An Extension of Alzer's Inequality

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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)。

Genearlizations of Precupanu's Inequality for Orthornormal Families of Vectors in Inner Pro

In the right-hand side or in the left-hand side of (1.1) we have equality if and only if there are λ, µ ∈ R such that (1.2) λ x, a x
2
·x+µ
y, #43; µb) . 2
where some information about a third vector x is known: Theorem 2. Let (H ; ·, · ) be an inner product space over the real ﬁeld R and x, y, z ∈ H such that: (1.8) | x, y | ≥ (1 − ε) x y , | x, z | ≥ (1 − ε) x z . z , where ε is a positive real number, reasonably small. Then √ (1.9) | y, z | ≥ max 1 − ε − 2ε, 1 − 4ε, 0 y x, a x, b x
In the complex case, apparently independent of Richard, M.L. Buzano obtained in [2] the following inequality (1.14) b + | a, b | · x 2, 2 provided x, a, b are vectors in the complex inner product space (H ; ·, · ) . In the same paper [6], Precupanu, without mentioning Buzano’s name in relation to the inequality (1.14), observed that, on utilising (1.14), one may obtain the following result of Moore type: | x, a x, b | ≤ a

不等式证明的若干方法英语

不等式证明的若干方法英语Inequality proofs are an integral part of mathematics, serving as a foundation for establishing the validity of various mathematical statements. There are several methods to prove inequalities, each with its own unique approach and applications. Here are some common techniques used in proving inequalities:1. Direct Comparison: This method involves comparing the two sides of the inequality directly. If it can be shown that one side is always greater than or equal to the other, the inequality is proven.2. Rearrangement: By rearranging terms and applying algebraic identities, one can often transform an inequality into a more familiar form that is easier to prove.3. Use of Inequality Properties: Properties such as transitivity (if \(a \leq b\) and \(b \leq c\), then \(a \leq c\)) and the properties of absolute values can be used to simplify and prove inequalities.4. Application of Inequalities: Utilizing well-known inequalities like the AM-GM inequality (Arithmetic Mean-Geometric Mean inequality), Cauchy-Schwarz inequality, or Jensen's inequality can be instrumental in proving new inequalities.5. Induction: For inequalities involving sequences or series, mathematical induction can be a powerful tool. By proving the base case and then showing that if the inequality holds for \(n\), it must also hold for \(n+1\), theinequality is proven for all natural numbers.6. Contradiction: This approach involves assuming the opposite of what is to be proven, leading to a contradiction. If assuming the inequality is false leads to an impossibility, then the original inequality must be true.7. Limiting Arguments: By taking limits, one can showthat as \(n\) approaches infinity, the inequality holds. This is particularly useful in proving inequalities that involve limits.8. Graphical Methods: Plotting functions and analyzingtheir behavior can sometimes provide intuitive insights into the truth of an inequality.9. Analytical Methods: Calculus can be used to analyzethe behavior of functions, such as finding derivatives to determine where a function is increasing or decreasing, which can help prove inequalities.10. Combinatorial Arguments: For inequalities involving combinatorial expressions, counting arguments and the use of combinatorial identities can be effective.Each method has its own strengths and is suited todifferent types of inequalities. The choice of method oftendepends on the specific form of the inequality and the context in which it appears. Mastery of these techniques is crucial for anyone looking to engage deeply with mathematical proofs and inequalities.。

10.1007%2Fs11537-013-1280-5

Japan.J.Math.8,147–183(2013)DOI:10.1007/s11537-013-1280-5About the Connes embedding conjectureAlgebraic approachesNarutaka Ozawa?Received:28December2012/Accepted:15January2013Published online:20March2013©The Mathematical Society of Japan and Springer Japan2013Communicated by:Yasuyuki KawahigashiAbstract.In his celebrated paper in1976,A.Connes casually remarked that anyﬁnite von Neu-mann algebra ought to be embedded into an ultraproduct of matrix algebras,which is now known as the Connes embedding conjecture or problem.This conjecture became one of the central open problems in theﬁeld of operator algebras since E.Kirchberg’s seminal work in1993that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the ﬁeld.Since then,many more equivalents of the conjecture have been found,also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum infor-mation theory.In this note,we present a survey of this conjecture with a focus on the algebraic aspects of it.Keywords and phrases:Connes embedding conjecture,Kirchberg’s conjecture,Tsirelson’s prob-lem,semi-pre-C -algebras,noncommutative real algebraic geometryMathematics Subject Classiﬁcation(2010):16W80,46L89,81P151.IntroductionThe Connes embedding conjecture([Co])is considered as one of the most im-portant open problems in theﬁeld of operator algebras.It asserts that anyﬁ-nite von Neumann algebra is approximable by matrix algebras in a suitable sense.It turns out,most notably by Kirchberg’s seminal work([Ki1]),that the N.O ZAWAResearch Institute for Mathematical Sciences,Kyoto University,Kyoto606-8502,Japan(e-mail:)?Partially supported by JSPS(23540233)and by the Danish National Research Foundation (DNRF)through the Centre for Symmetry and Deformation.148N.Ozawa Connes embedding conjecture is equivalent to a variety of other important con-jectures,which touches most of the subﬁelds of operator algebras,and also someother branches of mathematics such as noncommutative real algebraic geome-try([Sm])and quantum information theory.In this note,we look at the alge-braic aspects of this conjecture.(See[BO,Ki1,Oz1]for the analytic aspects.)This leads to a study of the C -algebraic aspect of noncommutative real alge-braic geometry in terms of semi-pre-C -algebras.Speciﬁcally,we treat someeasy parts of Positivstellensätze of Putinar([Pu]),Helton–McCullough([HM]),and Schmüdgen–Bakonyi–Timotin([BT]).We then treat their tracial analogueby Klep–Schweighofer([KS]),which is equivalent to the Connes embeddingconjecture.We give new proofs of Kirchberg’s theorems on the tensor productC F d˝B.`2/and on the equivalence between the Connes embedding conjec-ture and Kirchberg’s conjecture.We also look at Tsirelson’s problem in quantuminformation theory([Fr,J+,Ts]),and prove it is again equivalent to the Connesembedding conjecture.This paper is an expanded lecture note for the author’slecture for“Masterclass on soﬁc groups and applications to operator algebras”(University of Copenhagen,5–9November2012).The author gratefully ac-knowledges the kind hospitality provided by University of Copenhagen duringhis stay in Fall2012.He also would like to thank Professor Andreas Thom forvaluable comments on this note.2.Ground assumptionWe deal with unital -algebras over k2f C;R g,and every algebra is assumedto be unital,unless it is clearly not so.The unit of an algebra is simply denotedby1and all homomorphisms and representations between algebras are assumedto preserve the units.We denote by i the imaginary unit,and by the complexconjugate of 2C.In case k D R,one has D for all 2k.3.Semi-pre-C -algebrasWe will give the deﬁnition and examples of semi-pre-C -algebras.Recall thata unital algebra A is called a -algebra if it is equipped with a map x!xsatisfying the following properties:(i)1 D1and.x / D x for every x2A;(ii).xy/ D y x for every x;y2A;(iii). x C y/ D x C y for every x;y2A and 2k.The sets of hermitian elements and unitary(orthogonal)elements are writtenrespectively asA h WD f a2A W a D a g and A u WD f u2A W u u D1D uu g:About the Connes embedding conjecture149 Every element x2A decomposes uniquely as a sum x D a C b of an hermitian element a and a skew-hermitian element b.The set of hermitian elements is an R-vector space.We say a linear map'between -spaces is self-adjoint if ' D',where' is deﬁned by' .x/D'.x / .We call a subset A C A h a -positive cone(commonly known as a quadratic module)if it satisﬁes the following:(i)R 01 A C and a C b2A C for every a;b2A C and 2R 0; (ii)x ax2A C for every a2A C and x2A.For a;b2A h,we write aÄb if b a2A C.We say a linear map'be-tween spaces with positivity is positive if it sends positive elements to positive elements(and often it is also required self-adjoint),and a positive linear map' is faithful if a 0and'.a/D0implies a D0.Given a -positive cone A C, we deﬁne the -subalgebra of bounded elements byA bdd D f x2A W9R>0such that x xÄR1g:This is indeed a -subalgebra of A.For example,if x is bounded and x xÄR1,then x is also bounded and xx ÄR1,because0ÄR 1.R1 xx /2D R1 2xx C R 1x.x x/x ÄR1 xx : Thus,if A is generated(as a -algebra)by S,then S A bdd implies A D A bdd.Deﬁnition.A unital -algebra A is called a semi-pre-C -algebra if it comes together with a -positive cone A C satisfying the Combes axiom(also called the archimedean property)that A D A bdd.Since hÄ.1C h2/=2for h2A h,one has A h D A C A C for a semi-pre-C -algebra.We deﬁne the ideal of inﬁnitesimal elements byI.A/D f x2A W x xÄ"1for all">0gand the archimedean closure of the -positive cone A C(or any other cone)by arch.A C/D f a2A h W a C"12A C for all">0g:The cone A C is said to be archimedean closed if A C D arch.A C/.A C -alge-bra A is of course a semi-pre-C -algebra,with a zero inﬁnitesimal ideal and an archimedean closed -positive coneA C D f x x W x2A g:If A B.H/(here B.H/denotes the C -algebra of the bounded linear opera-tors on a Hilbert space H over k),then one also hasA C D f a2A h W h a ; i 0for all 2H g:150N.Ozawa Note that the condition a being hermitian cannot be dropped when k D R.Itwill be shown(Theorem1)that if A is a semi-pre-C -algebra,then A=I.A/is a pre-C -algebra with a -positive cone arch.A C/.Deﬁnition.We deﬁne the universal C -algebra of a semi-pre-C -algebra Aas the C -algebra C u.A/together with a positive -homomorphismÃW A!C u.A/which satisﬁes the following properties:Ã.A/is dense in C u.A/andevery positive -representation of A on a Hilbert space H extends to a -representation N W C u.A/!B.H/,i.e., D N ıÃ.In other words,C u.A/isthe separation and completion of A under the C -semi-normsup fk .a/k B.H/W a positive -representation on a Hilbert space H g: (We may restrict the dimension of H by the cardinality of A.)We emphasize that only positive -representations are considered.Everypositive -homomorphism between semi-pre-C -algebras extends to a posi-tive -homomorphism between their universal C -algebras.It may happen that A C D A h and C u.A/D f0g,which is still considered as a unital(?)C -algebra.Every -homomorphism between C -algebras is automatically pos-itive,has a norm-closed range,and maps the positive cone onto the positivecone of the range.However,this is not at all the case for semi-pre-C -algebras,as we will exhibit a prototypical example in Example1.On the other hand,we note that if A is a norm-dense -subalgebra of a C -algebra A such thatarch.A C/D A\A C,then every positive -representation of A extends to a-representation of A,i.e.,A D Cu .A/.(Indeed,if x2A has k x k A<1,then1 x x2A C and hence k .x/k<1for any positive -representation ofA.)It should be easy to see that the following examples satisfy the axiom of semi-pre-C -algebras.Example1.Let be a discrete group and kŒ be its group algebra over k:.f g/.s/DXt2f.st 1/g.t/and f .s/D f.s 1/ for f;g2kŒ :The canonical -positive cone of kŒ is deﬁned as the sums of hermitian squares,kŒ C Dn n Xi D1 i i W n2N; i2kŒo:Then,kŒ is a semi-pre-C -algebra such that C u.kŒ /D C ,the full group C -algebra of ,which is the universal C -algebra generated by the unitary representations of .There is another group C -algebra.Recall that the left regular representation of on`2 is deﬁned by .s/ıt Dıst for s;t2 , or equivalently by .f/ D f for f2kŒ and 2`2 .The reducedAbout the Connes embedding conjecture151 group C -algebra C r of is the C -algebra obtained as the norm-closure of .kŒ /in B.`2 /.The group algebra kŒ is equipped with the corresponding -positive conek rŒ C D f f2kŒ W9f n2kŒ C such that f n!f pointwise gD f f2kŒ W f is of positive type g;and the resultant semi-pre-C -algebra k rŒ satisﬁes C u.k rŒ /D C r .Indeed, if f2kŒ \ 1.C r C/,then for D .f/1=2ı12`2 ,one has f D and f is the pointwise limit of n n2kŒ C,where n2kŒ are such that k n k2!0.On the other hand,if f is of positive type(i.e.,the kernel .x;y/!f.x 1y/is positive semi-deﬁnite),then f D f and h .f/Á;Ái 0 for everyÁ2`2 ,which implies .f/2C r C.It follows that k rŒ C D arch.kŒ C/if and only if is amenable(see Theorem1).Example2.The -algebra kŒx1;:::;x d of polynomials in d commuting hermitian variables x1;:::;x d is a semi-pre-C -algebra,equipped with the -positive conekŒx1;:::;x d C D -positive cone generated by f1 x2i W i D1;:::;d g:One has C u.kŒx1;:::;x d /D C.Œ 1;1 d/,the algebra of the continuous func-tions onŒ 1;1 d,and x i is identiﬁed with the i-th coordinate projection.Example3.The -algebra k h x1;:::;x d i of polynomials in d non-commuting hermitian variables x1;:::;x d is a semi-pre-C -algebra,equipped with the -positive conek h x1;:::;x d i C D -positive cone generated by f1 x2i W i D1;:::;d g: One has C u.k h x1;:::;x d i/D C.Œ 1;1 / C.Œ 1;1 /,the unital full free product of d-copies of C.Œ 1;1 /.Example4.Let A and B be semi-pre-C -algebras.We denote by A˝B the algebraic tensor product over k.There are two standard ways to make A˝B into a semi-pre-C -algebra.Theﬁrst one,called the maximal tensor product and denoted by A˝max B,is A˝B equipped with.A˝max B/C D -positive cone generated by f a˝b W a2A C;b2B C g: The second one,called the minimal tensor product and denoted by A˝min B, is A˝B equipped with.A˝min B/C D.A˝B/h\.ÃA˝ÃB/ 1..C u.A/˝min C u.B//C/: (See Theorem14for a“better”description.)One has C u.A˝˛B/D C u.A/˝˛C.B/for˛2f max;min g.The right hand side is the C -algebra maximal u152N.Ozawa (resp.minimal)tensor product(see[BO,Pi1]).For A1 A2and B1 B2, one has.A1˝min B1/C D.A1˝min B1/\.A2˝min B2/C;but the similar identity need not hold for the maximal tensor product. Example5.The unital algebraic free product A B of semi-pre-C -algebras A and B,equipped with.A B/C D -positive cone generated by.A C[B C/;is a semi-pre-C -algebra,and C u.A B/D C u.A/ C u.B/,the unital full free product of the C -algebras C u.A/and C u.B/.The following is very basic(cf.[Ci]and Proposition15in[Sm]).Theorem1.Let A be a semi-pre-C -algebra andÃW A!C u.A/be the uni-versal C -algebra of A.Then,one has the following.kerÃD I.A/,the ideal of the inﬁnitesimal elements.A h\Ã 1.C.A/C/D arch.A C/,the archimedean closure of A C.uAlthough it follows from the above theorem,we give here a direct proof of the fact that arch.A C/\. arch.A C// I.A/.Indeed,if h2Ä1and "1<h<"1for"2.0;1/,then one has0Ä.1C h/." h/.1C h/D".1C h/2 h h.2C h/hÄ.4"C"/1 .2 "/h2; which implies h2<5"1.We postpone the proof and give corollaries to this theorem.4.PositivstellensätzeWe give a few results which say if an element a is positive in a certain class of representations,then it is positive for an obvious reason.Such results are re-ferred to as“Positivstellensätze.”Recall that a C -algebra A is said to be resid-uallyﬁnite dimensional(RFD)ifﬁnite-dimensional -representations separate the elements of A,i.e., .a/ 0for allﬁnite-dimensional -representations implies a 0in A.All abelian C -algebras and full group C -algebras of residuallyﬁnite amenable groups are RFD.Moreover,it is a well-known result of Choi that the full group C -algebra C F d of the free group F d of rank d is RFD(see Theorem26).In fact,ﬁnite representations(i.e.,the unitary repre-sentations such that .F d/isﬁnite)separate the elements of C F d([LS]). However,we note that the full group C -algebra of a residuallyﬁnite group need not be RFD([Be1]).We also note that the unital full free products of RFDAbout the Connes embedding conjecture153 C -algebras is again RFD([EL]).In particular,C u.k h x1;:::;x d i/is RFD.The results mentioned here have been proven for complex C -algebras,but they are equally valid for real cases.See Sect.7.Theorem1,when combined with resid-ualﬁnite dimensionality,immediately implies the following Positivstellensätze (cf.[Pu,HM]).Corollary2.The following are true.Let f2kŒ h.Then, .f/ 0for every unitary representation if and only if f2arch.kŒ C/.The full group C -algebra C of a group is RFD if and only if the fol-lowing statement holds.If f2kŒ h is such that .f/ 0for every ﬁnite-dimensional unitary representation ,then f2arch.kŒ C/.Let f2kŒx1;:::;x d h.Then,f.t1;:::;t d/ 0for all.t1;:::;t d/2Œ0;1 d if and only if f2arch.kŒx1;:::;x d C/.(See Example2.)Let f2k h x1;:::;x d i h.Then,f.X1;:::;X d/ 0for all contractive her-mitian matrices X1;:::;X d if and only if f2arch.k h x1;:::;x d i C/.(See Example3.)In some cases,the -positive cones are already archimedean closed.We will see later(Theorem26)this phenomenon for the free group algebras kŒF d . 5.Eidelheit–Kakutani separation theoremThe most basic tool in functional analysis is the Hahn–Banach theorem.In this note,we will need an algebraic form of it,the Eidelheit–Kakutani separation theorem.We recall the algebraic topology on an R-vector space V.Let C V be a convex subset.An element c2C is called an algebraic interior point of C if for every v2V there is">0such that c C v2C for all j j<". The convex cone C is said to be algebraically solid if the set Cıof algebraic interior points of C is non-empty.Notice that for every c2Cıand x2C, one has c C.1 /x2Cıfor every 2.0;1 .In particular,CııD Cıfor every convex subset C.We can equip V with a locally convex topology,called the algebraic topology,by declaring that any convex set that coincides with its algebraic interior is open.Then,every linear functional on V is continuous with respect to the algebraic topology.Now Hahn–Banach separation theorem reads as follows.Theorem3(Eidelheit–Kakutani([Ba])).Let V be an R-vector space,C an algebraically solid cone,and v2V n C.Then,there is a non-zero linear func-tional'W V!R such that'.c/:'.v/Äinfc2CIn particular,'.v/<'.c/for any algebraic interior point c2C.154N.Ozawa Notice that the Combes axiom A D A bdd is equivalent to that the unit1is an algebraic interior point of A C A h and arch.A C/is the algebraic closure of A C in A h.(This is where the Combes axiom is needed and it can be dispensed when the cone A C is algebraically closed.See Sect.3.4in[Sm].)Let A be a semi-pre-C -algebra.A unital -subspace S A is called a semi-operator system.Here,a -subspace is a subspace which is closed under the -operation. Existence of1in S ensures that S C D S\A C has enough elements to span S h.A linear functional'W S!k is called a state if'is self-adjoint, positive,and'.1/D1.Note that if k D C,then S is spanned by S C and every positive linear functional is automatically self-adjoint.However,this is not the case when k D R.In any case,every R-linear functional'W S h!R extends uniquely to a self-adjoint linear functional'W S!k.We write S.S/ for the set of states on S.Corollary4.Let A be a semi-pre-C -algebra.Let W A be a -subspace and v2A h n.A C C W h/.Then,there is a state'on A such that'.W/D f0g and'.v/Ä0.(Krein’s extension theorem)Let S A be a semi-operator system.Then every state on S extends to a state on A.Proof.Since A C C W h is an algebraically solid cone in A h,one mayﬁnd a non-zero linear functional'on A h such that'.v/Äinf f'.c/W c2A C C W h g:Since'is non-zero,'.1/>0and one may assume that'.1/D1.Thus the self-adjoint extension of'on A,still denoted by',is a state such that'.v/Ä0 and'.W h/D f0g.Let x2W.Then,for every 2k,one has'.x/C. '.x// D'.. x/C. x/ /D0:This implies'.W/D f0g in either case k2f C;R g.For the second assertion,let'2S.S/be given and consider the coneC D f x2S h W'.x/ 0g C A C:It is not too hard to see that C is an algebraically solid cone in A h and v…C for any v2S h such that'.v/<0.Hence,one mayﬁnd a state N'on A such that N'.C/ R 0.In the same way as above,one has that N'is zero on ker', which means N'j S D'.About the Connes embedding conjecture155 6.GNS constructionWe recall the celebrated GNS construction(Gelfand–Naimark–Segal construc-tion),which provides -representations out of states.Let a semi-pre-C -algebra A and a state'2S.A/be given.Then,A is equipped with a semi-inner product h y;x i D'.x y/,and it gives rise to a Hilbert space,which will bedenoted by L2.A;'/.We denote by O x the vector in L2.A;'/that correspondsto x2A.Thus,h O y;O x i D'.x y/and k O x k D'.x x/1=2.The left multiplica-tion x!ax by an element a2A extends to a bounded linear operator '.a/on L2.A;'/such that '.a/O x D ca x for a;x2A.(Observe that a aÄR1implies k '.a/k2ÄR.)It follows that 'W A!B.L2.A;'//is a positive -representation of A such that h '.a/O1;O1i D'.a/.If W A!B.H/is a positive -representation having a unit cyclic vector,then'.a/D h .a/ ; i is a state on A and .x/ !O x extends to a uni-tary isomorphism between H and L2.A;'/which intertwines and '.Sinceevery positive -representation decomposes into a direct sum of cyclic repre-sentations,one may obtain the universal C -algebra C u.A/of A as the closureof the image under the positive -representationM '2S.A/ 'W A !BM'2S.A/L2.A;'/Á:We also make an observation that.A˝min B/C in Example4coincides withc2.A˝B/h W .'˝ /.z cz/ 0for all'2S.A/;2S.B/;z2A˝B:7.Real versus complexWe describe here the relation between real and complex semi-pre-C -algebras. Because the majority of the researches on C -algebras are carried out for com-plex C -algebras,we look for a method of reducing real problems to complex problems.Suppose A R is a real semi-pre-C -algebra.Then,the complexiﬁca-tion of A R is the complex semi-pre-C -algebra A C D A R C i A R.The -algebra structure(over C)of A C is deﬁned in an obvious way,and.A C/C is deﬁned to be the -positive cone generated by.A R/C:.A C/C Dn n Xi D1z i a i z i W n2N;a i2.A R/C;z i2A Co:(This is a temporary deﬁnition,and the ofﬁcial one will be given later.See Lemma11.)Note that A R\.A C/C D.A R/C.The complexiﬁcation A C has an involutive and conjugate-linear -automorphism deﬁned by x C i y!156N.Ozawa x i y ,x;y 2A R .Every complex semi-pre-C -algebra with an involutive and conjugate-linear -automorphism arises in this way.Lemma 5.Let R W A R !B R be a -homomorphism between real semi-pre-C -algebras (resp.'R W A R !R be a self-adjoint linear functional).Then,the complexiﬁcation C W A C !B C (resp.'C W A C !C )is positive if and only if R (resp.'R )is so.Proof.Weonly prove that 'C is positive if 'R is so.The rest is trivial.Let b D P i z i a i z i 2.A C /C be arbitrary,where a i 2.A R /C and z i D x i C i y i .Then,b D P i .x i a i x i C y i a i y i /C i P i .x i a i y i y i a i x i /.Since x i a i y i y i a i x i is skew-hermitian,one has 'R .x i a i y i y i a i x i /D 0,and 'C .b/D'R .P i x i a i x i C y i a i y i / 0.This shows 'C is positive.We note that if H C denotes the complexiﬁcation of a real Hilbert space H R ,then B .H R /C D B .H C /.Thus every positive -representation of a real semi-pre-C -algebra A R on H R extends to a positive -representation of its complexiﬁcation A C on H C .Conversely,if is a positive -representation of A C on a complex Hilbert space H C ,then its restriction to A R is a positive -representation on the realiﬁcation of H C .The realiﬁcation of a complex Hilbert space H C is the real Hilbert space H C equipped with the real inner product h Á; i R D <h Á; i .Therefore,we arrive at the conclusion that C u .A R /C D C u .A C /.We also see that .R Œ /C D C Œ ,.A R ˝B R /C D A C ˝B C ,.A R B R /C D A C B C ,etc.8.Proof of Theorem 1We only prove the ﬁrst assertion of Theorem 1.The proof of the second is very similar.We will prove a stronger assertion thatk Ã.x/k C u.A /D inf f R >0W R 21 x x 2A C g :The inequality Ätrivially follows from the C -identity.For the converse,as-sume that the right hand side is non-zero,and choose >0such that 21 x x …A C .By Corollary 4,there is '2S.A /such that '. 21 x x/Ä0.Thus for the GNS representation ',one hask '.x/k k '.x/O 1k D '.x x/1=2 :It follows that k Ã.x/k .About the Connes embedding conjecture157 9.Trace positive elementsLet A be a semi-pre-C -algebra.A state on A is called a tracial state if .xy/D .yx/for all x;y2A,or equivalently if is zero on the -subspace K D span f xy yx W x;y2A g spanned by commutators in A.We denote by T.A/the set of tracial states on A(which may be empty).Associated with 2T.A/is aﬁnite von Neumann algebra. .A/00; /,which is the von Neumann algebra generated by .A/ B.L2.A; //with the faithful normal tracial state .a/D h a O1;O1i that extends the original .Recall that aﬁnite von Neumann algebra is a pair.M; /of a von Neumann algebra and a faithful normal tracial state on M.The following theorem is proved in[KS]for the algebra in Example3and in[JP]for the free group algebras,but the proof equally works in the general setting.We note that for some groups ,notably for D SL3.Z/([Be2]),it is possible to describe all the tracial states on kŒ . Theorem6([KS]).Let A be a semi-pre-C -algebra,and a2A h.Then,the following are equivalent..1/ .a/ 0for all 2T.A/..2/ . .a// 0for everyﬁnite von Neumann algebra.M; /and every positive -homomorphism W A!M..3/a2arch.A C C K h/,where K h D K\A h D span f x x xx W x2A g. Proof.The equivalence.1/,.2/follows from the GNS construction.We only prove.1/).3/,as the converse is trivial.Suppose a C"1…A C C K h for some">0.Then,by Corollary4,there is 2S.A/such that .K/D f0g (i.e., 2T.A/)and .a/Ä "<0.10.Connes embedding conjectureThe Connes embedding conjecture(CEC)asserts that anyﬁnite von Neumann algebra.M; /with separable predual is embeddable into the ultrapower R! of the hyperﬁnite II1-factor R(over k2f C;R g).Here an embedding means an injective -homomorphism which preserves the tracial state.We note that if is a tracial state on a semi-pre-C -algebra A andÂW A!N is a -preserving -homomorphism into aﬁnite von Neumann algebra.N; /,thenÂextends to a -preserving -isomorphism from .A/00onto the von Neumann subalgebra generated byÂ.A/in N(which coincides with the ultraweak clo-sure ofÂ.A/).Hence,.M; /satisﬁes CEC if there is an ultraweakly dense -subalgebra A M which has a -preserving embedding into R!.In particular, CEC is equivalent to that for every countably generated semi-pre-C -algebra A and 2T.A/,there is a -preserving -homomorphism from A into R!.We will see that this is equivalent to the tracial analogue of Positivstellensätze in158N.Ozawa Corollary 2.We ﬁrst state a few equivalent forms of CEC.We denote by tr the tracial state 1N Tr on M N .k /.Theorem 7.For a ﬁnite von Neumann algebra .M; /with separable predual,the following are equivalent..1/.M; /satisﬁes CEC,i.e.,M ,!R !..2/Let d 2N and x 1;:::;x d 2M be hermitian contractions.Then,forevery m 2N and ">0,there are N 2N and hermitian contractions X 1;:::;X d 2M N .k /such thatj .x i 1 x i k / tr .X i 1 X i k /j <"for all k Äm and i j 2f 1;:::;d g ..3/Assume k D C (or replace M with its complexiﬁcation in case k D R ).Letd 2N and u 1;:::;u d 2M be unitary elements.Then,for every ">0,there are N 2N and unitary matrices U 1;:::;U d 2M N .C /such thatj .u i u j / tr .U i U j /j <"for all i;j 2f 1;:::;d g .In particular,CEC holds true if and only if every .M; /satisﬁes condition .2/and/or .3/.The equivalence .1/,.2/is a rather routine consequence of the ultra-product construction.For the equivalence to (3),see Theorem 27.Note that the assumption k D C in condition (3)is essential because the real analogue of it is actually true ([DJ]).Since any ﬁnite von Neumann algebra M with sep-arable predual is embeddable into a II 1-factor which is generated by two her-mitian elements (namely .M R/N ˝R ),to prove CEC,it is enough to verify the conjecture (2)for every .M; /and d D 2.We observe that a real ﬁnite von Neumann algebra .M R ; R /is embeddable into R !R (i.e.,it satisﬁes CEC)if and only if its complexiﬁcation .M C ; C /is embeddable into R !C .The “only if”di-rection is trivial and the “if”direction follows from the real -homomorphism M N .C /,!M 2N .R /,a C i b ! a b b a .A complex ﬁnite von Neumann al-gebra .M; /need not be a complexiﬁcation of a real von Neumann algebra,but M ˚M op is (isomorphic to the complexiﬁcation of the realiﬁcation of M ).Therefore,M satisﬁes CEC if and only if its realiﬁcation satisﬁes it.For a ﬁnite von Neumann algebra .M; /and d 2N ,we denote by H d .M /the set of those f 2k h x 1;:::;x d i h such that .f.X 1;:::;X d // 0for all hermitian contractions X 1;:::;X d 2M .Further,let H d D\M H d .M /and H ﬁn d D \NH d .M N .k //D H d .R/:Notice that H d D arch .k h x 1;:::;x d i C C K h /(see Example 3and Theorem 6).About the Connes embedding conjecture 159Corollary 8([KS]).Let k 2f C ;R g .Then one has the following.Let .M; /be a ﬁnite von Neumann algebra with separable predual.Then,M satisﬁes CEC if and only if H ﬁn d H d .M /for all d .CEC holds true if and only if H ﬁn d D arch .k h x 1;:::;x d i C C K h /for all/some d 2.Proof.It is easy to see that condition (2)in Theorem 7implies H ﬁn d H d .M /.Conversely,suppose condition (2)does not hold for some d 2N ,x 1;:::;x d 2M ,m 2N ,and ">0.We introduce the multi-index notation.For i D .i 1;:::;i k /,i j 2f 1;:::;d g and k Äm ,we denote x i D x i 1 x i k .It may happen that i is the null string ;and x ;D 1.Then,C D closure f .tr .X i //i W N 2N ;X 1;:::;X d 2M N .k /h ;k X i k Ä1gis a convex set (consider a direct sum of matrices).Hence by Theorem 3,there are 2R and ˛i 2k such that <X i ˛i .x i /< Äinf 2C <X i˛i i :Replacing ˛i with .˛i C ˛ i/=2(here i is the reverse of i ),we may omit <from the above inequality.Further,arranging ˛;,we may assume D 0.Thus f D P i ˛i x i belongs to H ﬁn d ,but not to H d .M /.This completes the proof of the ﬁrst half.The second half follows from this and Theorem 6.An analogue to the above also holds for C ŒF d .Corollary 9([JP]).Let k D C .The following holds.Let .M; /be a ﬁnite von Neumann algebra with separable predual.Then,M satisﬁes CEC if and only if the following holds true:If d 2N and ˛2M d .C /h satisﬁes that tr .P ˛i;j U i U j / 0for every N 2N and U 1;:::;U d 2M N .C /u ,then it satisﬁes .P ˛i;j u i u j / 0for every uni-tary elements u 1;:::;u d 2M .CEC holds true if and only if for every d 2N and ˛2M d .C /h the follow-ing holds true:If tr .P ˛i;j U i U j / 0for every N 2N and U 1;:::;U d 2M N .C /u ,then P ˛i;j s i s j 2arch .C ŒF d C C K h /,where s 1;:::;s d are the free generators of F d .11.Matrix algebras over semi-pre-C -algebrasWe describe here how to make the n n matrix algebra M n .A /over a semi-pre-C -algebra A into a semi-pre-C -algebra.We note that x D Œx j;i i;j for x D Œx i;j i;j 2M n .A /.We often identify M n .A /with M n .k /˝A .There。

An extension theorem for separately holomorphic functions with singularities

(b) M = ∅: J. Siciak [Sic 2000] solved the case: N ≥ 2, k1 = · · · = kN = 1, D1 = · · · = DN = C, M = P −1(0), where P is a non-zero polynomial of N complex variables. The special subcase N = 2, P (z, w) := z − w had been studied in [O¨ kt 1998]. The general case for N = 2, k1 = k2 = 1 was solved in [Jar-Pﬂ 2001]; see also [O¨ kt 1999] for a partial discussion of the case N = 2, k1, k2 ≥ 1.
We say that a subset ∅ = A ⊂ Cn is locally pluriregular if h∗A∩Ω,Ω(a) = 0 for any a ∈ A and for any open neighborhood Ω of a (in particular, A ∩ Ω is non-pluripolar).
Abstract. Let Dj ⊂ Ckj be a pseudoconvex domain and let Aj ⊂ Dj be a locally pluripolar set, j = 1, . . . , N . Put
N
X :=
A1 × · · · × Aj−1 × Dj × Aj+1 × · · · × AN ⊂ Ck1+···+kN .
N
:= A1 × · · · × Aj−1 × Dj × Aj+1 × · · · × AN ⊂ Ck1+···+kN .

公平展新姿的英语作文

Fairness is a fundamental principle in society,and it is essential for maintaining social harmony and stability.In the modern world,where competition is fierce,fairness is more important than ever before.Here are some key points to consider when discussing the importance of fairness in an English essay:1.Equality of Opportunity:Fairness ensures that everyone has an equal chance to succeed.This means that no one is unfairly advantaged or disadvantaged based on their social,economic,or racial background.2.Justice in the Legal System:A fair legal system is crucial for the functioning of any society.It means that laws are applied equally to all individuals,regardless of their status or wealth.3.Meritocracy:Fairness promotes a meritocratic system where individuals are rewarded based on their abilities and achievements rather than their connections or privileges.4.Economic Fairness:This refers to the distribution of wealth and resources in a way that is equitable and does not favor a select few.It helps to reduce economic disparities and poverty.5.Fairness in Education:Access to quality education should be available to all, regardless of their financial situation.This helps to level the playing field and allows for social mobility.6.Workplace Fairness:Employees should be treated fairly in terms of wages,promotions, and working conditions.Discrimination based on gender,race,or age should not be tolerated.7.Fair Trade and Business Practices:Fairness extends to international trade,where businesses should operate ethically and not exploit workers in developing countries.8.Environmental Justice:This involves ensuring that all communities have access to clean air,water,and a healthy environment,without being disproportionately affected by pollution or climate change.9.Cultural Fairness:Recognizing and respecting the cultural differences and rights of various groups within a society can lead to a more inclusive and harmonious community.10.The Role of Government:Governments play a pivotal role in implementing policies that promote fairness and address any existing inequalities.In conclusion,fairness is a multifaceted concept that touches on various aspects of life.It is not just about treating people equally but also about creating an environment where everyone has the opportunity to thrive.By promoting fairness,societies can foster a sense of unity,trust,and cooperation,which are vital for progress and development.。

A theory of weak bisimulation for core cml

J.Functional Programming1(1):1–000,January1993c1993Cambridge University Press1A Theory of Weak Bisimulation for Core CMLWILLIAM FERREIRA†Computing LaboratoryUniversity of CambridgeMATTHEW HENNESSY AND ALAN JEFFREY‡School of Cognitive and Computing SciencesUniversity of Sussex1IntroductionThere have been various attempts to extend standard programming languages with con-current or distributed features,(Giacalone et al.,1989;Holmstr¨o m,1983;Nikhil,1990). Concurrent ML(CML)(Reppy,1991a;Reppy,1992;Panangaden&Reppy,1996)is a practical and elegant example.The language Standard ML is extended with two new type constructors,one for generating communication channels,and the other for delayed com-putations,and a new function for spawning concurrent threads of computation.Thus the language has all the functional and higher-order features of ML,but in addition pro-grams also have the ability to communicate with each other by transmitting values along communication channels.In(Reppy,1992),a reduction style operational semantics is given for a subset of CML calledλcv,which may be viewed as a concurrent version of the call-by-valueλ-calculus of(Plotkin,1975).Reppy’s semantics gives reduction rules for whole programs,not for program fragments.It is not compositional,in that the semantics of a program is not deﬁned in terms of the semantics of its subterms.Reppy’s semantics is designed to prove properties about programs(for example type safety),and not about program fragments(for example equational reasoning).In this paper we construct a compositional operational semantics in terms of a labelled †William Ferreira was funded by a CASE studentship from British Telecom.‡This work is carried out in the context of EC BRA7166CONCUR2.2W.Ferreira,M.Hennessy and A.S.A.Jeffreytransition system,for a core subset of CML which we callµCML.This semantics not only describes the evaluation steps of programs,as in(Reppy,1992),but also their communi-cation potentials in terms of their ability to input and output values along communication channels.This semantics extends the semantics of higher-order processes(Thomsen,1995) with types andﬁrst-class functions.We then proceed to demonstrate the usefulness of this semantics by using it to deﬁne a version of weak bisimulation,(Milner,1989),suitable forµCML.We prove that,modulo the usual problems associated with the choice operator of CCS,our chosen equivalence is preserved by allµCML contexts and therefore may be used as the basis for reasoning about CML programs.In this paper we do not investigate in detail the resulting theory but conﬁne ourselves to pointing out some of its salient features;for example standard identities one would expect of a call-by-valueλ-calculus are given and we also show that certain algebraic laws common to process algebras,(Milner,1989),hold.We now explain in more detail the contents of the remainder of the paper.In Section2we describeµCML,a monomorphically typed core subset of CML,which nonetheless includes base types for channel names,booleans and integers,and type con-structors for pairs,functions,and delayed computations which are known as events.µCML also includes a selection of the constructs and constants for manipulating event types,such as and for constructing basic events for sending and receiving values, for combining delayed computations,for selecting between delayed compu-tations,and a function for launching new concurrent threads of computation within a program.The major omission is thatµCML has no facility for generating new channel names.However we believe that this can be remedied by using techniques common to the π-calculus,(Milner,1991;Milner et al.,1992;Sangiorgi,1992).In the remainder of this section we present the operational semantics ofµCML in terms of a labelled transition system.In order to describe all possible states which can arise dur-ing the computation of a well-typedµCML program we need to extend the language.This extension is twofold.Theﬁrst consists in adding the constants of event type used by Reppy in(Reppy,1992)to deﬁneλcv,i.e.constants to denote certain delayed computations.This extended language,which we callµCML cv,essentially coincides with theλcv,the lan-guage used in(Reppy,1992),except for the omissions cited above.However to obtain a compositional semantics we make further extensions toµCML cv.We add a parallel oper-ator,commonly used in process algebras,which allows us to use programs in place of the multisets of programs of(Reppy,1992).Theﬁnal addition is more subtle;we include inµCML cv expressions which correspond to the ed versions of Reppy’s constants for representing delayed computations.Thus the labelled transition system uses as states programs from a language which we call µCML.This language is a superset ofµCML cv,which is our version of Reppy’sλcv, which in turn is a superset ofµCML,our mini-version of CML.The following diagramA Theory of Weak Bisimulation for Core CML3indicates the relationships between these languages:µCMLλcvCMLIn Section3we discuss semantic equivalences deﬁned on the labelled transition of Sec-tion2.We demonstrate the inadequacies of the obvious adaptations of strong and weak bisimulation equivalence,(Milner,1989),and then consider adaptations of higher-order and irreﬂexive bisimulations from(Thomsen,1995).Finally we suggest a new variation called hereditary bisimulation equivalence which overcomes some of the problems en-countered with using higher-order and irreﬂexive bisimulations.In Section4we show that hereditary bisimulation is preserved by allµCML contexts.This is an application of the proof method originally suggested in(Howe,1989)but the proof is further complicated by the fact that hereditary bisimulations are deﬁned in terms of pairs of relations satisfying mutually dependent properties.In Section5we brieﬂy discuss the resulting algebraic theory ofµCML expressions.This paper is intended only to lay the foundations of this theory and so here we simply indicate that our theory extends both that of call-by-valueλ-calculus(Plotkin,1975)and process algebras(Milner,1989).In Section6we show that,up to weak bisimulation equivalence,our semantics coincides with the reduction semantics forλcv presented in(Reppy,1992).This technical result ap-plies only to the common sub-language,namelyµCML cv.In Section7we brieﬂy consider other approaches to the semantics of CML and related languages and we end with some suggestions for further work.2The LanguageIn this section we introduce our languageµCML,a subset of Concurrent ML(Reppy, 1991a;Reppy,1992;Panangaden&Reppy,1996).We describe the syntax,including a typing system,and an operational semantics in terms of a labelled transition system. Unfortunately,there is not enough space in this paper to provide an introduction to pro-gramming in CML:see(Panangaden&Reppy,1996)for a discussion of the design and philosophy of CML.The type expressions for our language are given by:A::A A A A A AThus we have three base types,,and;the latter two are simply examples of useful base types and one could easily include more.These types are closed under four con-structors:pairing,function space,and the less common and type constructors.4W.Ferreira,M.Hennessy and A.S.A.JeffreyOur language may be viewed as a typedλ-calculus augmented with the type constructors A for communication channels sending and receiving data of type A,and A for constructing delayed computations of type A.Let Chan A be a type-indexed family of disjoint sets of channel names,ranged over by k, and let Var denote a set of variables ranged over by x,y and z.The expressions ofµCML are given by the following abstract syntax:e f g Exp::v ce e e e e e x e e eev w Val::x y e x k01c Const::The main syntactic category is that of Exp which look very much like the set of expressions for an applied call-by-value version of theλ-calculus.There are the usual pairing,let-binding and branching constructors,and two forms of application:the application of one expression to another,ee,the application of a constant to an expression,ce.There is also a syntactic category of value expressions Val,used in giving a semantics to call-by-value functions and communicate-by-value channels.They are restricted in form: either a variable,a recursively deﬁned function,x y e,or a predeﬁned literal value for the base types.We will use some syntax sugar,writing y e for x y e when x does not occur in e,and e;f for x e f when x does not occur in f. Finally there are a small collection of constant functions.These consist of a representa-tive sample of constants for manipulating objects of base type,,which could easily be extended,the projection functions and,together with the set of constants for manipulating delayed computations taken directly from(Reppy,1992):and,for constructing delayed computations which can send and receive values,,for constructing alternatives between delayed computations,,for spawning new computational threads,,for launching delayed computations,,for combining delayed computations,,for a delayed computation which always deadlocks,and,for a delayed computation which immediately terminates with a value. Note that there is no method for generating channel names other than using the predeﬁned set of names Chan A.There are two constructs in the language which bind occurrences of variables,xe1e2where free occurrences of x in e2are bound and x y e where free oc-currences of both x and y in e are bound.We will not dwell on the precise deﬁnitions of free and bound variables but simply use f v e to denote the set of variables which have free occurrences in e.If f v e/0then e is said to be a closed expression,which we sometimes refer to as a program.We also use the standard notation of e v x to denote the substitution of the value v for all free occurrences of x in e where bound names may be changed in order to avoid the capture of free variables in v.(Since we are modelling aA Theory of Weak Bisimulation for Core CML5:A B A:A A:A B B:A A::A A A::::A A B B:A A:A:A AFigure1a.Type rules forµCML constant functionsx yΓx:A y:BΓ:Γ:Γx y e:A BΓe:AΓf:BΓe:AΓe f:BΓe:AΓx:A f:BΓe f g:A6W.Ferreira,M.Hennessy and A.S.A.Jeffreythis reduction semantics are of the form:CτCwhere C C are conﬁgurations which combine a closed expression with a run-time envi-ronment necessary for its evaluation,andτis Milner’s notation for a silent action.However this semantics is not compositional as the reductions of an expression can not be deduced directly from the reductions of it constituent components.Here we give a compositional operational semantics with four kinds of judgements:eτe,representing a one step evaluation or reduction,e v e,representing the production of the value v,with a side effect e,e k?x e,representing the potential to input a value x along the channel k,ande k!v e,representing the output of the value v along the channel k.These are formally deﬁned in Figure2,but weﬁrst give an informal overview.In order to deﬁne these relations we introduce extra syntactic constructs.These are introduced as required in the overview but are summarized in Figure3.The rules for one step evaluation or reduction have much in common with those for a standard call-by-valueλ-calculus.But in addition a closed expression e of type A should evaluate to a value of type A and it is this production of values which is the subject of the second kind of judgement.HoweverµCML expressions can spawn subprocesses before returning a value,so we have to allow expressions to continue evaluation even after they have returned a result.For example in the expression:0a;aone possible reduction is(whereτindicates a sequence ofτ-reductions):0a;aτa?11a!0where the process returns the value1before outputting0.For this reason we need a reduc-tion e v e rather than the more usual termination e v.The following diagram illustrates all of the possible transitions from this expression:0a;aτa!0τa?vva!0vA Theory of Weak Bisimulation for Core CML7 judgements of the operational semantics apply to these conﬁgurations.The second,more common in work on process algebras,(Bergstra&Klop,1985;Milner,1989),extends the syntax of the language being interpreted to encompass conﬁgurations.We choose the latter approach and one extra construct we add to the language is a parallel operator,e f.This has the same operational rules as in CCS,allowing reduction of both processes:eαee fαe fand communication between the processes:e k!v ef k?x fe fτe v x fThe assymetry is introduced by termination(a feature missing from CCS).A CML process has a main thread of control,and only the main thread can return a value.By convention, we write the main thread on the right,so the rule is:f v feαeSecondly,e may have spawned some concurrent processes before returning a function,and these should carry on evaluation,so we use the silent rule for constant application:e v e8W.Ferreira,M.Hennessy and A.S.A.JeffreyThe well-typedness of the operational semantics will ensure that v is a function of the appropriate type,.With this method of representing newly created computation threads more of the rules corresponding toβ-reduction in a call-by-valueλ-calculus may now be given.To evaluate an application expression e f,ﬁrst e is evaluated to a value of functional form and then the evaluation of f is initiated.This is represented by the rules:eαee fτe yf g(In fact we use a slightly more complicated version of the latter rule as functions are al-lowed to be recursive.)Continuing with the evaluation of e f,we now evaluate f to a value which is then substituted into g for y.This is represented by the two rules:fτfx f gτf g v xThe evaluation of the application expression c f is similar;f is evaluated to a value and then the constant c is applied to the resulting value.This is represented by the two rulesfτfc fτfδc vHere,borrowing the notation of(Reppy,1992),we use the functionδto represent the effect of applying the constant c to the value v.This effect depends on the constant in question and we have already seen one instance of this rule,for the constant,which result from the fact thatδv v.The deﬁnition ofδfor all constants in the language is given in Figure2f.For the constants associated with the base types this is self-explanatory; the others will be explained below as the constant in question is considered.Note that because of the introduction of into the language we can treat all constants uniformly, unlike(Reppy,1992)where and have to considered in a special manner.In order to implement the standard left-to-right evaluation of pairs of expressions we introduce a new value v w representing a pair which has been fully evaluated.Then to evaluate e f:ﬁrst allow e to evaluate:eαee fτe xf v xThese value pairs may then be used by being applied to functions of type A B.For example the following inferences result from the deﬁnition of the functionδfor the constants and:e v w eeτe m nIt remains to explain how delayed computations,i.e.programs of type A,are han-dled.It is important to realise that expressions of type A represent potential rather than actual computations and this potential can only be activated by an application of theA Theory of Weak Bisimulation for Core CML9eαee fαe feαee f gαe f geαee fαef fαfe f v e fFigure2a.Operational semantics:static rules ge1αege1ge2αegeαeceτeδc v e ee f gτe ge v ee fτe yfg v xv x y g e v ee fτef v xe k?x ef k!v fv vΛk?k?x x10W.Ferreira,M.Hennessy and A.S.A.Jeffreye f g Exp::v ce e e e e e x e e eev w Val::x y e x k01c Const::Figure3a.Syntax ofµCMLv w Val::v v gege GExp::v!v v?ge v ge geΛA vFigure3b.Syntax ofµCML cve f g Exp::ge e eFigure3c.Syntax ofµCMLconstant,of type A A.Thus for example the expression k is of type A and represents a delayed computation which has the potential to receive a value of type A along the channel k.The expression k can actually receive such a value v along channel k,or more accurately can evaluate to such a value,provided some other computation thread can send the value along channel k.The semantics of is handled by introducing a new constructor for values.For certain kinds of expressions ge of type A,which we call guarded expressions,let ge be a value of type A;this represents a delayed computation which when launched initiates a new computation thread which evaluates the expression ge.Then the expression ge reduces in one step to the expression ge.More generally the evaluation of the expressione proceeds as follows:First evaluate e until it can produce a value:eτeeτe geNote that here,as always,the production of a value may have as a side-effect the generation of a new computation thread e and this is launched concurrently with the delayed compu-tation ge.Also both of these rules are instances of more general rules already considered. Theﬁrst is obtained from the rule for the evaluation of applications of the form ce and the second by deﬁningδge to be ge.The precise syntax for guarded expressions will emerge by considering what types of values of the form e can result from the evaluation of expressions of type from the basic languageµCML.The constant is of type A A and thereforethe evaluation of the expression e proceeds byﬁrst evaluating e to a value of type A until it returns a value k,and then returning a delayed computation consisting of an event which can receive any value of type A on the channel k.To represent this event we extend the syntax further by letting k?be a guarded expression for any k and A,with the associated rule:e k eeτe k!vIt is these two new expressions k?and k!v which perform communication between compu-tation threads.Formally k!v is of type and we have the axiom:k?k?x xTherefore in general input moves are of the form e k?x f where e:B and x:A f:B. Communication can now be modelled as in CCS by the simultaneous occurrence of input and output actions:e k?x ef k!v feτeΛobtained,once more,by deﬁningδto beΛ.The constant is of type A A B B.The evaluation of e proceeds in the standard way by evaluating e until it produces a value,which must be of the form ge v,where ge is a guarded expression of type A and v has type A B.Then the evaluation of e continues by the construction of the new delayed computation ge v.Bearing in mind the fact that the production of values can generate new computation threads,this is formally represented by the inference rule:e ge v ege vαveThe construct,of type A A,evaluates its argument to a value v,and thenreturns a trivial a delayed computation;this computation,when activated,immediately evaluates to the value v.In order to represent these trivial computations we introduce a new constructor for guarded expressions,A and the semantics of is then captured by the rule:e v eA vτvThe choice construct e is a choice between delayed computations as has the type A A A.To interpret it we introduce a new choice constructor ge1ge2where ge1and ge2are guarded expressions of the same type.Then e pro-ceeds by evaluating e until it can produce a value,which must be of the form ge1ge2, and the evaluation continues by constructing the delayed computation ge1ge2.This is represented by the rule:e ge1ge2ege2αege1ge2αeΓv:AΓw:BΓge:AΓv:AΓw:AΓv?:AΓge:AΓv:A BΓge1ge2:AΓA v:AΓe:AΓf:BΓτ:A Γv:AΓk?x:AΓw:Bof the form e k ?xf where f may be an open expression we need to consider relations over open expressions.Let an open type-indexed relation R be a family of relations R ΓA such that if e R ΓA f then Γe :A and Γf :A .We will often elide the subscripts from relations,for example writing e R f for e R ΓA f when context makes the type obvious.Let a closed type-indexed relation R be an open type-indexed relation where Γis everywhere the empty context,and can therefore be elided.For any closed type-indexed relation R ,let its open extension R be deﬁned as:e R x :A Bf iff e v x R B f v x for allv :AA closed type-indexed relation R is structure preserving iff:if v R A w and A is a base type then v w ,if v 1v 2R A 1A 2w 1w 2then v i R A i w i ,if ge 1R A ge 2then ge 1R A ge 2,andif v R A B v then for all w :A we have vw R B v w .With this notation we can now deﬁne strong bisimulations over µCML expressions.A closed type-indexed relation R is a ﬁrst-order strong simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2ase 1lRe 2lsince the deﬁnition of strong bisimulation demands that the actions performed by expres-sions match up to syntactic identity.This counter-example can also be reproduced using only µCML contexts:kx121kx21since the left hand side can perform the move:kx12τk !x12but this can only be matched by the right hand side up to strong bisimulation:kx21τk !x21In fact,it is easy to verify that the only ﬁrst-order strong bisimulation which is a congruence for µCML is the identity relation.To ﬁnd a satisfactory treatment of bisimulation for µCML,we need to look to higher-order bisimulation ,where the structure of the labels is accounted for.To this end,given a closed type-indexed relation R ,deﬁne its extension to labels R l as:v R l A wk !v R l A k !wkChan BThen R is a higher-order strong simulation iff it is structure-preserving and the followingdiagram can be completed:e 1R e 2e 1R e 2aswhere l 1R l l 2e 1l 1Re 2l 2lotherwise.Then R is a ﬁrst-order weak simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2ase 1lRe 2ˆl Let1be the largest ﬁrst-order weak bisimulation.Proposition 3.31is an equivalence.ProofSimilar to the proof of Proposition 3.1.Unfortunately,1is not a congruence,for the same reason as 1,and so we can attempt the same modiﬁcation.R is a higher-order weak simulation iff it is structure-preserving and the following diagram can be completed:e 1R e 2e 1R e 2aswhere l 1R ll 2e 1l 1Re 2ˆl 2Lethbe the largest higher-order weak bisimulation.Proposition 3.4h is an equivalence.ProofSimilar to the proof of Proposition 3.1.However,h is still not a congruence,for the usual reason that weak bisimulation equiva-lence is not a congruence for CCS summation.Recall from (Milner,1989)that in CCS 0τ0but a 00a 0τ0.We can duplicate this counter-example in µCML since the CCS operator corresponds to the µCML operator and 0corresponds to Λ.However may only be applied to guarded expressions and therefore we need a guarded expressionwhich behaves like τ0;the required expression is A Λ.Thus:ΛhA Λsince the right hand side has only one reduction:A ΛτΛτΛbut:Λk !0hA Λk !0because the only reduction of Λk !0is Λk !0k !0ΛΛand:A Λk !0τΛτΛThis counter-example can also be replicated using the restricted syntax of µCML.We have:hsince the left hand side has only one reduction:ΛΛand the right hand side can match this with:A ΛΛand we have seen:ΛhA ΛHowever:k 0hk 0since the left hand side has only one reduction:k 0τΛk !0whereas the right hand side has the reduction:k 0τA Λk !0A ﬁrst attempt to rectify this is to adapt Milner’s observational equivalence for µCML,and to deﬁne h as the smallest symmetric relation such that the following diagram can be completed:e 1he 2e 1he 2aswhere l 1h ll 2e 1l 1he 2l 2Proposition 3.5h is an equivalence.ProofSimilar to the proof of Proposition 3.1.This attempt fails,however,since it only looks at the ﬁrst move of a process,and not at theﬁrst moves of any processes in its transitions.Thus,the above µCML counter-example for h being a congruence also applies to h ;i.e.hbut:k 0hk 0This failure was ﬁrst noted in (Thomsen,1995)for CHOCS.Thomsen’s solution to this problem is to require that τ-moves can always be matched by at least one τ-move,which produces his deﬁnition of an irreﬂexive simulation as a structure-preserving relation where the following diagram can be completed:e 1R e 2e 1R e 2aswhere l 1R l l 2e 1l 1Re 2l 2Letibe the largest irreﬂexive bisimulation.Proposition 3.6iis a congruence.ProofThe proof that i is an equivalence is similar to the proof of Proposition 3.1.The proof that it is a congruence is similar to the proof of Theorem 4.7in the next section.However this relation is rather too strong for many purposes,for example 12i111since the right hand side can perform more τ-moves than the left hand side.This is similar to the problem in CHOCS where a τP i a P .In order to ﬁnd an appropriate deﬁnition of bisimulation for µCML,we observe that µCML only allows to be used on guarded expressions ,and not on arbitrary expressions.We can thus ignore the initial τ-moves of all expressions except for guarded expressions.For this reason,we have to provide two equivalences:one on terms where we are not interested in initial τ-moves,and one on terms where we are.A pair of closed type-indexed relations R R n R s form a hereditary simulation (we call R n an insensitive simulation and R s a sensitive simulation )iff R s is structure-preserving and we can complete the following diagrams:e 1R ne 2e 1R ne 2aswhere l 1R sll 2e 1l 1R ne 2ˆl 2and:e 1R se 2e 1R se 2aswhere l 1R s l l 2e 1l 1R ne 2l 2Let n sbe the largest hereditary bisimulation.Note that we require R s to be structure-preserving because it is used to compare the labels in transitions,which may contain ab-stractions or guarded events.In the operational semantics of µCML expressions,guarded expressions can only appear in labels,and not as the residuals of transitions.This explains why in the deﬁnition of n labels are compared with respect to the sensitive relation s whereas the insensitive relation is used for the residuals.For example,if ge 1n s ge 2then we have:xge 1nxge 2since once either side is applied to an argument,their ﬁrst action will be a τ-step.On the other hand:ge 1nge 2sinceis precisely the construct which allows us to embed ge 1and ge 2in acontext.Theorem 3.7s is a congruence for µCML ,andnis a congruence for µCML.ProofThe proof that s and n are equivalences is similar to the proof of Proposition 3.1.The proof that they form congruences is the subject of the next section.Proposition 3.8The equivalences on µCML have the following strict inclusions:1shh111x1xk k i h k12s i1111n s x1xh n1h h x1xwhere:x x(Note that this settles an open question(Thomsen,1995)as to whether i is the largestcongruence contained in h.)It is the operator which differentiates between the two equivalences n and h.Howeverin order to demonstrate the difference we need to be able to apply to guarded expressionswhich can spontaneously evolve,i.e.performτ-moves.The onlyµCML constructor for guarded expressions which allows this is A,and in turn occurrences of this can only begenerated by theµCML constructor.Therefore:Proposition3.9For the subset ofµCML without and A,n is the same as h,and s is the same as h.ProofFrom Proposition3.8n h.For the subset ofµCML without and A,deﬁne R s as:v w v h w ge1ge2ge1h ge2v1w v2w v1h v2Then since no event without A can perform aτ-move,and since the only initial moves ofv i w areβ-reductions,we can show that h R s forms an hereditary bisimulation,and so h n.From this it is routine to show that s h.Unfortunately we have not been able to show that n is the largestµCML congruence con-tained in weak higher-order bisimulation equivalence.However we do have the following characterisation:Theorem3.10n is the largest higher-order weak bisimulation which respectsµCML contexts.ProofBy deﬁnition,n is a higher-order weak bisimulation,and we have shown that it respectsµCML contexts.All that remains is to show that it is the largest such.Let R be a higher-order weak bisimulation which respectsµCML contexts.Then deﬁne: R n R v1w e2v1R v2v2wτe2e1v2w v1R v2v1wτe1R s v w v R w ge1ge2ge1R ge2v1w v2w v1R v2We will now show that R n R s forms an hereditary simulation,from which we can de-duce R R n n.。

#x #

Abstract We consider uniqueness theorems in classical analysis having the form (+) ∀u ∈ U, v1 , v2 ∈ Vu G(u, v1 ) = 0 = G(u, v2 ) → v1 = v2 , where U, V are complete separable metric spaces, Vu is compact in V and G : U × V → IR is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ∀x ∈ X ∃y ∈ Yx ∀z ∈ Z F (x, y, z ) = 0 only (where X, Y, Z are complete separable metric spaces, Yx ⊂ Y is compact and F : X × Y × Z → IR constructive), then we can extract from the proof of (++) → (+) an eﬀective modulus of uniqueness, i.e. (+ + +) ∀u ∈ U, v1 , v2 ∈ Vu , k ∈ IN |G(u, v1 )|, |G(u, v2 )| ≤ 2−Φuk → dV (v1 , v2 ) ≤ 2−k . Such a modulus Φ can e.g. be used to give a ﬁnite algorithm which computes the (uniquely determined) zero of G(u, ·) on Vu with prescribed precision if it exists classically. The extraction of Φ uses a proof–theoretic combination of functional interpretation and pointwise majorization. If the proof of (++) → (+) uses only simple instances of induction, then Φ is a simple mathematical operation (on a convenient standard representation of X , e.g. on f together with a modulus of uniform continuity for X = C [0, 1]). Various uniqueness theorems in best approximation theory have the form (+) and are proved using only analytical tools of the form (++). We analyse the most common proof of uniqueness for the best Chebycheﬀ approximation of f ∈ C [0, 1] by polynomials of degree ≤ n given by de La Vall´ ee Poussin and obtain explicit moduli of uniqueness and uniform constants of strong unicity. In a subsequent paper two further proofs of this uniqueness will be analysed yielding better estimates (due to the fact that mainly (++)–lemmas are used) which allow us to improve results obtained prior by D. Bridges signiﬁcantly.

功能对等理论视角下的英汉习语翻译1

本文共分为六个章节。第一章主要介绍撰写本论文的基本要素:研究背景、研究问题、研究目的和意义、研究方法以及论文结构。第二章主要通过文献综述，了解国内外习语翻译的研究概况。同时帮助读者理解习语的一些基本知识，包括习语的定义、特点、分类及奈达的功能对等翻译理论。第三章从文化对等讨论英汉习语的翻译问题。涉及文化、习语和翻译的关系,习语的文化内涵以及具体的翻译方法等。第四章从语用对等讨论习语的翻译问题。涉及习语的语用功能及其误译现象从意义对等讨论习语的翻译问题，第五章讨论在习语翻译中应该注意的问题。最后，对本文作总结，得出本文的结论，指出本研究的不足以及今后研究的方向。
Manyscholarsat home and abroad have proposed various translation theories in the process of idioms translating, and Nida’s functionalequivalence translationtheory is much flexible in dealing with the target language and source language in translating because of basing on functionalequivalence. Additionally, the theory also considers thereceptorin the target language, providing a new perspective for Nida’s translation theory. And the thesis will discuss thetranslation of idioms from English to Chinese, Chinese to English based on Nida’s functionalequivalence. The idioms to be discussed in the paper will bedefinedin broad, including set phrases, common sayings, proverbs, allusions, etc. It is hoped that themisunderstandingand mistranslation of idioms can be reduced through comprehensively idioms study,meanwhile, it will help us to learn the excellent cultures in English and Chinese nations to promote intercultural communication in the world.However, due to the different cultural systems between the east and the west, Chinese and English idioms have similarities and obvious differences in meanings and expression forms. So in the real intercultural communication there will surely appear misunderstanding, culture conflict, even hatred among people from different regions and countries, leading the failure of intercultural communication among different nations.

The first-order theory of ordering constraints over feature trees

Discrete Mathematics and Theoretical Computer Science4,2001,193–234The First-Order Theory ofOrdering Constraints over FeatureT reesMartin Müller1and Joachim Niehren1and Ralf Treinen21Programming Systems Lab,Universität des Saarlandes,Saarbrücken,Germany. http://www.ps.uni-sb.de/~mmueller,http://www.ps.uni-sb.de/~niehren 2Laboratoire de Recherche en Informatique,UniversitéParis-Sud,Orsay,France. http://www.lri.fr/~treinenreceived April19,1999,revised February2001,accepted Aug15,2001.1Introduction194 2Ordering Constraints196 3Expressiveness of the First-Order Theory197 4Undecidability Results199 5Entailment with Existential Quantiﬁers203 6Correctness of the Entailment Test217 7Completeness of the Entailment Test2261365–8050c2001Maison de l’Informatique et des Mathématiques Discrètes(MIMD),Paris,France194Martin Müller and Joachim Niehren and Ralf Treinen 1IntroductionFeature constraints have been used for describing records in constraint programming[1, 30,31,36]and record-like structures in computational linguistics[14,12,23,26].Feature constraints also occur naturally in type inference for programming languages with object types or record types[22,5,24].Following[2,4,3],we consider feature constraints as predicate logic formulas interpreted in the structure of feature trees.A feature tree is a tree with unordered edges labeled by features and with possibly labeled nodes.Features are functional in that the features label-ing the edges departing from the same node must be pairwise different.The structure of feature trees gives rise to an ordering in a very natural way which is called weak subsump-tion ordering in[7].Consider the following example where an unlabeled node is indicated as:Here,the left treeτ1is said to weakly subsume the right treeτ2sinceτ1has fewer edges and node labels thanτ2.In other words,every positive assertion about the presence of labels or features that holds forτ1also holds forτ2.In general,a treeτ1weakly subsumes a treeτ2, writtenτ1τ2,ifevery word of features in the tree domain ofτ1belongs to the tree domain ofτ2and the(partial)labeling function ofτ1is contained in the labeling function ofτ2. We consider the system FT of ordering constraints over feature trees[18,19,17].Its constraintsϕare given by the following abstract syntaxϕ::x x x f x a xϕϕwhere f denotes a feature symbol and a a label symbol.The constraints of FT are inter-preted in the structure of feature trees with the weak subsumption ordering.We distinguish two cases,the structure ofﬁnite feature trees and the structure of possibly inﬁnite feature trees.A constraint x x holds if the denotation of x weakly subsumes the denotation of x, x f x is valid if the denotation of x has an edge at the root that is labeled with the feature f and leads to the denotation of x,and a x means that the root of the denotation of x is labeled with a.The constraint system FT is an extension of the well-investigated constraint system FT[2, 4],which provides for equality constraints x y rather than more general ordering con-straints x y.The system FT can be seen as a sub-system of FT since x y can be expressed as x y y x thanks to anti-symmetry of the weak subsumption order.The fullﬁrst-order theory of FT is decidable[4]and has non-elementary complexity[37]. The decidability question for theﬁrst-order theory of FT has been raised in[17].There, two indications in favour of decidability have been formulated:its analogy to FT and its relationship to second-order monadic logic.However,we show in this paper that the theﬁrst-order theory of FT is undecidable.Our result holds in the structure of possibly inﬁnite feature trees and,more surprisingly,even in the structure ofﬁnite feature trees. Our proof is based on an encoding of the Post Correspondence Problem using a technique of[33].Once the undecidability of theﬁrst-order theory of FT is settled,it remains to distin-guish decidable fragments and their complexity.It is well-known that the satisﬁabilityThe First-Order Theory of Ordering Constraints over Feature Trees195FTﬁnn3[18]Entailment w/o quantiﬁers n3[18]Co-NP hard[17]PSPACE complete[here]Full theory undecidable[here]Fig.1:Fragments of theﬁrst-order theories of FT and FTﬁnproblem of FT,its entailment problemϕϕ,and its entailment problem with existential quantiﬁersϕx1x nϕcan be solved in quasi-linear time[31].The investigation of ordering constraints was initiated by Dörre[7]who gave an O n5-algorithm for deciding satisﬁability of FT-constraints.This result was improved to O n3in[18],where also the entailment problem of FT concerning quantiﬁer-free judgmentsϕϕwas shown decidable in cubic time.The next step towards larger fragments of the theory of FT was to consider entailment judgments with existential quantiﬁcationϕx1x nϕwhich are equivalent to unsatisﬁability judgmentsϕx1x nϕwith quantiﬁcation below nega-tion.As shown in[17],this problem is decidable,coNP-hard in case ofﬁnite trees,and PSPACE-hard in case of arbitrary trees.Decidability is proved by reduction to(weak)sec-ond order monadic logic(W)S2S.In aﬁrst reduction step,it is shown how to substitute the structure of feature trees by the related structure of so-called sufﬁciently labeled fea-ture trees.We note that this step cannot be generalized to arbritraryﬁrst-order formulas beyond entailment with existential quantiﬁers.Since the fullﬁrst-order theory of ordering constraints over sufﬁciently labeled(ﬁnite)feature trees can easily be encoded in(weak) second order monadic logic,decidability of entailment of FT with existential quantiﬁers follows from the classical results on(W)S2S[32,25].This paper contributes the exact complexity of the entailment problem of FT with existen-tial quantiﬁcation.We prove PSPACE-completeness,both in the structure ofﬁnite trees and in the structure of possibly inﬁnite trees.This result is obtained by reducing the entailment problem of FT with existential quantiﬁers to the inclusion problem of non-deterministic ﬁnite automata(NFA),and vice versa.Our reduction of entailment is based on the fol-lowing idea:Given an existential formula196Martin Müller and Joachim Niehren and Ralf Treinen feature trees and all feature structures).Following[8],ordering constraints interpreted with respect to the subsumption(resp.weak subsumption)ordering of arbitrary feature algebras are called subsumption(resp.weak subsumption)constraints.Syntactically,subsumption constraints,weak subsumption constraints,and FT constraints coincide but semanticallythey differ.As proved in[8],the satisﬁability problem of subsumption constraints is un-decidable.The satisﬁability problem of weak subsumption constraints is equivalent to the satisﬁability problem of FT constraints[7,18]and hence decidable in cubic time. Structure of the Paper.Section2reviews the deﬁnitions of feature trees and weak sub-sumption constraints.We demonstrate the expressivity of the constraint language in Sec-tion3and introduce some formulas used in later sections.Undecidability of theﬁrst-order theory of weak subsumption constraints is shown in Section4.Finally,we show the entail-ment problem of existentially quantiﬁed constraints to be PSPACE-complete in Section5. We prove the correctness of our algorithm in Section6and its completeness in Section7.A short version of this paper has been published as[21].2Ordering ConstraintsThe constraint system FT is deﬁned by a set of constraints,the structure of feature trees, and an interpretation of constraints over feature trees.We assume an inﬁnite set V of variables ranged over by x y z,a set F of at least two features ranged over by f g and a set L of labels ranged over by a b.2.1Feature T reesA pathπis a word of features.The empty path is denoted byεand the free-monoid concatenation of pathsπandπasππ.We haveεππεπ.A pathπis called a preﬁx ofπifπππfor some pathπ.A tree domain is a non-empty preﬁx closed set of paths.A feature treeτis a pair D L consisting of a tree domain D and a partial function L: D L that we call labeling function ofτ.Given a feature treeτ,we write Dτfor its tree domain and Lτfor its labeling function.For instance,τ0εf f ais a feature tree with domain Dτ0εf and Lτf a.A feature treeτ0a fisﬁnite if its tree domain isﬁnite,and inﬁnite otherwise.A node ofτis anelement of Dτ.A nodeπofτis labeled with a ifπa Lτ.A node ofτis unlabeled if it is not labeled with any a.The root ofτis the nodeε.The root label ofτis Lτε,and f F is a root feature ofτif f Dτ.A feature treeτis fully labeled if Lτis a total function with domain Dτ.Given a treeτwithπDτ,we write asτπthe subtree ofτat pathπ;formally DτππππDτand Lτππaππa Lτ.2.2Syntax and SemanticsAn FT constraintϕis deﬁned by the abstract syntaxϕ::x y a x x f yϕ1ϕ2where a L and f F.In other words,an FT constraint is a conjunction of basic con-straints which are either ordering constraints x y,labeling constraints a x,or selection constraints x f y.We deﬁne the structure FT over feature trees in which we interpret FT constraints.ItsThe First-Order Theory of Ordering Constraints over Feature Trees197 universe consists of the set of all feature trees.The constraints are interpreted as follows:τ1τ2iff Dτ1Dτ2and Lτ1Lτ2τ1fτ2iff Dτ2πfπDτ1and Lτ2πa fπa Lτ1aτiffεa LτThe substructure of FT whose universe contains only theﬁnite trees is denoted by FTﬁn. We will often use the following decomposition property without further mention:Proposition2.1Ifτ1τ2andτ1fτ1andτ2fτ2thenτ1τ2.2.3First-Order FormulasIf not speciﬁed otherwise,a formula is said to be valid(satisﬁable)if it is valid(satisﬁable) both in FT and FTﬁn.Our intention here is to treat both cases simultaneously and to note a distinction when needed only.LetΦandΦbeﬁrst-order formulas built from FT constraints with the usualﬁrst-order connectives and quantiﬁers.We say thatΦentailsΦ, writtenΦΦ,ifΦΦis valid,and thatΦis equivalent toΦifΦΦis valid.We denote with VΦthe set of variables occurring free inΦ,and with FΦand LΦthe set of features and labels occurring inΦ.3Expressiveness of the First-Order TheoryIn this section we introduce some abbreviations of formulas needed in Section4.We use the usual abbreviations for ordering constraints,for instance we write x y for x y,x y for x y x y,x y for y x and x y z for x y y z.3.1Minimal and Maximal ValuesWe can construct,for any formulaϕ,formulasµxϕandνxϕexpressing that x is minimal (maximal)with the propertyϕ:µxϕ:ϕyϕy x y xνxϕ:ϕyϕy x y xHere,y is a fresh variable not occurring inϕ,andϕy x denotes the formula where every free occurrence of x is replaced by y.Typically,x occurs free inϕbut this is not required. Note that,in contrast to x and x,µx andνx are no variable binders that restrict the scope of the variable x;hence x is free inµxϕand inνxϕif it is free inϕ.The formulaµxϕexpresses that x denotes a minimal tree satisfyingϕ,which is not necas-sarily a smallest tree with this property.In analogy,νxϕexpresses that x denotes a maximal but not necessary greatest tree satisfyingϕ.Example1The sentence xµx true is valid in FT and in FTﬁn(there even exists a smallest tree,namelyε.The formulaνx true is not satisﬁable in FTﬁn but is satisﬁed in FT by any fully labeled tree with domain F.The difference between smallest and minimal trees is important for the formula atom x which expresses that x denotes an atom in the lattice-theoretic sense,i.e.that it is a tree strictly greater than the smallest treeεbut with minimal distance(one feature or one label more):one-dist x y:µy x yatom y:xµx true one-dist x y198Martin Müller and Joachim Niehren and Ralf Treinen Example2The formulaµx x0x x1x is satisﬁed in FT by01,that is the full binary and everywhere unlabeled tree,and is not satisﬁable in FTﬁn since FTﬁn contains no inﬁnite trees.3.2Label RestrictionsThe formula x y reads x and y are consistent,that is wheneverπa Lτandπa Lτthen a a:x y:z x z y zFor any label a L we write x a to express that the root of x is either unlabeled or labeled with a:x a:y x y a yThe following formula expresses that the root of a tree is unlabeled:not-root-labeled x:x a x bwhere a and b are two arbitrary different label symbols.We obtain aﬁrst-class status of labels by encoding a label a as the feature treeεεa.label-atom x:atom x not-root-labeled xWe can now express that x and y either have the same root label or are both unlabeled at the root by:same-root-label x y:z label-atom z x z y z3.3Arity RestrictionsWe can simulate aﬁrst-class status of feature symbols by encoding a feature f by the tree εf/0.feature-atom x:atom x not-root-labeled xWe can express that y has at least all the root features of x byz feature-atom z z x z yThe following formula expresses that x has exactly the root features f1f n: x f1f n:x1x n x f1x1x f n x ny y f1x1y f n x n same-root-label x y x yThese so-called arity constraints have been introduced in[31].A decidable feature logic where feature symbols haveﬁrst class status has been investigated in[34].3.4Inductive PropertiesWe start this section by a demonstration of the expressivity of FT and show that we can express in FT“inductive properties”of trees,that is properties that require an inductive construction(for instance an automaton)to deﬁne.We conclude the section by the deﬁni-tion of the predicate string-c x that we will need in the undecidabability proof of Section4. In the case of inﬁnite trees it is in fact quite simple to express“inductive properties”of a tree.For instance,we can express that the domain of x contains the set01byy y0y y1y y xThe First-Order Theory of Ordering Constraints over Feature Trees199 The following formula says that the tree denoted by x has domain01and that exactely one of its nodes is labeled with a whereas all its remaining nodes are unlabeled: a-singleton x:y zµy y0y y1y not-root-labeled yµz z0z z1z a zy x z one-dist y xIf a-singleton x is satisﬁed then y denotes the complete binary,everywhere unlabeled tree (with domain01),and z denotes the complete binary,everywhere a-labeled tree.The formula b-singleton is deﬁned analogously.We can now express that x denotes a tree with domain01and that all its nodes are labeled with either a or b by:µx y z a-singleton y b-singleton z y z y x z xThe idea behind this formula is the following:an a-singleton and a b-singleton are in-consistent iff they have their label at the same position.Hence,the formula says that 01D x and every node of x which is reachable via a01-path is either labeled with a or with b.The minimality of x yields D x01.In case ofﬁnite trees we have to use another trick(which works also in case of inﬁnite trees).The next formula is crucial for our undecidability proof.A treeτsatisﬁes this formula iffεc Dτc and all its nodes are unlabeled:string-c x:x c not-root-labeled x y x c y y xThe correctness of this deﬁnition of string-c x with respect to the above stated semantics follows from the following lemma where we write c n for the word c c consisting of n letters c.Lemma3.1The formula y x c y y x is satisﬁed byτiff c Dτand for all k m0, whenever c m k Dτthenτc m kτc mProof.Letτcτandττ.Obviously,c Dτ.The inequality follows by induction: For any m,if c m Dτthenτc mτc m.Furthermore,for any k with c m k1Dτand τc m kτc m we have thatτc m k1τc c m kτc m kτc m kτc mFor the other direction,since c Dτthere is aτsuch thatτcτ.From the above inequality we get by setting m0and k1thatττc1τc0τετ4Undecidability ResultsTheorem4.1Theﬁrst-order theories of FTﬁn and of FT are undecidable.The result holds for arbitrary(even empty)L and for F of cardinality 2.For the sake of clarity we use in the proof distinct label symbols a b e and pairwise distinct feature sym-bols s c p l r.We prove Theorem4.1by reduction of the Post Correspondence Problem (PCP).The choice of PCP is motivated by the fact that our proof works by simulation of an iterative construction,and that PCP uses a technically very simple iteration.This is200Martin Müller and Joachim Niehren and Ralf Treinen different in nature to the technique in [8]for the proof of undecidability of the satisﬁabil-ity of strong subsumption constraints.There,Thue-systems could be used by exploiting a correspondence between word equations and the algebraic properties of feature structures.See [33]for a discussion of the proof technique employed in this chapter.An instance of PCP is a ﬁnite sequence P p i q i i 1m of pairs of words from a b .Such an instance is solvable if there is a nonempty sequence i 1i n ,1i j m ,such that p i 1p i n q i 1q i n .According to a classical result due to Post,it is undecidable whether an instance of the PCP is solvable.In the following,let Pp i q i i 1m be a ﬁxed instance of PCP.We say that a pair v w is P-constructed from a pair of words v w if,for some j ,v p j v and w q j w .We say that a set X of pairs of words is P-constructed if every pair in X is either εεor is P -constructed from some other pair in X .To encode solvability of P into the theory of FT ﬁn ,resp.FT ,we employ the following equivalent deﬁnition of solvability:Proposition 4.2P is solvable iff there is a P-constructed set X of pairs of words containing a pair w w with w ε.4.1Words and T reesGiven a word wa b over labels a b L ﬁxed above we denote its length by w and for a natural number 1j w we write w j for the j ’th letter of w .There is an obvious one-to-one encoding function γfrom words w a b to feature trees for which we use the feature symbol s and label e that we also ﬁxed above:γw D w L w where D w εs s w ,L w s j w j for 0j w 1,and L w s w e (see Figure 2(a)).We deﬁne a left-inverse function ¯γ,that is ¯γγw w ,from feature trees to (possibly inﬁnite)words in a b ωas follows:If τdoes not have root feature s ,or if its root isunlabeled or has label different from a and from b then ¯γτε.Otherwise let τbe such that τs τ.We deﬁne ¯γτa ¯γτif τhas root label a ,and ¯γτb ¯γτif τhas root label b .To express that y denotes the ﬁxed word πappended with the denotation of x ,we deﬁne for any πa b a formula app πx y ,such that1.if app πττthen π¯γτ¯γτ2.app πγw γπw is validfor all words w and feature trees ττ,by induction on π:app εx y :x yapp a πx y:a y z y s z app πx zapp b πx y :b y z y s z app πx zFurthermore,we deﬁne eps x ,expressing that x denotes a tree τwith ¯γτε,byeps x :y x s y a x b xFinally,the following formula expresses that x denotes a ﬁnite string:ﬁnite x :y y s y y xIn case of FT ﬁn this formula is,of course,equivalent to true .The First-Order Theory of Ordering Constraints over Feature Trees201 asase(a)The string abaa.pl rτl1τr1cpl rτl2τr2c202Martin Müller and Joachim Niehren and Ralf Treinen ccFig.3:A possible value for x such that one-branch x x,where x is as in Figure2(b). Since we know already how to encode words as trees,we now have to deﬁne an appropriate encoding of an arbitrary set of pairs of trees as a feature tree,together with a corresponding formula in.The representation of a sequenceτl iτr i i1n is given in Figure2(b).We deﬁne,for any formulaϕ,a formulaµ!xϕexpressing that x denotes the smallest element satisfyingϕ.This formula is stronger thanµxϕin that it requires the existence of a smallest tree satisfyingϕin addition:µ!xϕ:ϕyϕy x x yIf x denotes a tree as given in Figure2(b),then the formula one-branch x x given below expresses that x denotes a tree as given in Figure3.one-branch x x:x cνx c string-c x c x c xx c x xνx zµ!z x c z xIn this formula,x is smaller than x but is strictly greater than the c-spine x c of x.The tree x can have only one of the p-edges of x since the set of trees between x c and x must have a smallest element.By the maximality of x,the tree x contains x c plus exactly one of the subtrees of x starting with a p-edge(see Figure3).The following formula selectτlτrσ,whereσis as in Figure3,expresses thatτl is the treeτl i andτr is the treeτr i:select y l y r x:xµx x x x x c x x xz x p z z l y l z r y rFrom a treeσas given in Figure3,we get the treeσ(denoted by x)containing at all nodes c j with j i a pairτl jτr j such thatτl iτl j andτr iτr j(by Lemma3.1).By the minimality ofσwe get thatτl iτl j andτr iτr j for all j i,hence in particular for j0(see Figure4).Combination of the two formulas yieldsin y l y r x:x one-branch x x select y l y r xNow,it is easy to verify the conditions announced at the beginning of the proof.pl r τl iτr icpl rτl iτr icFig.4:The value of x in the formula select y l y r x where x is as in Figure3.Note that this proof did not make use of the fact that the feature trees considered here are partial.The proof of Theorem4.1transfers immediately to the structures of completelylabeled trees(both in the case ofﬁnite and of arbitrary trees),where a tree D L is called completely labeled if L is a total function with domain D.In this case,the trees depicted inFigures2(b),3and4have to be completed by giving some label to the nodes.5Entailment with Existential QuantiﬁersIn[17]it is shown that the entailment problem of FT with existential quantiﬁersϕxϕis PSPACE-complete for both structures FT and FTﬁn.In Section5.3we modify the PSPACE-hardness proof given in[17]for the case of inﬁ-nite trees such that it proves PSPACE-hardness for both cases(Theorem5.2).In particu-lar,we show that we can encode the Kleene-star operator without need for inﬁnite trees. Containment in PSPACE is shown(Theorem5.9)by reducing in polynomial time the en-tailment problem to an inclusion problem between the languages accepted by nondeter-ministicﬁnite state automata(NFA).Language equivalence for NFA(and hence inclusion, since A B B A B)is known to be PSPACE-complete if the alphabet contains at least two distinct symbols[9].5.1Path ConstraintsWe characterize existential FT formulasy y z z x yy za z x?f g afgy y z z x y z y a zfgFig.5:Graphical Presentation of Example4The semantics of path constraints is given by extension of the structure FT through the following predicates,which are deﬁned on basis of the subtree selection functionτπin-troduced above.τπiffπDτaτπiffπa Lττ?πa iffπDτimpliesτπaτ?πτ?πiffπDτandπDτimplyτπτπτ?πτ?πiffπDτandπDτimplyτπτπIn the Section5.2,we use path constraints for presenting typical examples of entailment judgements.Path constraints are also helpful for proving PSPACE-hardness in Section5.3. In Section5.5we will construct aﬁnite automaton that accepts all path constraintsψen-tailed byxy xxyzb xc zf g fg x ?f gaFig.6:Graphical Presentation of Example 5uxvyuvyxyf ffFig.7:Graphical Representation of Example 6holds.In other words,if αis a solution of the constraint displayed on the left hand side and if f g D αx then the subtree of αx at f g is compatible with any label a,and hence is unlabeled.Example 6(see Figure 7)The following situation illustrates a non-trivial example for entailment of selection constraints without existential quantiﬁers.yuu f uuxxv v f vvyx f yThe right-hand side x f y is equivalent to the conjunction y ?εx ?f x fx ?f y ?εof path constraints which are entailed by the ﬁrst and second part of the left-hand side,respectively.5.3Entailment is PSP ACE-hardIn this section we show how the PSPACE completeness proof of [17]can be modiﬁed such that it applies to the structure of ﬁnite feature trees as well.The formulas used in the earlier proof require the existence x πof all paths πin some regular language R ;every solution of the formula for an inﬁnite language R has to map x to an inﬁnite pared to this earlier proof,the trick is here to use conditional path constraints which may constrain inﬁnitely many paths without requiring their existence.Theorem 5.2The entailment problem for existentially quantiﬁed FT -constraints is PSPACE-hard in both the ﬁnite and the inﬁnite tree case.This follows from Proposition 5.6(see below),which claims a polynomial reduction of the inclusion problem between regular languages over the alphabet F to an entailment problem between two existential FT formulas.Notice that we have assumed F to contain at least two features.Our PSPACE-hardness proof is based on the fact that a satisﬁable ordering constraint ϕmay entail an inﬁnite conjunction of path constraints,even in case of ﬁnite trees:Example 71.for all n :x f y y x a xx ?f na.2.for all n m :x f y y xx ?f mnx ?f n .3.for all πf g:x xx f xx g xx ?πx ?ε.For this reason the entailment problem for FT ﬁndoes not necessarily reduce to an inclusion problem between ﬁnite regular languages (which is decidable in coNP [9]).We ﬁx a ﬁnite subset F F of features and consider regular expressions of the following form:R ::εfRR 1R 2R 1R 2where fFFor encoding a regular expression R the main idea is to deﬁne an existential formula Θx R y for fresh variables x y such that Θx R y is equivalent to πL R x ?πy ?ε.Once this is done,it will follow immediately that L RL R iff Θx R y Θx R y .It is not obvious,however,how to deﬁne such a formula.The reader might notice,that a naive deﬁnition of Θx R y yields some unintended compatibility relations to be entailed too.Hence,we have to reﬁne our main idea.We deﬁne the formula com F x expressing that all subtrees of x reachable via an F -path are compatible with each other,i.e.they have a common upper bound:com F x :y x yf Fy y f yyyLemma 5.3(Comon upper bound)com F xy πF x ?πy ?ε.For encoding a regular expression R ,a reﬁned idea is to deﬁne an existential formula Θx R y such that Θx R y is equivalent to com F x πL R x ?πy ?ε.We de-ﬁne for all regular expressions R over F and variables x and y ,the existential formulas Θx R y and Θx R y recursively as follows.Θx R y com F xΘx R yΘx εy x y Θx f yz x z z f yΘx R 1R 2y Θx R 1y Θx R 2y Θx R 1R 2y z Θx R 1z Θz R 2y Θx R yz x zΘz R z z yApparently,Θx R y has size linear in the size of R .Lemma 5.4For all regular expressions Rcom F xΘx R yπL Rx ?πy ?εProof.We proceed by induction on R .ε:Θx εyx yx ?εy ?επL εx ?πy ?ε.f :Θx f yz x z z f y x ?f y ?επL fx ?πy ?ε.R 1R 2:By induction hypothesis com F x entails the equivalences Θx R 1yπL R 1x ?πy ?εand Θx R 2y πL R 2x ?πy ?ε.Hence,com F x en-tails Θx R 1R 2y πL R 1R 2x ?πy ?εalso.R 1R 2:By deﬁnition Θx R 1R 2y z Θx R 1z com F z Θz R 2y .By in-duction hypothesis,com F x entails Θx R 1z π1L R 1x ?π1z ?εandcom F z entails Θz R 2yπ2L R 2z ?π2y ?ε.Hence,com F x entails thatΘx R 1R 2y is equivalent to (1):zπ1L R 1x ?π1z ?εcom F zπ2L R 2z ?π2y ?ε(1)It remains to show that com F x entails the equivalence between (1)and (2):πL R 1R 2x ?πy ?ε(2)Since (1)obviously entails (2),it is sufﬁcient to prove the validity of com F x 21.Let αbe an FT -valuation which satisﬁes both com F x and (2).We deﬁne a tree τsuch that αz τsatisﬁes the matrix of (1).For this deﬁnition we use a least upper bound operator on feature trees denoted by :τπ1L R 1D αxαx π1Since αsolves com F x there exists an upper bound of αx ππF as stated by Lemma 5.3and thus the least upper bound τexists.We next demonstrate that αz τsatisﬁes the matrix of (1).The deﬁnition of τyields αx π1τfor all π1L R 1D αx ,i.e.the variable assignment αz τsatisﬁes the ﬁrst conjunc-tion in (1).From com F αx it follows that com F τholds,i.e.αz τsatis-ﬁes com F z .Furthermore,all π2D τsatisfy:τπ2π1L R 1D αx αx π1π2.Since αis a solution of (2),αx π1π2αy is satisﬁed by all π2L R 2.Thus τπ2αy is valid for all π2L R 2,i.e.αz τsatisﬁes π2L R 2z ?π2y ?ε,the remaining conjunct in (1).R :By deﬁnition Θx R y z com F z x z Θz R z z y .The inductionassumption yields that com F z entails Θz R z πL R z ?πz ?ε.Hence,com F x entails that Θx R y is equivalent to (3):z com F zx zπL Rz ?πz ?εz y(3)It remains to show that com F z entails the equivalence between (3)and (4):πL Rx ?πy ?ε(4)In order to show the non-trivial implication,we assume an FT -valuation αwhich satisﬁes both com F x and (4).We deﬁne a tree τsuch that αz τsatisﬁes the matrix of (3)as follows:τπL RD αxαx πNote that τis well-deﬁned for the same reason as in the preceeding case.Our assump-tions on the choice of αyields:com F τ,αx τ(since εL R )and ταy .In order to show that αz τis a solution of (3)it remains to prove τπτfor all πL R D τ:τππL RD αxαx πππL RD αxαx πτ。

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The Power of Effective English LearningIn the era of globalization, the importance ofeffective English learning has become increasingly apparent. English, as the lingua franca of the world, is the key to accessing a wide range of opportunities and resources. However, mastering English is not a feat that can be achieved overnight. It requires dedication, perseverance, and a strategic approach.To achieve effective English learning, it is crucial to identify one's learning style and preferences. Different individuals learn in different ways, and understandingone's own learning strengths and weaknesses can greatly enhance the learning process. For example, some people find it easier to learn through auditory stimulation, while others prefer visual aids. Identifying one's learning style and catering to it can make the learning experience more enjoyable and less stressful.Another key to effective English learning is regular practice. Consistency is essential in language learning, as it helps to ingrain new vocabulary, grammar rules, and pronunciation patterns. Regular practice can be achievedthrough various activities such as reading, writing, speaking, and listening. Engaging in these activities daily, even if it's just for a few minutes, can significantly improve one's English proficiency.Moreover, exposure to native speakers and authentic English materials is crucial. Immersing oneself in an English-speaking environment, whether it's through travel, language exchange programs, or online communities, can help improve language skills significantly. Additionally,reading books, newspapers, and other materials written in English can provide valuable insights into the language's nuances and idioms.Technology has also revolutionized English learning, providing access to a wealth of resources and tools. Online learning platforms, language exchange apps, and interactive games can make learning more engaging and effective. These tools allow learners to customize their learning experience, track their progress, and receive feedback in real-time.However, effective English learning goes beyond mere proficiency. It involves cultivating a love for the language and its culture. Understanding the history,literature, and social context of English can deepen one's understanding and appreciation of the language. This, in turn, can enhance one's motivation to learn and improve.In conclusion, effective English learning is a journey that requires dedication, strategy, and regular practice.By understanding one's learning style, engaging in regular practice, immersing oneself in an English-speaking environment, and leveraging technology, learners canachieve significant progress in their English proficiency. Furthermore, cultivating a love for the language and its culture can make the learning experience more rewarding and fulfilling.**有效英语学习的力量**在全球化的时代，有效英语学习的重要性日益凸显。

仲裁裁决既判力的英文表达

仲裁裁决既判力的英文表达Conclusive Effect of Arbitral Awards.Arbitration is a form of alternative dispute resolution (ADR) in which the parties to a dispute agree to submittheir dispute to a neutral third party, known as an arbitrator, for a binding decision. Arbitral awards are generally considered to be final and binding on the parties, and they have the same effect as a judgment of a court.The doctrine of res judicata, also known as claim preclusion, bars the relitigation of a claim that has already been finally adjudicated by a court or other competent tribunal. The doctrine applies to arbitral awards as well as to court judgments. This means that once an arbitral award has been issued, the parties cannotrelitigate the same claim in a subsequent proceeding.The conclusive effect of arbitral awards is based on several factors, including:The parties' agreement to arbitrate: The parties' agreement to arbitrate is a binding contract, and it creates a duty on the parties to submit their dispute to arbitration and to abide by the arbitrator's decision.The arbitrator's impartiality and expertise: Arbitrators are generally neutral third parties who are experts in the subject matter of the dispute. This ensures that the arbitrator's decision is fair and impartial.The finality of the arbitral award: Arbitral awards are generally considered to be final and binding on the parties. This means that the parties cannot appeal the arbitrator's decision to a court, except in very limited circumstances.The conclusive effect of arbitral awards is essential to the effective functioning of the arbitration process. It ensures that arbitral awards are respected and enforced, and it prevents parties from relitigating the same claim multiple times.Exceptions to the Conclusive Effect of Arbitral Awards.There are a few exceptions to the conclusive effect of arbitral awards. These exceptions include:Fraud or corruption: If the arbitral award was obtained through fraud or corruption, it may be set aside by a court.Manifest disregard of the law: If the arbitrator manifestly disregarded the law in reaching his or her decision, the award may be set aside by a court.Public policy: If the arbitral award violates public policy, it may be set aside by a court.These exceptions are narrow, and they are only applied in rare cases. In general, arbitral awards are considered to be final and binding on the parties.Enforcement of Arbitral Awards.Arbitral awards can be enforced in the same manner as court judgments. The party who wins the arbitration canfile a petition with the court to confirm the award. Once the award is confirmed, it becomes a judgment of the court and can be enforced through the court's usual enforcement procedures.Conclusion.Arbitral awards are generally considered to be finaland binding on the parties. This means that the parties cannot relitigate the same claim in a subsequent proceeding. The conclusive effect of arbitral awards is essential tothe effective functioning of the arbitration process. It ensures that arbitral awards are respected and enforced,and it prevents parties from relitigating the same claim multiple times.。

MildorfOlympiad inequalities

ω1f (x1) + · · · + ωnf (xn) ≥ (ω1 + · · · + ωn) f
ω1x1 + · · · + ωnxn ω1 + · · · + ωn
If f is concave, then the inequality is ﬂipped.
Theorem 2 (Weighted Power Mean) If x1, . . . , xn are nonnegative reals and ω1, . . . , ωn are nonnegative reals with a postive sum, then
4/, where a seemingly inexhaustible supply of Olympiads is available.
1
1 The Standard Dozen
Throughout this lecture, we refer to convex and concave functions. Write I and I for the intervals [a, b] and (a, b) respectively. A function f is said to be convex on I if and only if λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y) for all x, y ∈ I and 0 ≤ λ ≤ 1. Conversely, if the inequality always holds in the opposite direction, the function is said to be concave on the interval. A function f that is continuous on I and twice diﬀerentiable on I is convex on I if and only if f (x) ≥ 0 for all x ∈ I (Concave if the inequality is ﬂipped.)

KYFAN’SINEQUALITYVIACONVEXITY（2007）

KY FAN’S INEQUALITY VIA CONVEXITYJAMAL ROOIND EPARTMENT OF M ATHEMATICSI NSTITUTE FOR A DVANCED S TUDIES IN B ASIC S CIENCESZ ANJAN,I RAN**************.irReceived20October,2007;accepted11December,2007Communicated by P.S.BullenA BSTRACT.In this note,using the strict convexity and concavity of the function f(x)=11+e x on[0,∞)and(−∞,0]respectively,we prove Ky Fan’s inequality by separating the left and right hands of it by1G n+Gn.Key words and phrases:Convexity,Ky Fan’s Inequality.2000Mathematics Subject Classiﬁcation.26D15.Let x1,...,x n in(0,1/2]andλ1,λ2,...,λn>0with ni=1λi=1.We denote by A n andG n,the arithmetic and geometric means of x1,...,x n respectively,i.e.(1)A n=ni=1λi x i,G n=ni=1xλii,and also by A n and G n,the arithmetic and geometric means of1−x1,...,1−x n respectively, i.e.(2)A n=ni=1λi(1−x i),Gn=ni=1(1−x i)λi.In1961the following remarkable inequality,due to Ky Fan,was published for theﬁrst time in the well-known book Inequalities by Beckenbach and Bellman[2,p.5]:If x i∈(0,1/2],then(3)AnGn≤A nG n,with equality holding if and only if x1=···=x n.Inequality(1)has evoked the interest of several mathematicians and in numerous articles new proofs,extensions,reﬁnements and various related results have been published;see the survey paper[1].Also,for some recent results,see[6]–[10].367-072J AMAL R OOINIn this note,using the strict convexity and concavity of the function f (x )=11+e xon [0,∞)and (−∞,0]respectively,we prove Ky Fan’s inequality (3)by separating the left and right hand sides of (3)by 1G n +G n:(4)A n G n ≤1G n +G n ≤A n G n.Moreover,we show equality holds in each inequality in (4),if and only x 1=···=x n .It is noted that,since for a,b,c,d >0the inequality a b ≤c d implies a b ≤a +c b +d ≤c d ,considering A n +A n =1,the inequalities (3)and (4)are equivalent.Indeed,since f (x )=ex (e x −1)(1+e x )3,the function f has the foregoing convexity properties.Now,using Jensen’s inequality f n i =1λi y i ≤n i =1λi f (y i ),for y i =ln 1−x i x i ≥0(1≤i ≤n ),we get the right hand of (4)with equality holding if and only if ln 1−x 1x 1=···=ln 1−x n x n ,or equivalently x 1=···=x n .The left hand of (4)is handled by using Jensen’s inequality for the convex function −f on (−∞,0]with y i =ln x i 1−x i ≤0(1≤i ≤n ).It might be noted that it sufﬁces to prove either of the two inequalities in (4)as a b ≤c d is equivalent to both a b ≤a +c b +d and a +c b +d ≤c d .It was pointed out by a referee that the use of the function f ,or rather its inverse g (x )=ln (1−x )/x ,to prove Ky Fan’s inequality can be found in the literature;see [4],[3,pp.31,154],[5].R EFERENCES[1]H.ALZER,The inequality of Ky Fan and related results,Acta Appl.Math.,38(1995),305–354.[2] E.F.BECKENBACH AND R.BELLMAN,Inequalities ,Springer-Verlag,Berlin,1961.[3]P.BILER AND A.WITKOWSKI,Problems in Mathematical Analysis ,Marcel Dekker,Inc.,1990.[4]K.K.CHONG,On Ky Fan’s inequality and some related inequalities between means,SoutheastAsian Bull.Math.,29(1998),363–372.[5] A.McD.MERCER,A short proof of Ky Fan’s arithmetic-geometric inequality,J.Math.Anal.Appl.,204(1996),940–942.[6]J.ROOIN,An approach to Ky Fan type inequalities from binomial expansion,(accepted).[7]J.ROOIN,Ky Fan’s inequality with binomial expansion,Elemente Der Mathematik ,60(2005),171–173.[8]J.ROOIN,On Ky Fan’s inequality and its additive analogues,Math.Inequal.&Applics.,6(2003),595–604.[9]J.ROOIN,Some new proofs of Ky Fan’s inequality,International Journal of Applied Mathematics20(2007),285–291.[10]J.ROOIN AND A.R.MEDGHALCHI,New proofs for Ky Fan’s inequality and two of its variants,International Journal of Applied Mathematics,10(2002),51–57.J.Inequal.Pure and Appl.Math.,9(1)(2008),Art.23,2pp..au/。

RICHANDPOORVSLIFEBOATETHICS：富人和穷人与救生艇伦理

RICH AND POOR VS LIFEBOAT ETHICSIntroductionThe gap between the rich and the poor in the world has continued to increase with no solution at sight. In rich and poor nations, the difference between the rich and the poor is still widening despite various efforts aimed at reducing the gap. The wealth nations continue to be rich while the poor nations continue to be poor. This has been brought by the inequitable distribution of resources in the world and lack of world policies that would lead to equitable distribution of resources. This has been due to lack of a justifiable distribution network that would give the rich and the poor equal rights in access of resources. In this is the rule of justice that is likely to determine how we achieve equity in distribution of resources which further accounts for the richness and poverty. Do we have the right to be rich or poor?Peter Singer provided an important insight about the right for one to be rich or poor. Singer explores different factors and theory, which relate to the state of poverty and richness. He argues on how the rich considers it their right to be rich while the poor looks upon the rich to assist them from poverty. In the context of social justice, we fail to achieve this equity based on its formulation and the way it addresses the needs of the poor and the rich.Inequality becomes a justifiable state of life in the society based on the resource distribution network. Singer looks at the Rawls principle of maximum rule which seeks to maximize the minimum level of welfare that exists in the society. This is a great contrast to the principle that guarantees maximum liberty for all. Since the concern is oneconomic redistribution, the Singer looks at Rawls maximum principle in this context of resources in the society. (Singer, 1975)The principle is equal to major inequality in the society. Taking an example of taxation in the society, those who are highly taxed would be justifiable to keep more as compared to those who work less. This means that there are those in the society who would be allowed to keep more even when others have less. This makes the worse-off to be worse than before and the rich to be richer than before.But what would happen when we come to the access of basic needs like food and medical care in the society? In light of maximum principle, the poor will be obliged to pay for the medical services that they receive. In this case they will be paying for the medical services at the same price with those who are rich. The bedrock of inequality in the society is the weakness in the redistribution law.Garret Hardin provides us with another outlook at the issue of the poor and the rich in the society. He takes the example of lifeboat ethics to justify the case against helping the poor. According to Garrett, environmentalists persuade countries, industries and people in general to take measures to protect the environment. They take the earth as a spaceship which carries us all. It is like a boat that ferries people over the ocean. In this case it is assumed that the earth belongs to us all. But does every one of us have an equal share of resources in the world? (Hardin, 1974)According to Garret, the rich nations are taken as a spaceship in which there are other poor people pleading to be admitted in the boat. Garret looks at the capacity of the wealth nations and those who are seeking to be admitted to share the wealth of the nation. If the rich nation is taken as a lifeboat that has a capacity of 50 people with an excesscapacity of 10 people, then it will have reached its maximum capacity by the time it admits the 10 people. If there are 50 others who are swimming in the water outside and begging for admission or for handouts, we will have to make a decision on the ones who we are going to admit. We may decide to admit only 10 people and leave the rest drowning or we decide to admit all people, and the boat will drown. This will be equal to a great justice but on the other hand a great tragedy since we will lose all the 150 people. This is just a moral issue regarding the poor and the rich. Does the rich nations have the right to admit immigrants from the poor nations?This brings us to our original argument of the rights of the poor and the rich. Does the rich nations have the right o keep away immigrants from the poor nation? Does the poor nations have the right to be admitted to the rich nation? This remains to be a moral issue.The utilitarian viewThere are several multifaceted arguments that have been used in light of the arguments on the state of being poor or rich. According to the utilitarian theory the price of justice constitute of rules that works for the common good of all people. Based on the principles of utility, it postulate for the need to have a distribution system that will meet the need of all people in the society. For example, if we take from the rich and give to the poor, we will be doing so not because the poor are entitled to some of what has been acquired by the rich, but because the rich will benefit more from the redistribution than the rich will suffer if is taken from them.If taken literally this may be seen as a way of confiscating from the rich and giving to the poor which again may be against moral justice. According to the utilitarian view,the principle of justice redistribution accounts for the need to have a system of social equity in which the rich will be helping the poor to meet their need. This is the view of a society which shows a sense of a man build man society as opposed to a man eat man society. This is a tendency top regard men as a means to the other man welfare. Against the Rawls principle of maximum, utilitarianism requires us to improve the living conditions of those who are poor in the society since poverty is not their own fault. If we are all given the choice of being poor or rich, all of us would choose to be rich. No one would opt to be poor. Poverty is therefore caused by faulty resource redistribution system. No one has the right to be rich or poor. The faulty world trade system has been held responsible for increasing gap between the rich and poor nations in the world. While the rich nations strive to remain rich, the poor nations are doing their best to move out of poverty.Arguing against Garret case against the poor, we would say that the world belongs to all of us. If we live on the same resources, we have the duty to protect the resources. We will not achieve equal distribution of resources only when we admit those outside the boat to the boat but we can hand out life savers to them. In th utilitarian view, the decision that we make should be able to give justice to all people in access of resources. ConclusionGiven option to be rich or poor, we would all choose to be rich. No one can choose to be poor. All of us are entitled to share of the wealth in the world and state of inequality is brought about by unjustifiable resource redistribution system. In utilitarianism view, the social justice system should help us to achieve equity in the distribution of resource.Resource distribution system must be aimed at achieving equity in the distribution system and should be good for all people in the society.ReferencesHardin, G. (1974). Lifeboat ethics: the case against helping the poor. Psychology Today, September 1974.Singer, P. (1975). The right to be rich or poor. New York Review of Books, Vol. 23(2)。

极限思想外文翻译pdf

极限思想外文翻译pdfBSHM Bulletin, 2014Did Weierstrass’s differential calculus have a limit-avoiding character? His,,,definition of a limit in styleMICHIYO NAKANENihon University Research Institute of Science & Technology, Japan In the 1820s, Cauchy founded his calculus on his original limit concept and,,,developed his the-ory by using inequalities, but he did not apply theseinequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distin-guished bytheir limit-avoiding character. Dugac’s partial publication of the 1861 lecturesmakes these differences clear. But in the unpublished portions of the lectures,,,,Weierstrass actu-ally defined his limit in terms ofinequalities. Weierstrass’slimit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not providethe basic structure for the modern e d style analysis. Thus it was Dini’s 1878 text-book that introduced the,,,definition of a limit in terms of inequalities.IntroductionAugustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass’s 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen? 2003, 185–186).Weierstrass’s adoption of full epsilonic arguments, however, didnot mean that he attained a prototype of the modern theory. Modern analysis theory is founded on limits defined in terms of e d inequalities. His lectures were not founded on Cauchy’s limit or his own original definition of limit (Dugac 1973). Therefore, in order to clarify the formation of the modern theory, it will be necessary toidentify where the e d definition of limit was introduced and used as a foundation.We do not find the word ‘limit’ in the pu blished part of the 1861 lectures.Accord-ingly, Grattan-Guinness (1986, 228) characterizesWeierstrass’s analysis aslimit-avoid-ing. However, Weierstrass actually defined his limit in terms of epsilonics in the unpublished portion of his lectures. Histheory involved his limit concept, although the concept did not function as the foundation of his theory. Based on this discovery, this paper re-examines the formation of e d calculus theory, noting mathematicians’ treat-ments of their limits. We restrict ourattention to the process of defining continuity and derivatives. Nonetheless, this focus provides sufficient information for our purposes.First, we confirm that epsilonics arguments cannot representCauchy’s limit,though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based2013 British Society for the History of Mathematics52 BSHM Bulletinon Cauchy’s limits but could have involved Cauchy’s resu lts. Thenwe confirmWeierstrass’s definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed inthe e d style.Cauchy’s limit and epsilonic argumentsCauchy’s series of textbooks on calculus, Cours d’analyse (1821), Resume deslecons? donnees a l’Ecole royale polytechnique sur le calcul infinitesimal tomepremier (1823), and Lecons? sur le calcul differentiel (1829), are often considered as the main referen-ces for modern analysis theory, the rigour of which is rooted more in the nineteenth than the twentieth century.At the beginning of his Cours d’analyse, Cauchy defined the limit concept as fol-lows: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all theothers’ (1821, 19; English translation fromGrabiner 1981, 80). Starting from this concept, Cauchy developed a theory of continuous func-tions, infinite series, derivatives, and integrals, constructing an analysis based on lim-its (Grabiner 1981, 77).When discussing the evolution of the limit concept, Grabiner writes:‘This con-cept, translated into the algebra of inequalities, was exactly what Ca uchy needed for his calculus’ (1981, 80). From the present-day point of view, Cauchy described rather than defined his kinetic concept of limits. According to his ‘definition’—which has the quality of a translation or description—he could develop any aspectof the theory by reducing it to the algebra of inequalities.Next, Cauchy introduced infinitely small quantities into his theory. ‘When the suc-cessive absolute values of a variable decrease indefinitely, in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. Such a variable has zero for its limit’ (1821, 19; English translationfrom Birkhoff and Merzbach 1973, 2). That is to say, in Cauchy’s framework ‘thelimit of variable x is c’ is intuitively understood as ‘x indefinitely approaches c’,and is represented as ‘jx cj is as little as desired’ or ‘jx cj is infinitesimal’.Cauchy’s idea of defining infinitesimals as variables of a special kind was original, because Leibniz and Euler, for example, had treated them as constants (Boyer 1989, 575; Lutzen? 2003, 164).In Cours d’analyse Cauchy at first gave a verbal definition of a continuous func-tion. Then, he rewrote it in terms of infinitesimals:[In other words,] the function f ðxÞ will rema in continuous relative to x in a given interval if (in this interval) an infinitesimalincrement in the variable always pro-duces an infinitesimal increment in the function itself. (1821, 43; English transla-tion from Birkhoff and Merzbach 1973, 2).He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 19–20) and Lecons? (1829, 278).Following Cauchy’s definition of infinitesimals, a continuous function can be defined as a function f ðxÞ in which ‘the variable f ðx þ aÞ f ðxÞ is an infinitelysmall quantity (as previously defined) whenever the variable a is, that is, that f ðx þ aÞ f ðxÞ approaches to zero as a does’, as notedby Edwards (1979, 311). Thus,the definition can be translated into the language of e dinequalities from a modern viewpoint. Cauchy’s infinitesimals are variables, and we can also takesuch an interpretation.Volume 29 (2014) 53Cauchy himself translated his limit concept in terms of e d inequalities. He changed ‘If the difference f ðx þ 1Þ f ðxÞ converges towards a certain limit k, for increasing values of x, (. . .)’ to‘First suppose that the quantity k has a finitevalue, and denote by e a number as small as we wish. . . . we cangive the number h a value large enough that, when x is equal to orgreater than h, the difference in question is always contained between the limits k e; k þ e’ (1821, 54; Englishtranslation from Bradley and Sandifer 2009, 35).In Resume , Cauchy gave a definition of a derivative: ‘if f ðxÞ is continuous, thenits derivative is the limit of the difference quotient,,yf(x，i),f(x), ,xias i tends to 0’ (1823, 22–23). He also translated the concept of derivative asfollows: ‘Designate by d and e two very small numbers; the first being chosen in such a way that, for numerical values of i less than d, [. . .], the ratio f ðx þ iÞ fðxÞ=i always remains greater than f ’ðx Þ e and less than f ’ðxÞ þ e’ (1823,44–45; English transla-tion from Grabiner 1981, 115).These examples show that Cauchy noted that relationships involving limits or infinitesimals could be rewritten in term of inequalities. Cauchy’s argumentsabout infinite series in Cours d’analyse, which dealt with the relationship betweenincreasing numbers and infinitesimals, had such a character. Laugwitz (1987, 264; 1999, 58) and Lutzen? (2003, 167) have noted Cauchy’s strict use of the e Ncharacterization of convergence in several of his proofs. Borovick and Katz (2012) indicate that there is room to question whether or not our representation using e d inequalities conveys messages different from Cauchy’s original intention. Butthis paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen?.Cauchy’s lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesi-mals, or between increasing numbers and infinitesimals. His contemporaries presum-ably recognized the possibility of developing analysis theory in terms of only e, d, and n0 inequalities. With a few notable exceptions, all of Cauchy’s lectures could be rewrit-ten in terms of e d inequalities. Cauchy’s limits and hisinfinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities.Cauchy’s limit concept was the foundation of his theory. Thus, Weierstrass’s fullepsilonic analysis theory has a different foundation from that of Cauchy.Weierstrass’s 1861 lecturesWeierstrass’s consistent use of e d argumentsWeierstrass delivered his lectures ‘On the differential calculus’ at the GewerbeInsti-tut Berlin2 in the summer semester of 1861. Notes of these lectures were taken by1Edwards (1979, 310), Laugwitz (1987, 260–261, 271–272), andFisher (1978, 16–318) point out tha t Cauchy’s infinitesimals equate to a dependent variablefunction or aðhÞ that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he intro-duced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchy’s is a vari-able in the parts that the present paperdiscusses.2A forerunner of the Technische Universit?at Berlin.54 BSHM BulletinHerman Amandus Schwarz, and some of them have been published in the original German by Dugac (1973). Noting the new aspects related to foundational concepts in analysis, full e d definitions of limit and continuous function, a new definition of derivative, and a modern definition of infinitesimals, Dugac considered that the nov-elty of Weierstrass’s lectures was incontestable (1978, 372, 1976, 6–7).3 After beginning his lectures by defining a variable magnitude, Weierstrass gave the definition of a function using the notion of correspondence. This brought him to the following important definition, which did not directly appear in Cauchy’s theory:(D1) If it is now possible to determine for h a bound d such thatfor all values of h which in their absolute value are smaller than d, f ðx þ hÞ f ðxÞ becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. (Dugac 1973, 119; English translation from Calinger 1995, 607)That is, Weierstrass defined not infinitely small changes of variables but ‘infinitelysmall changes of the arguments correspond(ing) to infinitely small changes of function’ that were presented in terms of e d inequalities. He founded his theory on this correspondence.Using this concept, he defined a continuous function as follows: (D2) If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument. (Dugac 1973, 119–120; English translation from Calinger 1995, 607)So we see that in accordance with his definition of correspondence, Weierstrass actually defined a continuous function on an interval in terms of epsiloni cs. Since (D2) is derived by merely changing Cauchy’s term ‘produce’ to, it seems that Weierstrass took the idea of this definition from‘correspond’Cauchy. However, Weierstrass’s definition was given in terms of epsilonics, whileCauchy’s definition c an only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchy’s limit.Luzten? (2003, 186) indicates that Weierstrass still used the concept of ‘infinitelysmall’ in his lectures. Until giving his definition of derivative, Weierstrass actuallya function continued to use the term ‘infinitesimally small’ and often wrote of ‘which becomes infinitely small with h’. But several instances of‘infinitesimallysmall’ appeared in forms of the relationships involving them. Definition (D1) gives the rela-tionship in terms of e d inequalities. We may therefore assume that Weierstrass’s lectures consistently used e d inequalities, even though his definitions were not directly written in terms of these inequalities.Weierstrass inserted sentences confirming that the relationships involving the term ‘infinitely small’ were defined in terms of e d inequalities as follows:ðhÞ is an (D3) If h denotes a magnitude which can assume infinitely small values, ’arbitrary function of h with the property that for an infinitely small value of h it3The present paper also quotes Kurt Bing’s translation included in Calinger’sClassics of mathematics.Volume 29 (2014) 55also becomes infinitely small (that is, that always, as soon as a definite arbitrary small magnitude e is chosen, a magnitude d can be determined such that for all values of h whose absolute value is smaller than d, ’ðhÞ becomes smaller than e).(Dugac 1973, 120; English translation from Calinger, 1995, 607)As Dugac (1973, 65) in dicates, some modern textbooks describe ’ðhÞ as infinitelysmall or infinitesimal.Weierstrass argued that the whole change of function can in general be decom-posed asDf ðxÞ ? f ðx þ hÞ f ðxÞ ? p:h þ hðhÞ; ð 1Þwhere the factor p is independent of h and ðh Þ is a magnitude that becomes infinitely small with h.4 However, he overlooked that such decomposition is not possible for all functions and inserted the term‘in general’. He rewrote h as dx.One can make the difference between Df ðxÞ and p:dx s maller than any magnitude with decreasing dx. Hence Weierstrass defined ‘differential’ as the changewhich a function undergoes when its argument changes by an infinitesimally small magnitude and denoted it as df ðxÞ. Then, df ðxÞ ? p:dx. Weierstrass pointed outthat the differential coefficient p is a function of x derived from f ðxÞ and called it a derivative (Dugac 1973, 120–121; English translation from Calinger 1995, 607–608). In accordance with Weierstrass’s definitions (D1) and (D3),he largelydefined a derivative in terms of epsilonics.Weierstrass did not adopt the term ‘infinitely small’ but directly used e dinequalities when he discussed properties of infinite seriesinvolving uniform conver-gence (Dugac 1973, 122–124). It may beinferred from the publishedportion of his notes that Cauchy’s limit has no place in Weierstrass’s lectures.Grattan-Guinness’s (1986, 228) description of the limit-avoiding character of his analysis represents this situation well.However, Weierstrass thought that his theory included most of the content of Cauchy’s theory. Cauchy first gave the definition of limits of variables andinfinitesi-mals. Then, he demonstrated notions and theorems that were written in terms of the relationships involving infinitesimals. From Weierstrass’s viewpoint,they were writ-ten in terms of e d inequalities. Analytical theory mainly examines properties of functions and series, which were described in the relationships involving Cauchy’s limits and infinitesimals. Weierstrass recognized this fact and had the idea of consis-tently developing his theory in terms of inequalities. Hence Weierstrass atfirst defined the relationships among infinitesimals in terms of e d inequalities. In accor-dance with this definition, Weierstrass rewrote Cauchy’sresults and naturally imported them into his own theory. This is a process that may be described as fol-lows: ‘Weierstrass completed the transformation away fromthe use of terms such as “infinitely small”’ (Katz 1998, 728).Weierstrass’s definition of limitDugac (1978, 370–372; 1976, 6–7) read (D1) as the first definition of limit withthe help of e d. But (D1) does not involve an endpoint thatvariables or functions4Dugac (1973, 65) indicated that ðhÞ corresponds to the modernnotion of oð1Þ. In addition, hðhÞ corre-sponds to the function that was introduced as ’ðhÞ in theformer quotation from Weierstrass’s sentences.。

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a r X i v :0710.3678v 1 [m a t h .C A ] 19 O c t 2007
An Extension of Alzer’s Inequality
by Convexity
J.Rooin
Department of Mathematics
Institute for Advanced Studies in Basic Sciences
P.O.Box 45195-1159
Zanjan,Iran rooin@iasbs.ac.ir
Abstract
In this article,we obtain two interesting general inequalities concerning Riemman sums of convex functions,which in particular,sharpen Alzer’s in-equality and give a suitable converse for it.
1Introduction
In [1],H.Alzer proved the following inequality
n
where r is a positive real and n is a natural number.In other words,the Riemman
sums1
n r(n=1,2,···)of the function x r is a decreasing sequence.The proof of Alzer[1]is technical,but quite complicated.So,in several articles Alzer’s
proof has been simpliﬁed,and also in many others,this inequality has been extended;
see e.g.[2-4].
In this article,using some trivial facts about convex functions,we obtain some valu-
able results concerning special kinds of Riemman sums of convex functions,from
which Alzer’s inequality with an its converse are handled at once.
2Main Results
Throughout this section,we suppose that f:[a,b]→R is an arbitrary function on a closed interval[a,b],and put
A n=b−a
n
n−1
i=0f x(n)i ,(2)
where
x(n)i=a+i
b−a
n(n+2)[A n+1−(b−a)f(a))]≤A n≤A n+1+
1
n(n+2)[B n+1−(b−a)f(b))]≤B n≤B n+1+
1
2(n+1)
f(a)+
n+2
2(n+1)
f(a)+
n
If f is concave,all the above inequalities reverse.Moreover,all these inequalities are strict in the case of strict convexity or concavity.
Proof.Since x(n+1)
i =i
n+1
x(n)i(1≤i≤n),by Jensen’s inequality,we
have
f(x(n+1)
i )≤
i
n+1
f(x(n)i)(1≤i≤n),
which by summing them up from i=1to i=n,with some calculations,we get the left hand side of(3).
Similarly,since x(n)i−1=n+1−i
n x(n+1)
i
(1≤i≤n+1),we have
f(x(n)i−1)≤n+1−i
n
f(x(n+1)
i
)(1≤i≤n+1),
which by summing them up from i=1to i=n+1,with some calculations,we get the right hand side of(4).
The right hand side of(3)and the left hand side of(4)are obtained from the right hand side of(4)and the left hand side of(3)respectively,by considering
A k−
B k=
(b−a)
Corollary2.3.If r≥1,then
n
n(n+2) 1/r≤ (n+1)n
i=1
i r/n
n+1
i=1i r 1/r(10)
≤n
n2 n+1i=1i r
1/r.
If0<r≤1the inequalities in(10)reverse.
Obviously,these give us a reﬁnement and a reverse of Alzer’s inequality(1). Proof.If r≥1,the function f(x)=x r(x≥0)is convex and so(10)follows from (3),by taking a=0and b=1.
If0<r≤1the function f is concave and so−f is convex.So,the inequalities in (3),and therefore the inequalities in(10)reverse.
REFERENCES
1.H.Alzer,On an inequality of H.Minc and L.Sathre,J.Math.Anal.Appl.,
179(1993),396-402.
2.F.Qi,Generalization of H.Alzer’s inequality,J.Math.Anal.Appl.240(1999),
no.1,294–297.
3.Elezovi´c,N.;Peˇc ari´c,J.On Alzer’s inequality.J.Math.Anal.Appl.223
(1998),no.1,366–369.
4.J.Sandor,On an inequality of Alzer,J.Math.Anal.Appl.192(1995),no.3,
1034-1035.
4。