−1)n+k+1 n
j ≥0
S (j, n)/k j +1 = (−1)n+k ∆n
1 x−k
x=0
where the S (j, n) are the Stirling numbers of the second kind, and ∆ is the forward difference operator ∆p(x) = p(x + 1) − p(x). Region 5 If n ≥ 0 > k , then n k 1 n−k n−k−1 1 = (−1)k n n k k−n −1 n − k − 1 1 = (−1)k+1 n+1 n+1 1 n k = (−1) ∆n = n−k x − n + k x=0 = −B (k − n, −k ) = (−1)k
−1 3 −3 1
5
Region 2 If k ≥ 0 > n, then Region 3 If 0 > n ≥ k , then
−3
1 −2 1
−2
−1 1
−1
1
Region 4 If k > n ≥ 0, then n k 1 k = (−1) n−k n 1 k−1 = (−1)n+k+1 n k
Dedicated to David and Maureen
Contents
1 Introduction 1.1 The Roman Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Roman Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Properties of Roman Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Generalizations of the Roman Coefficients . . . . . . . . . . . . . . . . . . . 1.5 Resistance of the n-cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 7 10 16
Table 2: Roman Coefficients, n\k 6 5 4 3 2 1 0 −1 −2 −3 −4 −5 −4
−1/840 −1/504 −1/280 −1/140 −1/80 −1/20 −1/4 −1 3 −3 1 −1/4
n k
−3
1/252 1/168 1/105 1/60 1/30 1/12 1/3 1 −2 1 −1/3 −1/12
Γ(a + 1)
when a is not a negative integer, and
(−1)a−1 /(−a − 1)! when a is a negative integer
where Γ(a) is the analytic Gamma function. Thus, for all a ⌊a⌉!/ ⌊a − 1⌉! = ⌊a⌉ where Roman a is defined to be ⌊a⌉ =
2
15 10 6 3 1 1/2 −1/2 1 3 6 10 15
3
20 10 4 1 1/3 −1/6 1/3 −1 −4 −10 −20 −35
4
15 5 1 1/4 −1/12 1/12 −1/4 1 5 15 35 70
5
6 1 1/5 −1/20 1/30 −1/20 1/5 −1 −6 −21 −56 −126
1
1/2 −1/2
1/3 −1/6 1/3
1/4 −1/12 1/12 −1/4
1
Abstract
arXiv:math/9502218v1 [math.CO] 9 Feb 1995
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. G´ en´ eralisation des Coefficients du Binˆ ome On cherche ici ` a d´ eterminer est la meilleure g´ en´ eralisation possible des factorielles et des coefficients du binˆ ome. On s’interesse ` a plusieurs exemples, ` a leurs propri´ et´ es combinatoires, et aux differentes relations qu’ils mettent en jeu.