[3] K. ˙Zyczkowski and H.–J. Sommers, Induced measures in the space of mixed quantum stat
可度量化空间
定理6.6.1作业§6.6可度量化空间本节重点:掌握三个定理的结论(前两个定理的证明不要求)先回忆一下在第二章中的可度量化空间的定义．一个拓扑空间称为是可度量化的，如果它的拓扑可以由它的某一个度量诱导出来．我们已经在许多章节中研究过度量空间的一些拓扑性质，这些拓扑性质当然也是可度量化空间所具有的．在这一章中我们部分地回答具有什么样的拓扑性质的拓扑空间是可度量化空间这个问题．定理6.6.1[Urysohn嵌入定理] 每一个满足第二可数性公理的空间都同胚于Hilbert空间H的某一个子空间．证明(略）定理6.6.2 Hilbert空间H是一个可分空间．证明(略)定理6.6.3 设X是一个拓扑空间．则下列条件等价：（1）X是一个满足第二可数性公理的空间；（2）X同胚于Hilbert空间H的某一个子空间；（3）X是一个可分的可度量化空间．证明（l）蕴涵（2）．此即定理6.6.1．（2）蕴涵（3）．由于Hilbert空间H是一个可分的度量空间，而可分的度量空间的每一个子空间都是可分的度量空间（参见推论5.2.5），与一个可分的度量空间同胚的拓扑空间是可分的（参见§5.2习题第4题），也是可以度量化的（参见§2.2习题12）．（3）蕴涵（1）．可分的度量空间满足第二可数性公理参见定理5.2.4），可度量化空间是一个空间（参见定理6.2.3）．因此更是一个空间．作业:P180 1.本章总结：（1）性质是描述点的分离性的,熟记各空间的定义、性质、与实数空间的区别.注意它们的充要条件，往往是证明的出发点．（2）正则、正规是描述点、闭集与闭集之间关系的性质．注意它们的充要条件．（3）完全正则、Tychonoff只有一种定义，一定要用映射来描述．（4）有了Urysohn引理，可将正规空间与实数空间联系起来，给证明提供了极大的方便．（完全正则与Tychonoff空间也是如此）（5）掌握它们的关系图及是否是连续映射所能保持的、有限可积的、可遗传的．从而会判断一个空间是哪种空间．。
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)。
相依随机序列滑动平均的强偏差定理
相依随机序列滑动平均的强偏差定理摘要本文主要研究任意相依连续型随机变量滑动平均的强偏差定理(即用不等式表示的强极限定理),全文共分四章.绪论部分,首先简明扼要地介绍了概率论极限理论背景以及发展现状.其次介绍了刘文创立的研究随机变量强偏差定理的基本思想与方法.最后引入了随机变量滑动平均,滑动似然比以及滑动相对熵概念.第二章.利用随机序列滑动似然比以及滑动相对熵概念作为任意随机序列联合分布与参考乘积分布(服从指数分布的独立随机变量序列)的偏差的随机性度量,并通过滑动相对熵给出了样本空间的一个子集.在此子集上得到了一类用不等式表示的强偏差定理,即小偏差定理.第三章.利用滑动似然比的概念和Laplace变换的方法,研究相依连续型非负随机变量序列滑动和的极限性质,得到了一类用不等式表示的强偏差定理,即小偏差定理.第四章.设{ξn,n≥1}是任意相依连续型随机变量序列,{B n,n≥1}是实直线上的Lebesgue可测集,1Bn (x)是B n的示性函数,本章研究{1Bn(ξn),n≥1}的极限性质,得到了一类用不等式表示的强偏差定理,并且得到关于{1Bn(ξn),n≥1}的强大数定律作为本节定理的推论.关键词:滑动平均;滑动似然比;滑动相对熵;指数分布;强偏差定理The Strong Deviation Theorems Concerning Moving Averageof Dependent Random VariablesAbstractThis dissertation makes a main study of strong deviation theorems of any con-tinuous random variable moving average(namely the strong limit theorem with the inequality).The article is divided into four chapters.Introduction:Wefirst briefly introduce probability theory limit theory background and development present situation.Secondly introduce the basic ideas and methods of small deviation theorems of random variable Liuwen founded.Finally this chapter introduces the random variable moving average、moving likelihood ratio and moving relative entropy concept.The second chapter:We use the concept of random sequence moving likelihood ratio and moving relative entropy to study the random measurement of the deviation of the random sequence joint distribution and the reference distribution.A subset of Sample space is constructed with the moving relative entropy and we get a class of strong deviation theorems represented with inequalities on this subset,that is the small deviation theorem.The third chapter is devoted to a study of the limit property of the moving sum of dependent continuous non-negative random variable sequences.With the concept of moving likelihood ratio and the method of Laplace transformation,we obtain a kind of strong deviation theorems represented with inequalities i.e.the Small Deviation Theorem In chapter4,we study the limit property of{1B(ξn),n≥1}for an arbitrary depen-ndent continuous random variable sequence{ξn,n≥1}where{B n,n≥1}is a sequence of Lebesgue sets of the real line and1Bis the characteristic of B n for each n∈ω.A classnof strong deviation theorem represented with inequalities is obtain.Moreover,we show that the strong law of large numbers of{1B(ξn),n≥1}is an immediate consequence ofnthe theorem in this section.安徽工业大学硕士毕业论文5Keywords:moving average;moving likelihood ratio;moving relative entropy;expo-nential distribution;strong deviation theorem文中部分缩写与符号说明µ, µ———概率测度a.s.———几乎必然µ−a.s.———依概率µ几乎必然L n(ω)———滑动似然比———滑动相对熵h µµ1(.)———示性函数log———自然对数R+———正实数R−———负实数↗———单调递增↘———单调递减目录第一章绪论 (1)第二章基于指数分布的随机序列滑动平均的强偏差定理 (5)第三章Laplace变换与随机变量滑动和的一类强偏差定理 (10)第四章连续型随机变量滑动平均的一类强偏差定理 (17)第五章总结与展望 (24)参考文献 (25)致谢 (28)附录 (29)安徽工业大学硕士毕业论文第一章绪论概率论是研究大量随机现象的统计规律性的数学学科。
微积分五讲-第五讲
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基于拟生灭过程的随机支付2×2双矩阵博弈演化模型
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带随机支付的双矩阵博弈模型及其纳什均衡
定义 # 设 /， …， ， …， 为 两个博 弈人的 纯策 略集合， 他 # 表示两个博弈人， ( / #｛ F， *， &｝ ( # #｛ F， *， ,｝ （ 1F ， …， …， 0/ # { 1* ， 1 & ）$ 2 & . D F， *， & ， 1. D F } ， G S 1 . ! E， %
（F）
" " 则称｛! （ "） ， （ "） ｝ 是一个拟生灭过程 （ 92#3( :(’;1 #$% %/#;1 <’&=/33 简记为 >.?） # ! 其中所有子块是 & 阶 方阵， 有负的对角线元素和非负的非对角线 元素， 并且 行和非 正 ! * $ 和 ) $ 都 是 非 负 阵， 满足 （ (E ($ ， $ !E， （ ($ G *$ G )$ ） G )E ） -D - D E， $ !F ! 定义 ! 如果一个拟生灭过程 （ >.?） 的状态空间有限且相互独立， 则这个拟生灭过程被称为广义拟生 灭过程 （ @>.?） ! 对于一般 >.? 尚未建立起平 行 于 经 典 生 灭 过 程 的 理 论 和 处 理 方 法， 关于经典生灭过程平稳分布的 "#’H($IA=@’/0&’ 定 理 目 前 还 不 能 推 广 到 >.? 过 程 ! 目 前 只 是 对 一 些 特 殊 的 >.? 有 些 相 关 的 研 究 结 果 !
［ %］ 法要么只能求解简单的博弈 （如几何图形算法 只能 求 解 $ 阶 以 下 的 矩 阵 博 弈） ， 要么算法的实现十分复 ［ !］ ［ $］ 杂 （如 \>=Y>&]6H16: 算法 和同伦路径法 等） ， 而带有随机支付博弈的均衡求解则更加困难 - 因此本文将
New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems
x ˙(t ) = y(t );
y(t ) =
i=1
Di y(t − hi ) +
i=0
Ai x(t − hi ):
(2)
The latter can be represented in the form of descriptor system with discrete and distributed delay in the “fast variable” y:
i=1
x(t ) y(t )
+ V 1 + V2 ;
(4)
I 0 ; 0 0
m t t −hi
P=
P1 P2
0 ; P3
T P1 = P1 ¿0;() (6)yT (s)Qi y(s) d s;
Qi ¿0
and
m
V2 =
i=1
0 −hi
t t +Â
yT (s)Ri y(s) d s d Â;
Ri ¿0:
E. Fridman ∗
Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel Received 10 September 2000; received in revised form 14 February 2001
m m m
x ˙(t ) = y(t );
0 = − y(t ) +
i=1
D i y (t − h i ) +
i=0
Ai
x(t ) −
i=1
Ai
t t −hi
y(s) d s:
Peskin & Schroesder-An Introduction to Quantum Field Theory 习题答案 Ch02
δµAν = ∂µAν .
(5)
Thus
T˜µν
=
∂L ∂(∂µAλ)
∂ν
Aλ
−
ηµν
L
=
−F
µλ∂ν Aλ
+
1 4
ηµν FλκF
λκ.
(6)
Obviously, this tensor is not symmetric. However, we can add an additional term ∂λKλµν to T˜µν with Kλµν being antisymmetric to its first two indices. It’s easy to see that this term does not affect the conservation of T˜µν. Thus if we choose Kλµν = F µλAν , then:
T µν
= T˜µν + ∂λKλµν
= F µλFλ ν +
1 4
ηµν FλκF
λκ.
(7)
Now this tensor is symmetric. It is called the Belinfante tensor in literature. We can also rewrite it in terms of Ei and Bi:
∂L ∂Aν
= −∂µF µν
(2)
This is sometimes called the “second pair” Maxwell’s equations. The so-called “first pair” comes directly from the definition of Fµν = ∂µAν − ∂νAµ, and reads
4H 和 6H-SiC的介电常数
II. BULK 4H AND 6H SiC
A. Samples
The 4H and 6H SiC samples studied here were obtained commercially from Cree Research. For spectroscopic ellipsometry, we used single-side polished ͑Si-terminated͒ wafers, 35 mm in diameter, and 0.42 mm thick. The miscut, i.e., the angle between the surface normal and the hexagonal axis, was about 8°, confirmed by x-ray diffraction. The wafers were not intentionally doped, but nitrogen impurities resulted in electron concentrations near 6 – 8ϫ1018 cmϪ3 in 4H SiC and 1ϫ1018 cmϪ3 in 6H SiC, determined by secondary ion mass spectrometry, Fourier-transform infrared ellipsometry, and electrical measurements by the supplier. The samples were measured as received. No surface preparation was performed. Therefore, we expect that the wafers are covered with a thin native oxide (SiO2). Samples used for transmission intensity measurements were similar, but two-side polished.
2018年中科院数学SCI期刊分区
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半序概率度量空间中混合单调算子的耦合不动点定理
2019年5月第45卷第3期西南民族大学学报(自然科学版)May.2019 Journal of Southwest Minzu University(Natural Science Edition)Vol.45No.3doi：10.11920/xnmdzk.2019.03.015半序概率度量空间中混合单调算子的耦合不动点定理朱奋秀(湖北经济学院法商学院，湖北武汉430205)摘要：不动点理论是非线性泛函分析的重要组成部分.对非线性微分和积分方程的研究有重要意义.通过泛函在概率度量空间中引入半序关系，建立半序概率度量空间，在满足两点压缩和拉伸条件下，弱化混合单调算子的连续性，且通过构造集合，并证明集合存在极大元，利用Zron引理，从而证明满足条件的混合单调算子的耦合不动点定理・关键词：半序概率度量空间；不动点；混合单调算子中图分类号:0177.91文献标志码:A 文章编号:2095^271(2019)03-0314-04Fixed point theorems of mixed monotone operators inpartially ordered probabilistic metric spacesZHU Fen-xiu(School of Law and Business,Hubei University of Economics,Wuhan430205,P.R.C.)Abstract:Fixed point theory is an indispensable part of nonlinear functional analysis and is of great significance in the research of differential and integral equations.A partial order is introduced by a function in probabilistic metric spaces.A class of mixed monotone operator is discussed under compression type and tension type in ordered metric spaces,and the continuity of operators is weakened.The Zron lemma is used to prove the coupled fixed point theorem of mixed monotone operators by constructinga set.Key words:partially ordered probabilistic metric space；fixed point；mixed monotone operator1引言与预备知识近年来，半序方法成为研究非线性算子不动点问题主要的方法，然而在概率度量空间中利用半序方法研究混合单调算子不动点问题的文献较少-1976年, Caristi J利用泛函在度量空间中引入了一个半序并得到几个不动点定理;文献［4］在概率度量空间中引入半序，讨论了序压缩算子的不动点理;文献［5］讨论了在半序概率度量空间中的一类增算子的不动点定理,并得到一类非线性算子方程的可解性理论;文献［6］讨论了概率度量空间中序压缩算子对的重合点定理;文献［7］讨论了在半序概率度量空间中满足一定条件的形如Lx=N(%,%)，—类非线性算子方程的可解性问题.下面将在概率度量空间中引入半序关系研究一类混合单调算子，在满足两点压缩和拉伸条件下，得到算子的耦合不动点的存在性.一些关于半序概率度量空间的基本概念和记号见文献［2-15］,设(E,F,A)是Menger概率度量空间，且t-范数4满足Nt’/EfR为泛函，定义E上的"V收稿日期:2018-04-24作者简介:朱奋秀(1983-),男，汉族，甘肃白银人，讲师，研究方向:非线性泛函分析.E-mail：zhufenxiu@ 基金项目：湖北经济学院法商学院科研资助项目(2018K12)第3期朱奋秀：半序概率度量空间中混合单调算子的耦合不动点定理315为，对任意的e E,x<y<=>e><p(x)-<p(y),V A >0,都有化,,(£)>1-入，可以验证"V为(E,F,A)上的半序关系，称此关系为由<p导出的半序•设x o»Jo e E且x0W y(),称[x0,y0]=｛xeE\x0<x<y0｝为序区间定义1设(E,F,A)是Menger概率度量空间，其中的半序由泛函卩所导出，设算子A：ExE-E,若对任意的如，％2』1』2wE,当坷<X2,y2</!时，有4(X|, y,)<A(x2.y2),则称4为混合单调算子.引理1⑶设"V是概率度量空间(E,F,4)上由(P导出的半序，若％今,则(p(x)><[P(y).2主要结果定理1设(E,F,A)是Menger概率度量空间，且t-范数连续且满足NtwE-R为连续泛函，"V是由<P导出的半序.设X o,y o e E,X O<y0,D=[x0, %]•又设A：DxD^E是关于t"的混合单调算子，且X o4(%0』0),A(y0,x0)<y0.则算子A存在耦合不动点(x,y)eDxD,使得A(x,y)=x,4(y,x)=y.证明设C=｛(x,y)eD xZ)：x0<x<A(x,y),A (y,x)<y<y0|，显然(%,%)wC,于是C为非空集合,在C上定义关系如下：(«i,Yi)<(x2,y2)¢=>x l<x2,y2<y t.易证(C,V)为半序空间，下面证明(C,V)有极大元•令｛(耳是(C,V)的任一全序子集，其中J为定向集，即e J,/jl<v当且仅当(％“,％)<对任意(JL,veJ,/JL。