Polynomials with the half-plane property and matroid theory

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一种用于高精度随动控制系统的轨迹预测方法

一种用于高精度随动控制系统的轨迹预测方法巫佩军;杨文韬;余驰;杨耕【摘要】For high precision servo control, discontinuities of target trajectory or/and long-periodic in-structions lead to slow response and bad characteristics in servo system. A trajectory prediction method was proposed. Targets were classified according to its differentiability and fitting polynomials with proper orders for different kinds of trajectories were chosen, and prediction with improved least square method based on trajectory classification was made. Simulation and experiment with two-phase hybrid stepper mo-tor have been actualized to analyze the prediction result and prove the feasibility of this method, which can reduce the maximum tracking error and tracking system latency.%针对一类高精度随动控制系统中，跟踪目标轨迹短时不连续、跟踪指令的给定周期过大，导致随动系统跟踪响应变慢、特性变差的问题，提出了一种跟踪轨迹预测算法。

该方法首先对被跟踪轨迹进行简单判定与分类，然后针对不同类型的轨迹，选择合适阶次的拟合多项式，采用改进最小二乘算法，对目标的未来轨迹进行预测。

Global robust output regulation for a class of

0167-6911/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 9 1 1 ( 9 9 ) 0 0 0 9 9 - 7
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A. Serrani, A. Isidori / Systems & Control Letters 39 (2000) 133–139
We state the global robust regulation problem for (2) as that to ÿnd a dynamic error-feedback controller, modeled by equations of the form ˙ = Á( ; e); u = Â( ; e ) (4)
[7, Chapter 8] to obtain a controller yielding semi-global robust regulation for a signiÿcant class of nonlinear systems which are not fully feedback linearizable, but are characterized by an input-to-state stable zero dynamics. While the latter works have solved, up to a certain extent, the case of robust semiglobal regulation, the question whether the global nonlinear regulator problem is indeed solvable in the general framework of [10] remained still unanswered. In this paper, we solve the problem of robust global regulation for the class of nonlinear systems which has been considered in [13,14]. The important feature of our scheme is an iterative procedure for the design of a robust stabilizer for the parallel connection of the plant and the internal model, which exploits the passivity property of this latter. The paper is organized as follows: in Section 2 the problem of global output regulation is brie y recalled. In Section 3, we deal with the transformation of coordinates and the plant augmentation which put the plant into a particular normal form, and we state the necessary assumptions for the solvability of the problem. The recursive procedure on which the controller design relies upon is discussed in Section 4. Finally, we draw some conclusions in Section 5. 2. The problem of output regulation The systems considered throughout this paper are modeled by equations of the form x ˙ = f(x; u; w; ); y = h(x; w; ); e = y − q(w; ) (1) (2)

力学专业英语考试重点整理

学而不思则惘，思而不学则殆力学专业英语考试重点整理一、单词：英译汉、汉译英Centroid 质心，形心Elasticity 弹力，弹性Linear 线性的，直线的Prismatic 棱镜的，棱柱形的Strain 应变Stress 应力Tension 张力，拉力，拉紧Alloy 合金Aluminium 铝Ductile 易延展的，韧性的Failure 失败，破坏，失效Lateral 侧面的，横向的Necking 颈缩Couple 力偶Cylinder 圆筒，圆筒状物Inertia 惯性，惯量，惰性Shaft 轴，杆状物Torsion 扭矩Cantilever 悬臂梁，伸臂Neutral 中性的Statics 静力学，静止状态，静态Symmetry 对称Transverse 横向的，横断的Astronomer 天文学家Galaxy 天河，银河，星系Planet 行星Collinear 共线的，在同一直线上的Dimension 尺寸，大小，维数Equate 使相等，等同Parameter 参量，参数Visualize 想象，形象化，使看得见Acceleration 加速度Dynamics 动力学Stationary 不动的，稳定的，定常的Vector 矢量Velocity 速度Angular 角的，角度的Coordinate 坐标Radian 弧度Shaft 轴Symbol 符号Conservation 守恒，保存Differentiation 微分Integration 积分，集成，一体化Interval 间隔，间隙，空隙Linear 直线的，线性的，一次的Moment 力矩，瞬间，片刻Vector 矢量Velocity 速度Derivative 导数，派生的Frequency 频率Friction 摩擦Magnitude 大小，量级Power 力量，乘方，幂Viscous 黏性的Continuum 连续体，连续统Rectangular 矩形的，成直角的Resultant 合成的，合力，合量Torque 扭转力，扭矩Dilatation 膨胀，扩张Distortion 扭曲，变形Isotropic 各向同性的Tensor 张量Coordinate 坐标Crack 裂缝Curvature 曲率Ellipse 椭圆Formula 公式Function 函数，功能Buckle 屈曲，皱曲，弄弯，翘曲Deflection 挠曲，偏向Wrinkle 皱纹，皱褶，起皱Factor 因数，系数Flexural 弯曲的，挠曲的Notch 缺口，凹槽，刻痕Vibrate 振动（v）Vibration 振动（n）Detector 发现者，侦察器，探测器，检波器Vacuum 真空，空间，真空的，产生真空的Other than 除了二、句子：英译汉1、The concepts of stress and strain can be illustrated in an elementary way byconsidering the extension of a prismatic bar. As shown in Fig. 1, a prismatic bar is one that has constant cross section throughout its length and a straight axis. In this illustration the bar is assumed to be loaded at its ends by axial forces P that produce a uniform stretching, or tension, of the bar.翻译：应力和应变的概念可以通过考虑一个棱柱形杆的拉伸这样一个简单的方式来说明。

Rational polynomials of simple type

a rX iv:mat h /88v1[mat h.AG ]1Aug200RATIONAL POLYNOMIALS OF SIMPLE TYPE W ALTER D.NEUMANN AND PAUL NORBURY Abstract.We classify two-variable polynomials which are rational of simple type.These are precisely the two-variable polynomials with trivial homological monodromy. 1.Introduction A polynomial map f :C 2→C is rational if its generic ﬁbre,and hence every ﬁbre,is of genus zero.It is of simple type if,when extended to a morphism ˜f :X →P 1of a compactiﬁcation X of C 2,the restriction of ˜f to each curve C of the compactiﬁcation divisor D =X −C 2is either degree 0or 1.The curves C on which ˜f is non-constant are called horizontal curves ,so one says brieﬂy “each horizontal curve is degree 1”.The classiﬁcation of rational polynomials of simple type gained some new interest through the result of Cassou-Nogues,Artal-Bartolo,and Dimca [4]that they are precisely the polynomials whose homological monodromy is trivial (it suﬃces that the homological monodromy at inﬁnity be trivial by an observation of Dimca).A classiﬁcation appeared in [12],but it is incomplete.It implicitly assumes trivial geometric monodromy (on page 346,lines 10–11).Trivial geometric mon-odromy implies isotriviality (generic ﬁbres pairwise isomorphic)and turns out to be equivalent to it for rational polynomials of simple type.The classiﬁcation in the non-isotrivial case was announced in the ﬁnal section of [17].The main purpose of this paper is to prove it.But we recently discovered that there are also isotrivial rational polynomials that are not in [12],so we have added a classiﬁcation for the isotrivial case using our methods.This case can also be derived from Kaliman’s classiﬁcation [9]of all isotrivial polynomials.The fact that his list includes rational polynomials of simple type that are not in [12]appears not to have been noticed before (it also includes rational polynomials not of simple type).In general,the classiﬁcation of polynomial maps f :C 2→C is an open problemwith extremely rich structure.One notable result is the theorem of Abhyankar-Moh and Suzuki [1,23]which classiﬁes all polynomials with one ﬁbre isomorphic to C .The analogous result for the next simplest case,where one ﬁbre is isomorphic to C ∗,is open except in special cases when the genus of the generic ﬁbre of the polynomial is given.Kaliman [10]classiﬁes all rational polynomials with one ﬁbre isomorphic to C ∗.The basic tool we use in our study of rational polynomials is to associate to any rational polynomial f :C 2→C a compactiﬁcation X of C 2on which f extends toa well-deﬁned map ˜f :X →P 1together with a map X →P 1×P 1.The map to2WALTER D.NEUMANN AND PAUL NORBURYP1×P1is not in general canonical.We will exploit the fact that for a particular class of rational polynomials,there is an almost canonical choice.Although we give explicit polynomials,the classiﬁcation is initially presented in terms of the splice diagram for the link at inﬁnity of a genericﬁbre of the polynomial (Theorem4.1).This is called the regular splice diagram for the polynomial(since genericﬁbres are also called“regular”).See[15]for a description of the link at inﬁnity and its splice diagram.The regular splice diagram determines the embedded topology of a genericﬁbre and the degree of each horizontal curve.Hence we can speak of a“rational splice diagram of simple type”.Theﬁrst author has asked if the moduli space of polynomials with given regular splice diagram is connected.For a rational splice diagram of simple type weﬁnd the answer is“yes”.We describe the moduli space for our polynomials in Theorem4.2 and use it to help give explicit normal forms for the polynomials.We also describe how the topology of the irregularﬁbres varies over the moduli space.The more general problem of classifying all rational polynomials,which would cover much of the work mentioned above,is still an open and interesting problem.It is closely related to the problem of classifying birational morphisms of the complex plane since a polynomial is rational if and only if it is one coordinate of a birational map of the complex plane.Russell[20]calls this a“ﬁeld generator”and deﬁnes a goodﬁeld generator to be a rational polynomial that is one coordinate of a birational morphism of the complex plane.A rational polynomial is good precisely when its resolution has at least one degree one horizontal curve,[20].Daigle[5] studies birational morphisms C2→C2by associating to a compactiﬁcation X of the domain plane a canonical map X→P2.A birational morphism is then given by a set of curves and points in P2indicating where the map is not one-to-one.The approach we use in this paper is similar.The full list of rational polynomials f:C2→C of simple type is as follows.We list them up to polynomial automorphisms of domain C2and range C(so-called “right-left equivalence”).Theorem1.1.Up to right-left equivalence a rational polynomial f(x,y)of simple type has one of the following forms f i(x,y),i=1,2,or3.f1(x,y)=x q1s q+x p1s p r−1i=1(βi−x q1s q)a i(r≥2)f2(x,y)=x p1s p r−1i=1(βi−x q1s q)a i(r≥1)f3(x,y)=y r−1i=1(βi−x)a i+h(x)(r≥1).Here:0≤q1<q,0≤p1<p, p p1q q1 =±1;s=yx k+P(x),with k≥1and P(x)a polynomial of degree<k;a1,...,a r−1are positive integers;β1,...,βr−1are distinct elements of C∗;h(x)is a polynomial of degree< r−11a i.RATIONAL POLYNOMIALS OF SIMPLE TYPE3 Moreover,if g1(x,y)=g2(x,y)=x q1s q and g3(x,y)=x then(f i,g i):C2→C2is a birational morphism for i=1,2,3.In fact,g i maps a genericﬁbref−1 i (t)biholomorphically to C−{0,t,β1,...,βr−1},C−{0,β1,...,βr−1},or C−{β1,...,βr−1},according as i=1,2,3.Thus f1is not isotrivial and f2and f3are.In[12]the isotrivial case is subdivided into seven subcases,but these do not include any f2(x,y)with p,q,p1,q1all>1.2.ResolutionGiven a polynomial f:C2→C,extend it to a map¯f:P2→P1and resolve the points of indeterminacy to get a regular map˜f:X→P1that coincides with f on C2⊂X.We call D=X−C2the divisor at inﬁnity.The divisor D consists of a connected union of rational curves.An irreducible component E of D is horizontal if the restriction of˜f to E is not a constant mapping.The degree of a horizontal curve E is the degree of the restriction˜f|E.Although the compactiﬁcation deﬁned above is not unique,the horizontal curves are essentially independent of choice.Note that a genericﬁbre F c:=f−1(c)is a punctured Riemann surface with punctures precisely whereF c meets each horizontal curve exactly once,so the number of punc-tures equals the number of horizontal curves.For non-simple type the number of punctures will exceed the number of horizontal curves.We say that a rational polynomial is ample if it has at least three degree one hori-zontal curves.Those polynomials with no degree one horizontal curves,or badﬁeld generators[20],are examples of polynomials that are not ample.The classiﬁcation of Kaliman[10]mentioned in the introduction gives examples of polynomials with exactly one degree one horizontal curve so they are also not ample.Nevertheless, ample rational polynomials will be the focus of our study in this paper.We will classify all ample rational polynomials that are also of simple type.3.Curves in P1×P1.If˜f:X→P1is a regular map with rationalﬁbres then X can be blown down to a Hirzebruch surface,S,so that˜f is given by the composition of the sequence of blow-downs X→S with the natural map S→P1;see[2]for details.Moreover, byﬁrst replacing X by a blown-up version of X if necessary,we may assume that S=P1×P1and the natural map to P1is projection onto theﬁrst factor.A rational polynomial f:C2→C,once compactiﬁed to˜f:X=C2∪D→P1, may thus be given by P1×P1together with instructions how to blow up P1×P1 to get X and how to determine D in X.For this we give the following data:•a collection C of irreducible rational curves in P1×P1including L∞:=∞×P1;•a set of instructions on how to blow up P1×P1to obtain X;•a sub-collection E of the curves of the exceptional divisor of X→P1×P1; satisfying the condition:•If D is the union of the curves of E and the proper transforms of the curves of C then X−D∼=C2;If C⊂P1×P1is an irreducible algebraic curve we associate to it the pair of integers(m,n)given by degrees of the two projections of C to the factors of P1×P1. Equivalently,(m,n)is the homology class of C in terms of H2(P1×P1)=Z⊕Z.4WALTER D.NEUMANN AND PAUL NORBURYWe call C an(m,n)curve.The intersection number of an(m,n)curve C and an (m′,n′)curve C′is C·C′=mn′+nm′.The above collection C of curves in P1×P1will consist of some vertical curves (that is,(0,1)curves;one of these is L∞)and some other curves.These non-vertical curves give the horizontal curves for f,so they all have m=1if f is of simple type. Note that a(1,n)curve is necessarily smooth and rational(since it is the graph of a morphism P1→P1).The image in P1×P1of theﬁbre over inﬁnity is the(0,1)curve L∞and the image of a degree m horizontal curve is an(m,n)curve.This view allows one to see as follows a geometric proof of the result of Russell[20]that a rational polynomial f is good precisely when its resolution has at least one degree one horizontal curve.A degree one horizontal curve for f has image in P1×P1given by a(1,n)curve. Call this image C and let P be its intersection with L∞.The(1,n)curves that do not intersect C−P form a C–family that sweeps out P1×P1−(L∞∪C)so they lead to a map X→P1which takes values in C at points that do not lie over L∞∪C.Restricting to C2=X−D we obtain a meromorphic function g1 that has poles only at points that belong to exceptional curves that were blown up on C(and do not belong to E).However the polynomial f is constant on each such curve,so if c1,...,c k are the values that f takes on these curves,then g:=g1(f−c1)a1...(f−c k)a k will have no poles,and hence be polynomial,for a1,...,a k suﬃciently large.Then(f,g)is the desired birational morphism C2→C2.For the converse,given a birational morphism(f,g):C2→C2,we compactify it to a morphism(˜f,˜g):X→P1×P1.Then the proper transform of P1×∞is the desired degree one horizontal curve for f.We shall use the usual encoding of the topology of D by the dual graph,which has a vertex for each component of D,an edge when two components intersect,and vertex weights given by self-intersection numbers of the components of D.We will sometimes speak of the valency of a component C of D to mean the valency of the corresponding vertex of the dual graph,that is,the number of other components that C meets.The approach we will take to get rational polynomials will be to start with any collection C of k curves in P1×P1and see if we can produce a divisor at inﬁnity D for a map from C2to C.In order to get a divisor at inﬁnity we must blow up P1×P1,say m times,and include some of the resulting exceptional curves in the collection so that this new collection gives a divisor D whose complement is C2. The exceptional curves that we“leave behind”(i.e.,do not include in D)will be called cutting divisors.Lemma3.1.(i)D must have m+2irreducible components,so we must include m−k+2of the exceptional divisors in the collection leaving k−2behind as cutting divisors;(ii)D must be connected and have no cycles;(iii)D must reduce to one of the“Morrow conﬁgurations”by a sequence of blow-downs.The Morrow conﬁgurations are the conﬁgurations of rational curves with dual graphs of one of the following three types,in which,in the last case,after replacing the central(n,0,−n−1)by a single(−1)vertex the result should blow down to a single(+1)vertex by a sequence of blow-downs:1◦RATIONAL POLYNOMIALS OF SIMPLE TYPE50l◦◦◦◦6WALTER D.NEUMANN AND PAUL NORBURY3.1.Horizontal curves.The next few lemmas will be devoted toﬁnding restric-tions on the horizontal curves in the conﬁguration C⊂P1×P1,culminating in Proposition3.9.Lemma3.3.A horizontal curve of type(1,n)in C must be of type(1,1). Proof.Assume we have a horizontal curve C∈C of type(1,n)with n>1.It intersects each of the three(1,0)curves n times(counting with multiplicity)so in order to break cycles—Lemma3.1(ii)—we have to blow up at least n times on each(1,0)horizontal curve,so the proper transforms of the three(1,0)curves have self-intersection at most−n and the proper transform of the(1,n)curve has self-intersection at most2n−3n=−n.By Lemma3.1(iii),D must reduce to a Morrow conﬁguration by a sequence of blow-downs.Thus D must contain a−1curve E that blows down.By Lemma3.2, the curve E must be a proper transform of a horizontal curve.The proper transform of each(1,0)curve has self-intersection at most−n<−1.Thus E must come from one of the(1,∗)horizontal curves.As mentioned above,the proper transform of a(1,k)curve has self-intersection≤−k so E must be the proper transform of a (1,1)curve,E0.But E0would intersect C,the(1,n)curve,2n times and hence E.E≤2−2n<−1since n>1.This is a contradiction so any horizontal curve of type(1,n)must be a(1,1)curve.Hence,the horizontal curves consist of a collection of(1,0)curves and(1,1) curves.Figure1shows an example of a possible conﬁguration of horizontal curves in P1×P1.u u uu u uu u uu u uu u uu u uu u uu u uu uFigure1.Conﬁguration of horizontal curves.Lemma3.4.˜L∞·˜L∞=−1.Proof.We blow up at a point on L∞precisely when at least two horizontal curves meet in a common point there.In general,if a horizontal curve meets L∞with a high degree of tangency then we blow up repeatedly there.But,since all horizontal curves are(1,0)and(1,1)curves,they meet L∞transversally,so a point on L∞will be blown up at most once.If there are two such points to be blown up,then after blowing up there will be(in the dual graph)two non-neighbouring−1curves with valency>2.The complement of such a conﬁguration cannot be C2.This is proven by Kaliman[11] as Corollary3.Actually the result is stated for two−1curves of valency3but it applies to valency≥3.Thus,at most one point on L∞is blown up and˜L∞·˜L∞=0or−1.We must show0cannot occur.RATIONAL POLYNOMIALS OF SIMPLE TYPE7B1e eAB28WALTER D.NEUMANN AND PAUL NORBURY(1,1)curves and contain at least two points where it intersects the(1,1)curves and thus have self-intersection<−1after blowing up to break cycles.We are once more at the situation of Lemma3.5where the valency>2curve is˜L∞which has self-intersection−1by Lemma3.4,and the branches B1and B2are the proper transform of D and any curves beyond it,respectively the proper transform of H and any curves beyond it.Thus we have a contradiction.Notice that both cases apply to two(1,1)curves that may intersect at a tangent point,and shows that this situation is impossible.Lemma3.7.If there is more than one(1,1)curve in C then there are exactly three (1,0)horizontal curves in C.Proof.Assume that there are more than three(1,0)horizontal curves in C and at least two(1,1)curves,say C1and C2.Case1:C1and C2meet on˜L∞.Then they meet each of at least two(1,0) curves in distinct points,so after blowing up to destroy cycles,these(1,0)curves have self-intersection number≤−2and Lemma3.5applies.Case2:C1and C2meet˜L∞at distinct points.Then one of them,say C1, meets˜L∞at a point not on a(1,0)curve by Lemma3.4.At least one(1,0)curve C3meets C1and C2in distinct points.After breaking cycles,C1and C3have self-intersections≤−2so Lemma3.5applies again.Lemma3.8.A family of(1,1)horizontal curves in C must pass through a common pair of points.Proof.The statement is trivial for one(1,1)horizontal curve so assume there are at least two(1,1)horizontal curves in C.By the previous lemma,there are exactly three(1,0)horizontal curves.If there are exactly two(1,1)horizontal curves in C then the lemma is clear since the curves cannot be tangent by Lemma3.6.When there are more than two(1,1)curves in C,apply Lemma3.6to two of them.If another(1,1)horizontal curve in C does not intersect these two(1,1) curves at their common two points of intersection then,by Lemma3.6,it must meet both these(1,1)curves at the third(1,0)horizontal curve of C.So theﬁrst two(1,1)curves would meet there,which is a contradiction.Proposition3.9.Any conﬁguration of horizontal curves in C is equivalent to one of the form in Figure1.Proof.By assumption and Lemma3.3there are at least three(1,0)horizontal curves and some(1,1)horizontal curves in C.If there is exactly one(1,1)horizontal curve then the proposition is clear.If there is more than one(1,1)horizontal curve, then by Lemmas3.7and3.8there are precisely three(1,0)horizontal curves and two of the(1,0)horizontal curves contain the common intersection of the(1,1) curves.Each(1,1)curve also contains a distinguished point where the curve meets the third(1,0)horizontal curve.A Cremona transformation can bring such a conﬁguration to that in Figure1by blowing up at the two points of intersection of the(1,1)curves and blowing down the two vertical lines containing the two points.This sends two of the(1,0)horizontal curves and each(1,1)curve to(1,0) horizontal curves and one of the(1,0)curves to a(1,1)curve that intersects eachRATIONAL POLYNOMIALS OF SIMPLE TYPE9 of the other horizontal curves exactly once.Note that since we blow up P1×P1to get the polynomial map,two conﬁgurations of curves C,C′in P1×P1related by a Cremona transformation give rise to the same polynomial,so we are done.3.2.The conﬁguration C.The image C of D⊂X→P1×P1will consist of the conﬁguration of horizontal curves in Figure1plus some(0,1)vertical curves.The next two lemmas show that in fact the only(0,1)vertical curve we need to include in C is L∞and furthermore that C can be given by Figure4.Lemma3.10.The conﬁguration C appears in Figure3or Figure4.Proof.Let r+2denote the number of horizontal curves and k+1denote the number of(0,1)vertical curves in C.Thus C consists of k+r+3irreducible components and by Lemma3.1(i),when blowing up to get D from C we must leave k+r+1 exceptional curves behind as cutting divisors.By Lemma3.1(ii)we must break all cycles.The minimum number of cutting divisors needed to do this is kr+k+r−2min{k,r}.This is because each of the k(0,1)vertical curves diﬀerent from L∞must be separated from all but one of the r+1(1,0)horizontal curves,so we need kr cutting divisors.Also,the(1,1) horizontal curve meets each of the r+1(1,0)horizontal curves and each of the k (0,1)vertical curves once,so that requires k+r cutting divisors(by Lemma3.4 the(1,1)curve must meet L∞at a triple point with a(1,0)horizontal curve,so this intersection does not produce a cycle to be broken).We would thus require kr+k+r cutting divisors except that the(1,1)curve may pass through intersections of the(1,0)horizontal curves and the(0,1)vertical curves,so some of the cutting divisors may coincide.The most such intersections possible is min{k,r}and we have then over-counted required cutting divisors by2min{k,r}.Hence we get at least kr+k+r−2min{k,r}cutting divisors.Since the number k+r+1of cutting divisors is at least kr+k+r−2min{k,r}, we have k+r+1≥kr+k+r−2min{k,r},so(1)1≥k(r−2)and1≥(k−2)r,k≥0,r≥2.The solutions of(1)are(k,r)={(0,r),(1,2),(1,3),(2,2)}.Recall by Lemma3.4that the(1,1)curve must meet L∞at a triple point with a(1,0)horizontal curve.Furthermore,by keeping track of when either inequality in(1)is an equality,or one away from an equality,we can see that the(1,1)curve must meet any other(0,1)vertical curves at a triple point with a(1,0)horizontal curve.Thus,the only possible conﬁgurations for C are given in Figures3and4.10WALTER D.NEUMANN AND PAUL NORBURY...RATIONAL POLYNOMIALS OF SIMPLE TYPE11 Proof.The proper transform of each horizontal curve has self-intersection less than or equal to−1and all curves in D beyond horizontal curves have self-intersection strictly less than−1.If the two horizontal curves that meet˜L∞,˜H1and˜H2, have self-intersection strictly less than−1,then since all curves beyond the two horizontal curves also have self-intersection strictly less than−1,and since˜L∞has self-intersection−1and valence3this gives a contradiction by Lemma3.5.The same argument applies to˜H3and˜H4together with E.Lemma3.14.˜H4·˜H4=−1if and only if˜H2·˜H2=−1.Proof.Since L1must be separated from at least one of H2and H3then at most one of˜H2·˜H2=−1and˜H3·˜H3=−1can be true.Similarly E1must be separated from at least one of H1and H4so at most one of˜H1·˜H1=−1and˜H4·˜H4=−1 can be true.By Lemma3.13,if˜H2·˜H2=−1then˜H1·˜H1=−1so˜H4·˜H4=−1. Similarly,˜H1·˜H1=−1implies that˜H2·˜H2=−1and˜H4·˜H4=−1.Lemma3.15.The conﬁguration from Figure3with(k,r)=(1,2)together with the requirement that E1is not a cutting divisor cannot occur.Proof.Suppose otherwise.Assume that˜H1·˜H1=−1and˜H3·˜H3=−1.If this is not the case,then by Lemmas3.13and3.14we may assume that˜H4·˜H4=−1 and˜H2·˜H2=−1and argue similarly.The curves beyond˜H1have self-intersection strictly less than−1.The curve immediately adjacent and beyond˜H1is˜E1and this has self-intersection strictly less than−2.This is because we must blow up between E1and H4to separate cycles,and also between˜E1and˜L1to break cycles and to maintain˜H1·˜H1=−1and˜H3·˜H3=−1.Thus if we blow down˜H1 the remaining branch beyond˜L∞consists of curves with self-intersection strictly less than−1.Also˜H2has self-intersection strictly less than−1since we have to blow up the intersection between H2and H4and the intersection between H2and L1in order to break cycles and maintain˜H3·˜H3=−1.After blowing down˜H1,˜Lhas self-intersection0and valency3with two branches consisting of curves ∞of self-intersection strictly less than−1.Thus we can use Lemma3.5to get a contradiction.4.Non-isotrivial rational polynomials of simple typeThe conﬁguration in Figure4is the starting point for any non-isotrivial rational polynomial of simple type.Notice that we canﬁll one puncture in eachﬁbre of any such map to get an isotrivial family of curves and the puncture varies linearly with c∈C.Notice also that there is an irregularﬁbre for each of the r intersection points of the(1,1)curve with(1,0)horizontal curves away from L∞.In fact there is at most one more irregularﬁbre which can only occur in rather special cases,as we discuss in subsection4.1.From now on the conﬁguration C is given by Figure4with r+2horizontal curves. Beginning with C we will list all of the rational polynomials of simple type generated from this conﬁguration.We shall give the splice diagrams for these polynomials ﬁrst.Although we compute the polynomials later,geometric information of interest is often more easily extracted from the splice diagram or from our construction of the polynomials than from an actual polynomial.The splice diagram encodes the topology of the polynomial.It represents the link at inﬁnity of the genericﬁbre,or it can be thought of as an eﬃcient plumbing12WALTER D.NEUMANN AND PAUL NORBURYgraph for the divisor at inﬁnity,D ⊂X .It encodes an entire parametrised family of polynomials with the same topology of their regular ﬁbres.See [7,15,16]for more details.Within this family,polynomials can still diﬀer in the topology of their irregular ﬁbres.Our methods also give all information about the irregular ﬁbres,as we describe in subsection 4.1.The conﬁguration C has r +3irreducible components so when we blow up to get D by Lemma 3.1(i)we will leave r +1exceptional curves behind as cutting divisors.By Lemma 3.1(ii)we must break the r cycles in C with multiple blow-ups at the points of intersection leaving r exceptional curves behind as cutting divisors.We blow up multiple times between the r th (1,0)horizontal curve and the (1,1)horizontal curve in order to break a cycle.Thus,we require those blow-ups to satisfy the condition that the exceptional curve will break the cycle if removed.Equivalently,each new blow-up takes place at the intersection of the most recent exceptional curve with an adjacent curve.We call such a multiple blow-up a separating blow-up sequence .We have one extra cutting divisor.This will arise as the last exceptional curve blown up in a sequence of blow-ups that does not break a cycle.We will call this se-quence of blow-ups a non-separating blow-up sequence .A priori,this non-separating blow-up sequence could be a sequence as in Figure 5,where the ﬁnal −1curve isP ◦◦ ◦ ◦◦◦ ◦ −1◦Figure 5.Sequence of blow-ups starting at P and ending at the−1curve.the cutting divisor.However,we shall see that the extra nodes this introduces in the dual graph prohibit D from blowing down to a Morrow conﬁguration,so the sequence is simply a string of −2exceptional curves followed by −1exceptional curve that is the cutting divisor.This arises from blowing up a point on a curve in the blow-up of C that does not lie on an intersection of irreducible components.Let us begin by just performing the separating blow-up sequences at the points of intersection,of C and leaving the non-separating blow-up sequence until later.This gives the dual graph in Figure 6with the proper transforms of the r +1(1,0)horizontal curves and the (1,1)horizontal curve indicated along with ˜L∞and the exceptional curve E arising from the blow-up of the triple point in C .There are r branches heading out from the proper transform of (1,1)consisting of curves of self-intersection less than −1and beyond each of the proper transforms of the r (1,0)horizontal curves the curves have self-intersection less than −1.The self-intersection of each of ( 1,0)0,E and ˜L∞is −1.The self-intersections of ( 1,1)and ( 1,0)i ,i =1,...,r are negative and depend on how we blow up at each point of intersection.Lemma 4.1.There is at most one branch in D beyond ( 1,1),and r −1of the horizontal curves ( 1,0)i (those with index i =1,...,r −1say)have self-intersection−1and only −2curves beyond.RATIONAL POLYNOMIALS OF SIMPLE TYPE 13◦t t t t t t t t ◦k k k k k k k k k k k k k k E˜L ∞( 1,0)r −1◦◦( 1,0)0◦( 1,0)1◦s s s s s s s s ◦◦{a r −1−1}◦−114WALTER D.NEUMANN AND PAUL NORBURY Lemma4.2.The non-separating blow-up sequence occurs beyond either( 1,1),( 1,0)r, or( 1,0)0and in the latter case( 1,1)·( 1,1)=−1.Proof.If the non-separating blow-up sequence occurs on the branch beyond( 1,0)i, i=1,...,r−1then that branch cannot be blown down.By the proof of lemma4.1,in order to obtain a linear graph we must blow down r−1of the branches beyond ( 1,0)i,i=1,...,r.Thus,if the non-separating blow-up sequence does occur beyond( 1,0)i for some i≤r−1,then the( 1,0)r branch blows down,so we simply swap the labels i and r.The non-separating blow-up sequence cannot occur on E or˜L∞because the resulting cutting divisor would not be sent to aﬁnite value.If the non-separating blow-up sequence occurs on the branch beyond( 1,0)0then we must be able to blow down the branch beyond( 1,1),hence the branch must consist of( 1,1)with self-intersection−1.Lemma4.3.We may assume the non-separating blow-up sequence does not occur beyond( 1,0)0.Proof.By Lemma4.2if the non-separating blow-up sequence occurs beyond( 1,0)0 then( 1,1)·( 1,1)=−1.In particular,1=A= r−11a i.Thus,r=2,a1=1. With only four horizontal curves,we can perform a Cremona transformation to make( 1,0)0the(1,1)curve and hence we are in theﬁrst case of Lemma4.2.Lemma4.4.The non-separating blow-up sequence occurs on either of the last curves beyond( 1,1)or( 1,0)r and is a string of−2curves followed by the−1 curve that is a cutting divisor.Proof.Arguing as previously,if the non-separating blow-up sequence occurs any-where else,or if it is more complicated,then it introduces a new branch preventing the divisor D from blowing down to a linear graph.We now know that our divisor D results from Figure7by doing a separating blow-up sequence between the(1,1)curve and the r-th(1,0)curve,leaving behind theﬁnal−1exceptional curve as a cutting divisor and then performing a non-separating blow-up sequence on a curve adjacent to this cutting divisor to produce second cutting divisor.A priori,it is not clear that this procedure always gives rise to a divisor D⊂X where X is a blow-up of P2and D is the pre-image of the line at inﬁnity.The classiﬁcation will be complete once we show it does.Lemma4.5.The above procedure always gives rise to a conﬁguration that blows down to a Morrow conﬁguration(see Lemma3.1)and hence determines a rational polynomial of simple type.Proof.The calculation involves the relation between plumbing graphs and splice diagrams described in[7]or[16],with which we assume familiarity.In particular, we use the continued fractions of weighted graphs described in[7].If one has a chain of vertices with weights−c0,−c1,...,−c t,its continued fraction based at the。

J. reine angew. Math. 583 (2005), 163—174 Journal für die reine und angewandte Mathematik

J.reine angew.Math.583(2005),163—174Journal fu¨r die reine undangewandte Mathematik(Walter de GruyterBerlinÁNew York2005Ehrhart polynomials,simplicial polytopes,magic squares and a conjecture of StanleyDedicated to Richard Stanley on the occasion of his sixtieth birthdayBy Christos A.Athanasiadis at HeraklionAbstract.It is proved that for a certain class of integer polytopes P the polynomial hðtÞwhich appears as the numerator in the Ehrhart series of P,when written as a rational function of t,is equal to the h-polynomial of a simplicial polytope and hence that its co-e‰cients satisfy the conditions of the g-theorem.This class includes the order polytopes of graded posets,previously studied by Reiner and Welker,and the Birkho¤polytope of doubly stochastic nÂn matrices.In the latter case the unimodality of the coe‰cients of hðtÞ,which follows,was conjectured by Stanley in1983.1.IntroductionLet P be an m-dimensional convex polytope in R q having integer vertices.We will be concerned with the function iðP;rÞcounting integer points in the r-fold dilate of P.It is a fundamental result due to Ehrhart[3],[4]that iðP;rÞis a polynomial in r of degree m,called the Ehrhart polynomial.Thus one can writeP r f0iðP;rÞt r¼h0þh1tþÁÁÁþh d t dð1ÀtÞð1Þfor certain integers h0;h1;...;h d.It was proved by Stanley[15]that the integers h i are nonnegative.In this paper we describe simple conditions on P which imply that the se-quenceðh0;h1;...;h dÞis equal to the h-vector of a d-dimensional simplicial polytope. Such vectors are characterized by McMullen’s g-theorem;see[1],[9],[14]and Section2. In particular,they are symmetric and satisfy the inequalitiesh0e h1eÁÁÁe h b d=2c:ð2ÞOur main result(Theorem3.5)applies to the Birkho¤polytope of doubly stochastic nÂn matrices and to order polytopes of graded posets.In the former case iðP;rÞis equal to thenumber of n Ân integer stochastic matrices,or magic squares,having line sums equal to r and the inequalities (2)prove a long-standing conjecture of Stanley [16],Section I.1.In the latter case the integers h i count linear extensions of a naturally labeled poset by the number of descents and our result specializes to a recent result of Reiner and Welker [12].Our method extends that in the work [12],which provided one of the main motivations for this paper.New elements in our approach are the concept of a ‘special simplex’,introduced in Section 3,and the employment of reverse lexicographic triangulations,which are com-bined to produce simplicial decompositions of integer polytopes more general than order polytopes of graded posets and to construct from these decompositions,as in [12],explicitly simplicial polytopes with prescribed h -vectors.In Section 4we state several corollaries of our main result in the context of a‰ne semigroup rings.I am grateful to Volkmar Welker for encouraging e-discussions and to Jesu´s DeLoera,Victor Reiner,Francisco Santos,Richard Stanley and Bernd Sturmfels for helpful com-ments.2.BackgroundIn this section we review some basic deﬁnitions and background on convex polytopes and their face numbers,triangulations and Ehrhart polynomials.We refer the reader to the texts by Stanley [16],[17],Sturmfels [20]and Ziegler [21]for more information on these topics.We denote by N the set of nonnegative integers.Face enumeration.Given a ﬁnite (abstract or geometric)simplicial complex D of dimension d À1,let f i denote the number of i -dimensional faces of D ,so that ðf 0;f 1;...;f d À1Þis the f -vector of D .The polynomialPd i ¼0f i À1ðx À1Þd Ài ¼P d i ¼0h i x d Ài ;ð3Þwhere f À1¼1unless D is empty,is the h -polynomial of D ,denoted h ðD ;t Þ.The h -vector of D is the sequence ðh 0;h 1;...;h d Þdeﬁned by (3).A polytopal complex F [21],Section 8.1,is a ﬁnite,nonempty collection of convex polytopes such that (i)any face of a polytope in F is also in F and (ii)the intersection of any two polytopes in F is either empty or a face of both.The elements of F are its faces and those of dimension 0are its vertices .The dimension of F is the maximum dimension of a face.The complex F is pure if all maximal faces of F have the same dimension.The collection F ðP Þof all faces of a polytope P and the collection F ðq P Þof its proper faces are pure polytopal complexes called the face complex and boundary complex of P ,respec-tively.Thus P is simplicial if F ðq P Þis a simplicial complex.The h -vectors of boundary complexes of simplicial polytopes are characterized by McMullen’s g -theorem [9],[16],Section III.1,[21],Section 8.6,as follows.A sequence ðg 0;g 1;...;g l Þof nonnegative in-tegers is said to be an M -vector if(i)g 0¼1and(ii)0e g i þ1e g hi i ifor 1e i e l À1,Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares164where 0h i i ¼0andn h i i ¼k i þ1i þ1 þk i À1þ1i þÁÁÁþk j þ1j þ1for the unique representationn ¼k ii þk i À1i À1 þÁÁÁþk jjwith k i >k i À1>ÁÁÁ>k j f j f 1,if n f 1.A sequence ðh 0;h 1;...;h d Þof nonnegative in-tegers is the h -vector of the boundary complex of a d -dimensional simplicial polytope if and only if(i)h i ¼h d Ài for all i and(ii)ðh 0;h 1Àh 0;...;h b d =2c Àh b d =2cÀ1Þis an M -vector.In particular ðh 0;h 1;...;h d Þis symmetric and unimodal and hence satisﬁes the inequalities(2),known as the Generalized Lower Bound Theorem for simplicial polytopes.Triangulations and Ehrhart polynomials.A triangulation of a polytopal complex F is a geometric simplicial complex D with vertices those of F and underlying space equal to the union of the faces of F ,such that every maximal face of D is contained in a face of F .A triangulation of the face complex F ðP Þof a polytope P is simply called a triangula-tion of P .For any set s consisting of vertices of the polytopal complex F we denote by F n s the subcomplex of faces of F which do not contain any of the vertices in s and write F n v for F n s if s consists of a single vertex v .Given a linear ordering t ¼ðv 1;v 2;...;v p Þof the set of vertices of F we deﬁne the reverse lexicographic triangulation or pulling triangulation D ðF Þ¼D t ðF Þwith respect to t [15],[8],[20],p.67,as D ðF Þ¼f v g if F consists of a single vertex v andD ðF Þ¼D ðF n v p ÞW S FÈconv ðf v p g W G Þ:G A D ÀF ðF ÞÁW f j gÉotherwise,where the union runs through the facets F not containing v p of the maximal faces of F which contain v p and D ðF n v p Þand D ÀF ðF ÞÁare deﬁned with respect to the linear orderings of the vertices of F n v p and F ,respectively,induced by t .Equivalently,for i 0<i 1<ÁÁÁ<i t the set f v i 0;v i 1;...;v i t g is the vertex set of a maximal simplex of D t ðF Þif there exists a maximal ﬂag F 0H F 1H ÁÁÁH F t of faces of F such that v i j is the last vertex of F j with respect to t for all j and v i j is not a vertex of F j À1for j f 1.A di¤erent way to deﬁne D t ðF Þis the following.For any vertex v of F letpull v ðF Þ¼ðF n v ÞW S Ff conv ðf vg W G Þ:G A F ðF ÞW f j gg ;where the union runs through the facets F not containing v of the maximal faces of F which contain v .If F 0¼F and F i ¼pull v p Ài þ1ðF i À1Þfor 1e i e p then F p is a triangula-Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares 165tion of F which coincides with D t ðF Þ.It follows from [10],Theorem 2.5.23(see also [6],p.80)that if F is the boundary complex of a polytope P then pull v ðF Þis the boundary complex of another polytope,obtained from P by moving its vertex v beyond the hyper-planes supporting exactly those facets of P which contain v .This observation implies the following lemma.Lemma 2.1.The reverse lexicographic triangulation of the boundary complex of a polytope with respect to any ordering of its vertices is abstractly isomorphic to the boundary complex of a simplicial polytope of the same dimension .A convex polytope P L R q is said to be a rational or an integer polytope if all its vertices have rational or integer coordinates,respectively.It is called a 0/1polytope if all its vertices are 0/1vectors in R q .If P is rational then the function deﬁned for nonnegative integers r by the formulai ðP ;r Þ¼K ðrP X Z q Þis a quasi-polynomial in r ,called the Ehrhart quasi-polynomial of P [17],Section 4.6.If P is an integer polytope then this quasi-polynomial is actually a polynomial in r .Let A L R q be the a‰ne span of the integer polytope P .A triangulation D of P is called unimodular if the vertex set of any maximal simplex of D is a basis of the a‰ne integer lattice A X Z q .We denote by D t the reverse lexicographic triangulation of an arbitrary polytope P with respect to the ordering t of its vertices.Following [15]we call such an ordering of the ver-tices of an integer polytope P compressed if D t is unimodular and call P itself compressed if so is any linear ordering of its vertices.The following lemma holds for any unimodular triangulation of P ,although we will not need this fact here.Lemma 2.2([15],Corollary 2.5).If P is an m -dimensional integer polytope in R q and t is a compressed ordering of its vertices thenPr f 0i ðP ;r Þt r ¼h ðD t ;t Þð1Àt Þm þ1:Two compressed polytopes.(a)A real n Ân matrix is said to be doubly stochastic if all its entries are nonnegative and all its rows and columns sum to 1.The set P of all real doubly stochastic n Ân matrices is a convex polytope in R n Ân of dimension ðn À1Þ2,called the Birkho¤polytope [21],Example 0.12.It follows from the classical Birkho¤-von Neu-mann theorem that the vertices of P are the n Ân permutation matrices,so that P is a 0/1polytope.The Birkho¤polytope was shown to be compressed by Stanley [15],Example 2.4(b)(see also [20],Corollary 14.9).(b)Let W be a poset (short for partially ordered set)on the ground set ½m :¼f 1;2;...;m g .Recall that an (order)ideal of W is a subset I L W for which i <W j and j A I imply that i A I .Let W 0be the poset obtained from W by adjoining a minimum element 0.The order polytope [18]of W ,denoted O ðW Þ,is the intersection of the hyperplane x 0¼1in R m þ1with the cone deﬁned by the inequalities x i f x j for i <j in W 0and x i f 0for all i .Thus O ðW Þis an m -dimensional convex polytope.The vertices of O ðW Þare the characteristic vectors of the nonempty ideals of W 0[18],Corollary 1.3,so,in particular,O ðW Þis a 0/1polytope (see [18],Theorem 1.2,for a complete description of the facial struc-Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares166Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares167 ture of OðWÞ).Order polytopes were shown to be compressed by Ohsugi and Hibi[11], Example1.3(b).Quotient polytopes.If P L R m is an m-dimensional polytope and V is any linear subspace of R m then the quotient polytope P=V L R m=V is the image of P under the ca-nonical surjection R m!R m=V.This is a convex polytope in R m=V linearly isomorphic to the image pðPÞof P under any linear surjection p:R m!R mÀdim V with kernel V.Re-call that the simplicial join D1ÃD2of two abstract simplicial complexes D1and D2on dis-joint vertex sets has faces the sets of the form s1W s2,where s1A D1and s2A D2and that hðD1ÃD2;tÞ¼hðD1;tÞhðD2;tÞ.The following proposition is essentially Proposition3.12in [12].Proposition2.3.Let P be an m-dimensional polytope in R m having a triangulation abstractly isomorphic to sÃD,where s is the vertex set of a simplex not contained in the boundary of P.Let V be the linear subspace of R m parallel to the a‰ne span of s.The boundary complex of the quotient polytope P=V L R m=V is abstractly isomorphic to FðPÞn s and inherits a triangulation abstractly isomorphic to D.3.Special simplicesThroughout this section P denotes an m-dimensional convex polytope in R q with face complex FðPÞ.Let S be a simplex spanned by n vertices of P.We call S a special simplex in P if each facet of P contains exactly nÀ1of the vertices of S.Note that,in particular,S is not contained in the boundary of P.Example3.1.Let P be the polytope of real doubly stochastic nÂn matrices.If v1;v2;...;v n are the nÂn permutation matrices corresponding to the elements of the cyclic subgroup of the symmetric group generated by the cycleð12ÁÁÁnÞ(or any n permu-tation matrices with pairwise disjoint supports)then v1;v2;...;v n are the vertices of a spe-cial simplex in P.Indeed,each facet of P is deﬁned by an equation of the form x ij¼0in R nÂn and misses exactly one of v1;v2;...;v n.Example3.2.Let W be a poset on the ground set½m :¼f1;2;...;m g which is graded of rank nÀ2(we refer to[17],Chapter3,for basic background and terminology on par-tially ordered sets)and P¼OðWÞbe the order polytope of W in R mþ1.Let W0be the poset obtained from W by adjoining a minimum element0and for1e i e n let v i be the char-acteristic vector of the ideal of elements of W0of rank at most iÀ1,so that v i is a vertex of P.Since a facet of P is deﬁned either by an equation of the form x i¼x j with i<j in W0 and i,j in successive ranks or by one of the form x i¼0for i A W0of rank nÀ1,it follows that v1;v2;...;v n are the vertices of a special simplex in P.Lemma3.3.Suppose that v1;v2;...;v n are the vertices of a special simplex in P.If F is a face of P of codimension k for some1e k e nÀ1and F does not contain any of v1;v2;...;v k then F must contain v i for all kþ1e i e n.Proof.Let S be the special simplex with vertices v1;v2;...;v n.Any codimension k face of a polytope can be written as the intersection of k facets,so we can writeF¼F1X F2XÁÁÁX F k where the F j are facets of P.For each1e i e k we have v i B F and hence v i B F j for some j¼j i.Since S is special the integers j1;j2;...;j k are all distinct and hence for each1e j e k we have v i B F j for some1e i e k,which in turn implies that v i A F j for all kþ1e i e n.It follows that v i A F1X F2XÁÁÁX F k¼F for all kþ1e i e n.rLemma3.4.Suppose that t¼ðv p;v pÀ1;...;v1Þis an ordering of the vertices of P such that s¼f v1;v2;...;v n g is the vertex set of a special simplex in P.Let D be the abstract simplicial complex on f v nþ1;...;v p g deﬁned by the reverse lexicographic triangulation of FðPÞn s with respect toðv p;v pÀ1;...;v nþ1Þ.(i)The reverse lexicographic triangulation D t of P is abstractly isomorphic to the sim-plicial join sÃD.(ii)D is abstractly isomorphic to the boundary complex of a simplicial polytope of dimension mÀnþ1.Proof.(i)Let s i¼f v1;...;v i g for0e i e n,so that s0¼j and s n¼s,and let D i denote the abstract simplicial complex on the set f v iþ1;...;v p g deﬁned by the reverse lexi-cographic triangulation of FðPÞn s i with respect to the orderingðv p;v pÀ1;...;v iþ1Þ.To prove that D0¼s nÃD n,which is the assertion in the lemma,we will prove that FðPÞn s i is pureðmÀiÞ-dimensional and that D0¼s iÃD i for all0e i e n by induction on i.This is obvious for i¼0so let1e i e n.By induction,any maximal face F of FðPÞn s iÀ1is a codimension iÀ1face of P.Since F does not contain any of the vertices v1;...;v iÀ1,by Lemma3.3we have v i A F.This implies that FðPÞn s i is pureðmÀiÞ-dimensional and that D iÀ1¼v iÃD i.The last equality and the induction hypothesis D0¼s iÀ1ÃD iÀ1imply that D0¼s iÃD i,which completes the induction.(ii)Let V be the linear subspace of R q parallel to the a‰ne span of the vertices in s and P=V be the corresponding quotient polytope of P,so that P=V has dimension mÀnþ1.Part(i)and Proposition2.3imply that D is abstractly isomorphic to a reverse lexicographic triangulation of the boundary complex of P=V.This is in turn isomorphic to the boundary complex of a simplicial polytope of dimension mÀnþ1by Lemma2.1.r The following theorem is the main result in this paper.Theorem3.5.Suppose that P is an integer polytope and t¼ðv p;v pÀ1;...;v1Þis an ordering of its vertices such that:(i)t is compressed and(ii)f v1;v2;...;v n g is the vertex set of a special simplex in P.ThenP r f0iðP;rÞt r¼hðtÞð1ÀtÞmþ1Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares 168where hðtÞ¼h0þh1tþÁÁÁþh d t d is the h-polynomial of the boundary complex of a sim-plicial polytope Q of dimension d¼mÀnþ1,so that hðtÞsatisﬁes the conditions in the g-theorem.In particular h i¼h dÀi for all i and1¼h0e h1eÁÁÁe h b d=2c.Moreover,Q can be chosen so that its boundary complex is abstractly isomorphic to the reverse lexicographic triangulation of FðPÞnf v1;...;v n g with respect to the ordering ðv p;v pÀ1;...;v nþ1Þ.Proof.Let s¼f v1;v2;...;v n g and let D denote the reverse lexicographic triangula-tion of FðPÞn s with respect to the orderingðv p;v pÀ1;...;v nþ1Þ.Lemma2.2guarantees that the proposed equation holds with hðtÞ¼hðD t;tÞ.Part(i)of Lemma3.4implies thathðD t;tÞ¼hðsÃD;tÞ¼hðs;tÞhðD;tÞ¼hðD;tÞ;since face complexes of simplices have h-polynomial equal to1,and the result follows from part(ii)of the same lemma.rWe now apply Theorem3.5to the Birkho¤polytope and to order polytopes of graded posets.Observe that our theorem does not apply to all integer polytopes,for instance to nonunimodular integer simplices.Magic squares and the Birkho¤polytope.An integer stochastic matrix,or magic square,is a square matrix with nonnegative integer entries having all line sums equal to each other,where a line is a row or a column.Let H nðrÞbe the number of nÂn magic squares with line sums equal to r.The fact that for anyﬁxed positive integer n the quantityH nðrÞis a polynomial in r of degreeðnÀ1Þ2wasﬁrst proved by Stanley[13](see also[5],[16],Section I.5,and[17],Section4.6).Let P be the polytope of real doubly stochastic nÂn matrices and observe that H nðrÞcoincides with the Ehrhart polynomial iðP;rÞ.Since P is a compressed integer polytope of dimensionðnÀ1Þ2,Theorem3.5and Example3.1 imply immediately the following corollary.Corollary3.6.For any positive integer n we haveP r f0H nðrÞt r¼hðtÞð1ÀtÞðnÀ1Þ2þ1where hðtÞ¼h0þh1tþÁÁÁþh d t d is the h-polynomial of the boundary complex of a sim-plicial polytope of dimension d¼n2À3nþ2,so that hðtÞsatisﬁes the conditions in the g-theorem.In particular h i¼h dÀi for all i and1¼h0e h1eÁÁÁe h b d=2c.In view of the last statement in Theorem3.5,the polytope in the previous corollary can be constructed by pulling in an arbitrary order the vertices of the quotient of P with respect to the a‰ne span of the vertices v1;v2;...;v n,chosen explicitly as in Example3.1.Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares169Eulerian polynomials and equatorial spheres.Let W be a graded poset on the ground set ½m :¼f 1;2;...;m g of rank n À2.Let W i be the set of elements of W of rank i À1for 1e i e n À1and L ðW Þbe the set of linear extensions of W ,meaning the set of permuta-tions w ¼ðw 1;w 2;...;w m Þof ½m for which w i <W w j implies i <j .We assume that W is naturally labeled ,meaning that the identity permutation ð1;2;...;m Þis a linear extension.The W -Eulerian polynomial is deﬁned asW ðW ;t Þ¼P w A L ðW Þt des ðw Þwheredes ðw Þ¼K f i A ½m À1 :w i >w i þ1gis the number of descents of w .Following [12]we call a function g :W !R equatorial if min a A W g ða Þ¼0and for each 2e i e n À1there exist a i À1A W i À1and a i A W i such that a i À1<W a i and g ða i À1Þ¼g ða i Þ.An ideal I or,more generally,a strictly increasing chain of ideals I 1H I 2H ÁÁÁH I k in W is equatorial if the characteristic function w I of I or the sum w I 1þw I 2þÁÁÁþw I k ,respectively,is equatorial.The equatorial complex D eq ðW Þ,in-troduced in [12],is the abstract simplicial complex on the vertex set of equatorial ideals of W whose simplices are the equatorial chains of ideals in W .The following theorem is proved in Corollary 3.8and Theorem 3.14of [12].Theorem 3.7(Reiner-Welker [12]).Let W be a naturally labeled ,graded poset on ½m having n À1ranks .The equatorial complex D eq ðW Þis abstractly isomorphic to the boundary complex of a simplicial polytope of dimension d ¼m Àn þ1which has h -polynomial equal to the W -Eulerian polynomial W ðW ;t Þ.Hence W ðW ;t Þsatisﬁes the conditions in the g -theorem and ,in particular ,it has sym-metric and unimodal coe‰cients .Let P be the order polytope of W and W 0be the poset obtained from W by adjoining a minimum element 0.Recall that the vertices of P are the characteristic vectors of the nonempty ideals of W 0.The order polytope comes with its canonical triangulation [18],[12],Proposition 2.1,which is a unimodular triangulation with maximal simplices bijecting to the linear extensions of W .This canonical triangulation is in fact the reverse lexicographic triangulation of O ðW Þwith respect to any ordering ðu p ;u p À1;...;u 1Þof its vertices such that i <j whenever the ideal of W 0deﬁned by u i is strictly contained in that deﬁned by u j .We will use the following lemma.Lemma 3.8.Let v i be the characteristic vector of the ideal of elements of W 0of rank at most i À1for 1e i e n .Let s ¼f v 1;...;v n g and t ¼ðv p ;...;v n þ1Þbe an ordering of the remaining vertices of P such that i <j whenever i ;j f n þ1and the ideal deﬁned by v i is strictly contained in that deﬁned by v j .The equatorial complex D eq ðW Þis the abstract simplicial complex deﬁned by the reverse lexicographic triangulation of F ðP Þn s with respect to t .Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares170Proof.Let F denote the face complex of P and let x^1¼0by convention.The maxi-mal faces of F n s are the faces of P deﬁned by systems of equations of the form x is ¼x jsfor0e s e nÀ1where(i)i0¼0and j nÀ1¼^1,(ii)i s A W s for1e s e nÀ1,j s A W sþ1 for0e s e nÀ2and i s<W j s for1e s e nÀ2and(iii)if j s¼i sþ1for consecutive values s¼a;aþ1;...;bÀ1of s then the interval½i a;j b in^W consists only of the elements of the chain i a<i aþ1<ÁÁÁ<i b<j b.The statement of the lemma follows from the description of the maximal faces of a reverse lexicographic triangulation DðFÞ(see Section2)and that of the maximal faces of D eqðWÞ(see[12],Proposition3.5).We omit the details.r Proof of Theorem3.7.Let P be the order polytope of W,as before.Observe that iðP;rÞis equal to the number of order reversing maps r:W!f0;1;...;r g.It follows from [17],Theorem4.5.14,thatP r f0iðP;rÞt r¼WðW;tÞð1ÀtÞmþ1:ð4ÞLet the vertices v1;v2;...;v p of P and t¼ðv p;...;v nþ1Þbe as in Lemma3.8.We checked in Example3.2that v1;v2;...;v n are the vertices of a special simplex in P.Since P is a compressed integer polytope(see Section2)Theorem3.5applies and we haveP r f0iðP;rÞt r¼hðtÞð1ÀtÞmþ1;where hðtÞ¼h0þh1tþÁÁÁþh d t d is the h-polynomial of a simplicial polytope of dimen-sion d¼mÀnþ1having,in view of Lemma3.8,boundary complex abstractly isomorphic to D eqðWÞ.Comparison with(4)yields hðtÞ¼WðW;tÞand completes the proof.r4.Rational polyhedral cones and semigroup ringsIn this section we consider the a‰ne semigroup ring corresponding to an integer polytope and state several corollaries of Theorem3.5,including a generalization of Corol-lary3.6to magic labelings of bipartite graphs.For terminology and background related to semigroup rings we refer the reader to[2],Chapter6.Let P be an m-dimensional integer polytope in R q.We denote by R P the subalgebraof the algebra k½x1;...;x q;xÀ11;...;xÀ1q;t of Laurant polynomials over aﬁeld k generatedby the monomials x a t r for positive integers r and a A Z q such that a=r A P.The algebra R P can be graded by letting x a t r have degree r.With this grading R P is a normal,Cohen-Macaulay,graded commutative ring whose Hilbert series is the Ehrhart series of P.Let ~P¼fð1;xÞ:x A P g be the lift of P in the hyperplane x¼1in R qþ1.We denote by C P the cone in R qþ1generated by~P and by E P the semigroup of integer points in C P.We also denote by E P the set of points of E P which lie in the relative interior of C P.It is known that the ring R P is Gorenstein if and only if E P has a unique minimal element,meaning an ele-ment b A E P such that E P¼bþE P;see for instance[2],Corollary6.3.8.Corollary4.1.Let P be an m-dimensional integer polytope in R q.If(i)R P is Gorenstein andAthanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares171(ii)there exists a compressed ordering t ¼ðv p ;v p À1;...;v 1Þof the vertices of ~Psuch that v 1þv 2þÁÁÁþv n is equal to the unique minimal element b of E P for some n ,then the conclusion of Theorem 3.5holds .Proof.We denote by L the linear span of C P in R q þ1and by A a set of integer linear forms in R q þ1,one for each facet of P ,deﬁning the cone C P as the set of points x A L satisfying g ðx Þf 0for all g A A .In view of Theorem 3.5it su‰ces to show that f v 1;v 2;...;v n g is the vertex set of a special simplex in ~P.Let F be a facet of ~P and let f be the linear form in A corresponding to F .We need to show that exactly one of v 1;v 2;...;v n satisﬁes f ðv i Þ>0.Clearly at least one of them has this property,since f ðb Þ>0.Assume that at least two of v 1;v 2;...;v n satisfy f ðv i Þ>0,say v j is one of them,and let f ðb Þ¼band f ðv j Þ¼c ,so that 1e c <b .Since F is a facet of ~P there exists a point x in the a‰ne span of ~P,which we may assume to have rational coordinates,satisfying f ðx Þ<0and g ðx Þ>0for all g A A other than f .By replacing x with a suitable positive integer multiple we ﬁnd an integer point a in L satisfying f ða Þ<0and g ða Þ>0for all g A A other than f .Letting a ¼f ða Þ,we may choose a nonnegative integer t so that 0<a þb þtc <b .Then g ¼a þb þtv j is in E P and satisﬁes f ðg Þ<f ðb Þ,which contradicts the minimality of b .rRecall that R P is standard if it is generated by its homogeneous elements of degree one or,equivalently,if E P is generated as a semigroup by the integer points in ~P.Clearly this holds if P has a unimodular triangulation.The next corollary provides an instance in which a conjecture of Hibi [7],Conjecture 1.5,and Stanley (see [19],Conjecture 4a),stating that the h -vector of a standard,graded,Gorenstein domain has unimodal coe‰cients,can be answered in the a‰rmative.Corollary 4.2.Let P be an m -dimensional integer polytope in R q .If(i)P is compressed and(ii)R P is Gorenstein ,then the conclusion of Theorem 3.5holds where ,in the statement of the theorem ,n is the x 0-coordinate of the unique minimal element of E P .Proof.Let b be the unique minimal element of E P ,whose existence is guaranteed by (ii).Since ~Phas unimodular triangulations (having vertex set that of ~P ),the vertices of ~P must be its only integer points.Hence the fact that R P is standard implies that b ¼v 1þv 2þÁÁÁþv n for some vertices v 1;v 2;...;v n of ~P ,which must be pairwise distinct by the minimality of b .Because of (i)any ordering ðv p ;v p À1;...;v 1Þof the vertices of ~P satisﬁes the assumptions of Corollary 4.1.The result follows from this corollary observing that b has x 0-coordinate equal to n .rGeneral conditions on P which guarantee that P is compressed were given by Ohsugi and Hibi [11].Let F be a p Âq matrix with integer entries.Let u A Z p and suppose that P is deﬁned by the linear equalities and inequalitiesF x ¼u and 0e x i e 1for 1e i e q ;ð5Þwhere x ¼ðx 1;x 2;...;x q ÞA R q .Athanasiadis,Ehrhart polynomials,simplicial polytopes and magic squares172。

专业英语课文翻译

School of chemical engineering and pharmaceuticaltest tubes 试管 test tube holder试管夹test tube brush 试管刷 test tube rack试管架beaker烧杯stirring搅拌棒thermometer温度计 boiling flask长颈烧瓶 Florence flask平底烧瓶flask,round bottom,two-neck boiling flask,three-neckconical flask锥形瓶 wide-mouth bottle广口瓶graduated cylinder量筒gas measuring tube气体检测管volumetric flask容量瓶transfer pipette移液管Geiser burette(stopcock)酸式滴定管funnel漏斗Mohr burette(with pinchcock)碱式滴定管watch glass表面皿 evaporating dish蒸发皿 ground joint磨口连接Petri dish有盖培养皿desiccators干燥皿long-stem funnel长颈漏斗filter funnel过滤漏斗Büchner funnel瓷漏斗 separatory funnel分液漏斗Hirsh funnel赫尔什漏斗 filter flask 吸滤瓶Thiele melting point tube蒂勒熔点管plastic squeeze bottle塑料洗瓶medicine dropper药用滴管rubber pipette bulb 吸球microspatula微型压舌板pipet吸量管mortar and pestle研体及研钵filter paper滤纸Bunsenburner煤气灯burette stand滴定管架support ring支撑环ring stand环架 distilling head蒸馏头side-arm distillation flask侧臂蒸馏烧瓶air condenser空气冷凝器centrifuge tube离心管fractionating column精(分)馏管Graham condenser蛇形冷凝器crucible坩埚 crucible tongs坩埚钳beaker tong烧杯钳economy extension clamp经济扩展夹extension clamp牵引夹utility clamp铁试管夹 hose clamp软管夹burette clamppinchcock;pinch clamp弹簧夹 screw clamp 螺丝钳ring clamp 环形夹 goggles护目镜stopcock活塞wire gauze铁丝网analytical balance分析天平分析化学absolute error绝对误差 accuracy准确度 assay化验analyte(被)分析物 calibration校准constituent成分coefficient of variation变异系数confidence level置信水平detection limit检出限 determination测定 estimation 估算equivalent point等当点 gross error总误差impurity杂质indicator指示剂interference干扰internal standard 内标level of significance显着性水平 limit of quantitation定量限 masking掩蔽matrix基体 precision精确度primary standard原始标准物purity纯度qualitative analysis定性分析quantitative analysis定量分析random error偶然误差 reagent试剂relative error相对误差 robustness耐用性 sample样品relative standard deviation相对标准偏差selectivity选择性sensitivity灵敏度 specificity专属性 titration滴定significant figure有效数字solubility product溶度积standard addition标准加入法standard deviation标准偏差standardization标定法stoichiometric point化学计量点systematic error系统误差有机化学acid anhydride 酸酐 acyl halide 酰卤alcohol 醇aldehyde 醛aliphatic 脂肪族的alkene 烯烃alkyne炔 allyl烯丙基amide氨基化合物 amino acid 氨基酸aromatic compound 芳香烃化合物amine胺 butyl 丁基aromatic ring芳环,苯环branched-chain支链 chain链carbonyl羰基carboxyl羧基chelate螯合chiral center手性中心conformers构象copolymer共聚物derivative 衍生物dextrorotatary右旋性的diazotization重氮化作用dichloromethane二氯甲烷ester酯ethyl乙基 fatty acid脂肪酸functional group 官能团general formula 通式 glycerol 甘油，丙三醇heptyl 庚基heterocyclie 杂环的hexyl 己基 homolog 同系物hydrocarbon 烃,碳氢化合物hydrophilic 亲水的hydrophobic 疏水的hydroxide 烃基ketone 酮 levorotatory左旋性的methyl 甲基molecular formula分子式monomer单体 octyl辛基open chain开链optical activity旋光性（度）organic 有机的organic chemistry 有机化学organic compounds有机化合物pentyl戊基 phenol苯酚phenyl苯基polymer 聚合物，聚合体 propyl丙基ring-shaped环状结构zwitterion兼性离子saturated compound饱和化合物side chain侧链straight chain 直链tautomer互变（异构）体structural formula结构式triglyceride甘油三酸脂unsaturated compound不饱和化合物物理化学activation energy活化能 adiabat绝热线 amplitude振幅collision theory碰撞理论empirical temperature假定温度enthalpy焓enthalpy of combustion燃烧焓enthalpy of fusion熔化热 enthalpy of hydration水合热 enthalpy of reaction 反应热enthalpy o f sublimation升华热enthalpy of vaporization汽化热entropy熵first law热力学第一定律first order reaction一级反应free energy自由能 Hess’s law 盖斯定律Gibbs free energy offormation吉布斯生成能heat capacity热容 internal energy 内能 isobar等压线 isochore等容线isotherm等温线 kinetic energy动能latent heat潜能Planck’s constant 普朗克常数 potential energy势能quantum量子quantum mechanics量子力学rate law 速率定律specific heat比热spontaneous自发的standard enthalpy change标准焓变standard entropy of reaction标准反应熵standard molar entropy标准摩尔熵standard pressure标压state function状态函数thermal energy热能thermochemical equation热化学方程式thermodynamic equilibrium热力学平衡uncertainty principle测不准定理zero order reaction零级反应 zero point energy零点能课文词汇实验安全及记录：eye wash眼药水 first-aid kit急救箱gas line输气管safety shower紧急冲淋房water faucet水龙头flow chart流程图 loose leaf活页单元操作分类：heat transfer传热Liquid-liquid extraction液液萃取liquid-solid leaching过滤 vapor pressure蒸气压membrane separation薄膜分离空气污染：carbon dioxide 二氧化碳carbon monoxide一氧化碳particulate matter颗粒物质photochemical smog光化烟雾primary pollutants一次污染物secondary pollutants二次污染物stratospheric ozone depletion平流层臭氧消耗sulfur dioxide二氧化硫 volcanic eruption火山爆发食品化学：amino acid氨基酸,胺 amino group 氨基empirical formula实验式,经验式fatty acid脂肪酸peptide bonds肽键polyphenol oxidase 多酚氧化酶salivary amylase唾液淀粉酶 steroid hormone甾类激素table sugar蔗糖 triacylglycerol 三酰甘油,甘油三酯食品添加剂：acesulfame-K乙酰磺胺酸钾,一种甜味剂adrenal gland肾上腺ionizing radiation致电离辐射food additives食品添加剂monosodium glutamate味精,谷氨酸一钠（味精的化学成分） natural flavors天然食用香料,天然食用调料nutrasweet天冬甜素potassium bromide 溴化钾propyl gallate没食子酸丙酯 sodium chloride氯化钠sodium nitraten硝酸钠 sodium nitrite亚硝酸钠trans fats反式脂肪genetic food转基因食品food poisoning 食物中毒hazard analysis and critical control points (HACCP)危害分析关键控制点技术maternal and child health care妇幼保健护理national patriotic health campaign committee(NPHCC) 全国爱国卫生运动委员会 rural health农村卫生管理the state food and drugadministration (SFDA)国家食品药品监督管理局光谱：Astronomical Spectroscopy天文光谱学Laser Spectroscopy激光光谱学 Mass Spectrometry质谱Atomic Absorption Spectroscopy原子吸收光谱Attenuated Total Reflectance Spectroscopy衰减全反射光谱Electron Paramagnetic Spectroscopy 电子顺磁谱Electron Spectroscopy电子光谱Infrared Spectroscopy红外光谱Fourier Transform Spectrosopy傅里叶变换光谱Gamma-ray Spectroscopy伽玛射线光谱Multiplex or Frequency-Modulated Spectroscopy 复用或频率调制光谱X-ray SpectroscopyX射线光谱色谱：Gas Chromatography气相色谱High Performance Liquid Chromatography高效液相色谱Thin-Layer Chromatography薄层色谱magnesium silicate gel硅酸镁凝胶retention time保留时间mobile phase流动相 stationary phase固定相反应类型：agitated tank搅拌槽catalytic reactor催化反应器batch stirred tank reactor间歇搅拌反应釜continuous stirred tank 连续搅拌釜exothermic reactions放热反应pilot plant试验工厂fluidized bed Reactor流动床反应釜multiphase chemical reactions 多相化学反应packed bed reactor填充床反应器redox reaction氧化还原反应reductant-oxidant氧化还原剂acid base reaction酸碱反应additionreaction加成反应chemical equation化学方程式 valence electron 价电子combination reaction化合反应 hybrid orbital 杂化轨道decomposition reaction分解反应substitution reaction取代(置换)反应Lesson5 Classification of Unit Operations单元操作Fluid flow流体流动它涉及的原理是确定任一流体从一个点到另一个点的流动和输送。

MediterraneanJournalofMathematics

Mediterranean Journal of MathematicsVol. 10, No. 4(2013), pages 1609-2028ContentsS. Capparelli and P. MarosciaOn Two Sequences of Orthogonal Polynomials Relatedto Jordan Blocks ….…………………………………………………………………….1609-1630 G. Aranda Pino and L. VašNoetherian Leavitt Path Algebras and Their Regular Algebras………………………..1631-1654 S. Ali, A. Fošner, M. Fošner and M. S. KhanOn Generalized Jordan Triple ( α,β )*- Derivations and related Mappings……….…..1655-1666 H. LarkiLeavitt Path Algebras of Edge-Colored Graphs……………….…..…… . …………….1667-1682 F. Qi and C. BergComplete Monotonicity of a Difference Between the Exponential and TrigammaFunctions and Properties Related to a Modified Bessel Function ……………………..1683-1694 M. Krnić, J. Pečarić and P. VukovićA Unified Treatment of Half-Discrete Hilbert-Type Inequalities witha Homogeneous Kernel…………………………………………………………………..1695-1714 B. LisenaAsymptotic Properties in a Delay Differential Inequality withPeriodic Coefficients…………………………………………………………………….1715-1728 Y. Jalilian and R. JalilianExistence of Solution for Delay Fractional Differential Equations…………………….1729-1746 J. Llibre and C. VallsGeneralized Weierstrass Integrability of the Abel Differential Equations………………1747-1758 P. Álvarez-Caudevilla, J. D. Evans and V. A. GalaktionovThe Cauchy problem for a tenth-order thin film equation I. Bifurcationof oscillatory fundamental solutions…………………………………………..…………1759-1790 A. BoughammouraHomogenization of a Highly Heterogeneous Elastic-ViscoelaticComposite Materials……………………………………………………………………...1791-1810M. StojanovicWave Equation Driven by Fractional Generalized Stochastic Processes………..………1811-1829 C. Bardaro and I. MantelliniOn Linear Combinations of Multivariate Generalized Sampling Type Series……………1831-1850 C. Ferreira, J. L. López and E.Pérez SinusíaThe Second Appell Function for one Large Variable…………………………….……….1851-1864 B. WróbelLaplace Type Multipliers for Laguerre Expansions of Hermite Type…………………….1865-1879 A. OsękowskiSharp Logarithmic Bounds for Beurling-Ahlfors Operator Restrictedto the Class of Radial Functions……………………………………………………..……1881-1892 M. Baronti and P. L. PapiniOn Some Types of Rigid Sets in Banach Spaces…………………………………………..1893-1901 Y. Yagoub-ZidiSome Isometric Properties of Subspaces of Function Spaces…………………………….1903-1913 S. Lambert, K. Lee and A. LuttmanOn the Generality of Assuming that a Family of ContinuousFunctions Separates Points…………………………………………………………..……1915-1933 I. Akbarbaglu and S. MaghsoudiBanach-Orlicz Algebras on a Locally Compact Group…………………………………...1935-1945 M. Abel and R. M. Pérez-TiscareñoLocally Pseudoconvex Inductive Limit of Topological Algebras………………….………1947-1961 P. Aiena, J. R. Guillén and P. PeñaLocalized SVEP, property (b) and property (ab)………………………………………….1963-1974 I. K. Erken and C. MurathanA Class of 3-Dimensional Contact Metric Manifolds…………………………………..…1975-1991 D. E. Dobbs and J. ShapiroOn the Strong (A)-Rings of Mahdou and Hassani…………………………………………1993-1995 M. De Falco, F. de Giovanni and C. MusellaGroups whose Proper Subgroups of Infinite Rank Have a TransitiveNormality Relation…………………………………………………………………………………. 1997-2004 S. SamkoVariable Exponent Herz Spaces………………………………………………………..….2005-2023S. SamkoErratum to “Variable exponent Herz spaces”, Mediterr. J. Math.DOI: 10.1007/s00009-013-0285-x, 2013………………………………………………….2025-2028。

PolesandZerosofTransferFunctions：传递函数的零点和极点

Poles and Zeros of Transfer FunctionsMost circuits we consider, in practice, are discrete (circuit parameters such as R s, L s and C s, appear in discrete lumps, rather than smears, as they do intransmission lines), time invariant (the values of circuit parameters such as R s, L s and C s, so not change with time), and linear (the i-v relationships of the circuit elements, such as R s, L s and C s, are given by mathematically linear relations of i , v and their derivatives). For such ordinary circuits, all transfer functions can be written as the ratio of polynomials in the variable, s :()()()N s T s D s = where ()N s is the numerator polynomial and ()D s is the denominator polynomial.A function such as ()T s which can be expressed as a ratio of polynomials iscalled a rational function . A polynomial in s , recall, is a weighted sum of the powers of s up to some finite maximum power of s . For example,()212102P s s s =++is a polynomial. Because its highest power of s is 2, this particular polynomial is said to be a second-degree polynomial. (The function s e -, as an example, is not a polynomial because its Taylor series expansion is an infinite series of terms of increasingly higher powers of s .)The degree of the denominator polynomial, ()D s , for a transfer function basically is given by the number of energy storage elements (inductors and capacitors) in the circuit, although series or parallel combinations of inductors or capacitors that can be reduced to a single element count only once. The degree of the numerator polynomial, ()N s , is usually less than or equal to the degree of the denominator polynomial.Any polynomial of degree n can be factored into the product of n terms of the form ()s a +, where a is a constant. For example, we can factor thepolynomial ()212102P s s s =++ as follows:()()()212102223P s s s s s =++=++In this example, the constants in each factor are real integers. This circumstance is atypical. Often, the constants are complex irrational numbers. For example, the polynomial ()21P s s s =-+ requires complex constants in its factors:()211122P s s s s s ⎛=-+=-+-- ⎝⎭⎝⎭ If ()N s is a polynomial of degree m and if ()D s is a polynomial of degree n , then we can write transfer functions in factored form:()()()()()()()1212m n s z s z s z T s K s p s p s p --⋅⋅⋅-=--⋅⋅⋅- where the constants 12,,...,m z z z are called the zeros of the transfer function, ()T s , the constants 12,,...,n p p p are called the poles of the transfer function and K is some constant. The poles and zeros are not necessarily unique. That is,some factors may be repeated. Note that ()T s is completely specified by its poles, its zeros and the constant K . That is, ()T s can be reconstructed exactly if we know these 1m n ++ quantities.Should the value of s ever equal any one of the zeros, 12,,...,m z z z , say k z , of ()T s , note that()0k T z =Hence, the name zeros. Of course some of the zeros may be complex, so for these, ()T s achieves zero value only when s takes on complex values. We thus will find it necessary to consider s to be a complex variable. That is, it can take on values in the complex s -plane, a plane with real and imaginary axes. In thiscomplex s-plane, for example, we can plot the zeros112z j =+ 212z=-that we found earlier for the polynomial ()211122P s s s s s ⎛=-+=-+-- ⎝⎭⎝⎭as follows:In this example, note that the two zeros of ()P s lie in symmetric positions with respect to the horizontal axis. In the language of complex numbers, z 1 and z 2 are said to be complex conjugates : their real parts are identical and their imaginary parts have the same magnitude but opposite signs. If two numbers, z 1 and z 2, are complex conjugates, the customary notation is that 21*z z = or, of course, 12*z z =. If we think of conjugation as an operator, it has the effect of reversing the sign ofthe imaginary part of a complex number. For example, if112z j =+and 212z =-then12*11*2222z j j z ⎛=+=- ⎝⎭We now demonstrate an important result known as the conjugate law for transfer functions : the poles and zeros of a transfer function,()()()N s T s D s = either (1) lie on the real axis of the s -plane or (2) occur in complex conjugate pairs.To begin demonstration of this result, note that the coefficients of s in ()N s and ()D s are real numbers because they are just combinations of the various (real) circuit parameters (R s, L s, Cs and so forth). For any polynomial, ()P s , with real coefficients for the powers of s , it is true that()()**P s P s = For a specific example, suppose()246116P s s s =++in which, we note, all coefficients of s are real. Thus, we see()()()2***46116P s s s =++ ()()**246116P s s s =++()()()()***P s P s P s =≡ The key to obtaining this result is that the coefficients of s in the polynomial are real.We now apply this result separately to the polynomials ()N s and ()D s . For any of the zeros of the transfer function, say, k z , we have()0k N z =If we take the complex conjugate of this equation, we see()*0k N z =From the conjugate law, we have()*0k N z = Thus, we have the result that if k z is a zero of ()T s , then *k z also is a zero of()T s . Thus, unless k z is real, it is a member of a complex conjugate pair of zeros. We conclude that the zeros of ()T s either are real or occur in complex conjugate pairs.What about the poles of ()T s ? For any of the poles, say, k p , we have()0k D p =If we take the complex conjugate of this equation, we see()*0k D p =From the conjugate law, we have()*0k D p =Thus, we have the result that if k p is a pole of ()T s , then *k p also is a pole of ()T s . Thus, unless k p is real, it is a member of a complex conjugate pair of poles.We conclude that the poles of ()T s either are real or occur in complex conjugate pairs.These two results, for the poles and zeros of ()T s , demonstrate the conjugate law for ()T s . Because of the conjugate law, a plot in the s -plane of the locations of the poles and zeros of ()T s are always symmetric about the real axis in the s -plane. Consider, for example,Because a pole-zero plot such as this shows the location of all poles and zeros of ()T s and because ()T s can be constructed from its poles and zeros (plus the constant, K , which can be determined if the value of ()T s is known for a single value of s ), the pole-zero plot can serve as an alternative representation to the rational (ratio of polynomials) function representation of ()T s .We have already mentioned that inverting ()out V s to obtain ()out v t can involve a lot of effort. This difficulty provides strong motivation to learn as much as possible about the behavior of ()out v t from investigating ()out V s without going through the trouble of inverting it. After all, ()out V s contains precisely the same information as ()out v t since ()out v t can be reconstructed from ()out V s . Laplace transform theoryprovides two results, the initial value and the final value theorems, that give bits of information about ()out v t directly from ()out V s : ()()0lim out out s v sV s +→∞= ()()0lim out out s v sV s +→∞=These results, valid when the limits exist, give the initial and final values of ()out v t directly from ()out V s without inverting ()out V s . The final value theorem can give information about the ultimate stability of an amplifier – if ()out v ∞→∞, then the amplifier clearly is unstable. Unfortunately, the output of unstable amplifiers typically is an exponentially growing sinusoid, which oscillates between ±∞. In such cases, the limit in the final value theorem does not exist, so that theorem provides no information. We obtain a more useful test for stability by applying a partial fraction expansion to ()out V s .For present purposes, we consider an amplifier to be stable if, in response to an impulse, ()t δ, to its input, the output, ()out v t , of the amplifier eventuallyapproaches zero. By applying an impulse to the input, we are applying all possible frequencies simultaneously. We will therefore excite any instability in the amplifier, regardless of its frequency. If the input is an impulse at 0t =, then we have()(){}(){}1in in V s L v t L t δ===where {}L ⋅ indicates Laplace transformation. Thus, for the purpose of examining stability of an amplifier, we have:()()()()()()()()()()1212m out in n s z s z s z V s T s V s T s K s p s p s p --⋅⋅⋅-===--⋅⋅⋅- The idea of a partial fraction expansion is that a rational function, such as ()T s , can be expanded as the sum of simpler rational functions, each of which has only one pole (although, in certain cases the pole can be repeated). If, for simplicity, we assume that none of the poles of ()T s is repeated, its partial fraction expansion takes the form()1212......k n k nA A A A T s s p s p s p s p ≡+++++---- where ,1,2,...,k A k n =, are complex constants that are chosen so that the identity holds for all s . If ()T s has repeated poles, then its partial fractionexpansion is slightly more complicated, but our main results do not change. As we’ve seen, each of the poles 12,,...,n p p p can have both real and imaginaryparts: k k k p j οω=+Thus, we can write()121122......k n out k k n nA A A A V s s j s j s j s j οωοωοωοω=+++++-------- By taking the inverse Laplace transform (from a table of transforms), we see()112212......k k n n t j t t j t t j t t j t out k n v t Ae A e A e A e οωοωοωοω++++=+++++ Because the poles of ()T s are either real or occur in complex conjugate pairs, more detailed calculations show that the terms can be grouped to show explicitly that the right hand side of this equation is real, as required (since ()out v t must be real). Those details are of little concern to our stability analysis, however. We note that if all of the 0,1,2,...,k k n ο<=, then each term produces damped oscillations, so that the output, ()out v t , approaches zero for long times. That is, an amplifier is stable if all poles of its transfer function, ()T s , lie in the left half of the s-plane . If even one 0k ο≥, however, then the corresponding term produces exponentially growing oscillations for long times. That is , an amplifier is unstable if at least one pole of its transfer function lies in the right half s-plane . Incidentally, we consider a transfer function with poles on the imaginary axis to be unstable, even though the corresponding terms do not approach infinity because ()out v t does not approach zero after long times. It oscillates.。

最全高分子英语课文翻译

课文翻译第一单元什么是高聚物？什么是高聚物？首先，他们是合成物和大分子，而且不同于低分子化合物，譬如说普通的盐。

与低分子化合物不同的是，普通盐的分子量仅仅是58.5，而高聚物的分子量高于105 ，甚至大于106 。

这些大分子或“高分子”由许多小分子组成。

小分子相互结合形成大分子，大分子能够是一种或多种化合物。

举例说明，想象一组大小相同并由相同的材料制成的环。

当这些环相互连接起来，可以把形成的链看成是具有同种分子量化合物组成的高聚物。

另一方面，独特的环可以大小不同、材料不同，相连接后形成具有不同分子量化合物组成的聚合物。

许多单元相连接给予了聚合物一个名称，poly意味着“多、聚、重复”，mer意味着“链节、基体”（希腊语中）。

例如：称为丁二烯的气态化合物，分子量为54，化合将近4000次，得到分子量大约为200000被称作聚丁二烯（合成橡胶）的高聚物。

形成高聚物的低分子化合物称为单体。

下面简单地描述一下形成过程：丁二烯＋丁二烯＋…＋丁二烯——→聚丁二烯（4000次）因而能够看到分子量仅为54的小分子物质（单体）如何逐渐形成分子量为200000的大分子（高聚物）。

实质上，正是由于聚合物的巨大的分子尺寸才使其性能不同于象苯这样的一般化合物。

例如，固态苯，在5.5℃熔融成液态苯，进一步加热，煮沸成气态苯。

与这类简单化合物明确的行为相比，像聚乙烯这样的聚合物不能在某一特定的温度快速地熔融成纯净的液体。

而聚合物变得越来越软，最终，变成十分粘稠的聚合物熔融体。

将这种热而粘稠的聚合物熔融体进一步加热，不会转变成各种气体，但它不再是聚乙烯（如图1.1）。

固态苯——→液态苯——→气态苯加热，5.5℃加热，80℃固体聚乙烯——→熔化的聚乙烯——→各种分解产物-但不是聚乙烯加热加热图1.1 低分子量化合物（苯）和聚合物（聚乙烯）受热后的不同行为发现另一种不同的聚合物行为和低分子量化合物行为是关于溶解过程。

例如，让我们研究一下，将氯化钠慢慢地添加到固定量的水中。

英语数学词汇

Algebraic Equation代数方程Elementary Operations-Addition基础混算-加法Elementary Operations-Subtaction基础混算-减法Elementary Operations-Multiplication 基础混算-乘法Elementary Operations-Division基础混算-除法Elementary Operation基础四则混算Decimal Operations小数混算Fractional Operations分数混算Convert fractional no. into decimal no. 分数转小数Convert fractional no. into percentage. 分数转百分数Convert decimal no. into percentage. 小数转百分数Convert percentage into decimal no. 百分数转小数Percentage百分数Numerals数字符号Common factors and multiples公因子及公倍数Sorting数字排序Area图形面积Perimeter图形周界Change Units : Time单位转换-时间Change Units : Weight单位转换-重量Change Units :Length单位转换-长度Directed Numbers有向数Fractional Operations分数混算Decimal Operations小数混算Convert fractional no. into decimal no. 分数转小数Convert fractional no. into percentage.分数转百分数Convert decimal no. into percentage.小数转百分数Convert percentage into decimal no.百分数转小数Percentage百分数Indices指数Algebraic Substitution代数代入Polynomials多项式Co-Geometry坐标几何学Solving Linear Equation解一元线性方程Solving Simultaneous Equation解联立方程Slope直线斜率Equation of Straight Line直线方程x-intercept ( Equation of St. Line )直线x轴截距y-intercept ( Equation of St. Line )直线y轴截距Factorization因式分解Quadratic Equation二次方程x-intercept ( Quadratic Equation )二次曲线x轴截距Geometry几何学Inequalities不等式Rate and Ratio比和比例Bearing方位角Trigonometry三角学Probability概率Statistics-Graph统计学-统计图表Statistics-Measure of central tendency统计学-量度集中趋势Salary Tax薪俸税Bridging Game汉英对对碰Indices指数Function函数Rate and Ratio比和比例Trigonometry三角学Inequalities不等式Linear Programming线性规划Co-Geometry坐标几何学Slope直线斜率Equation of Straight Line直线方程x-intercept ( Equation of St. Line )直线x轴截距y-intercept ( Equation of St. Line )直线y轴截距Factorization因式分解Quadratic Equation二次方程x-intercept ( Quadratic Equation )二次曲线x轴截距Method of Bisection分半方法Polynomials多项式Probability概率Statistics-Graph统计学-统计图表Statistics-Measure of central tendency统计学-量度集中趋势Statistics-Measure of dispersion统计学-量度分布Statistics-Normal Distribution统计学-正态分布Surds根式Probability概率Statistics-Measure of dispersion 统计学-量度离差Statistics-Normal Distribution统计学-正态分布Statistics-Binomial Distribution 统计学Statistics-Poisson Distribution 统计学Statistics-Geometric Distribution 统计学Co-Geometry坐标几何学Sequence序列十万Hundred thousand三位数3-digit number千Thousand千万Ten million小数Decimal分子Numerator分母Denominator分数Fraction五位数5-digit number公因子Common factor公倍数Common multiple中国数字Chinese numeral平方Square平方根Square root古代计时工具Ancient timing device古代记时工具Ancient time-recording device 古代记数方法Ancient counting method古代数字Ancient numeral包含Grouping四位数4-digit number四则计算Mixed operations (The four operations)加Plus加法Addition加法交换性质Commutative property of addition未知数Unknown百分数Percentage百万Million合成数Composite number多位数Large number因子Factor折扣Discount近似值Approximation阿拉伯数字Hindu-Arabic numeral定价Marked price括号Bracket计算器Calculator差Difference真分数Proper fraction退位Decomposition除Divide除法Division除数Divisor乘Multiply乘法Multiplication乘法交换性质Commutative property of multiplication乘法表Multiplication table乘法结合性质Associative property of multiplication被除数Dividend珠算Computation using Chinese abacus倍数Multiple假分数Improper fraction带分数mixed number现代计算工具Modern calculating devices售价Selling price万Ten thousand最大公因子Highest Common Factor (H.C.F.)最小公倍数Lowest Common Multiple (L.C.M.)减Minus / Subtract减少Decrease减法Subtraction等分Sharing等于Equal进位Carrying短除法Short division单数Odd number循环小数Recurring decimal零Zero算盘Chinese abacus亿Hundred million增加Increase质数Prime number积Product整除性Divisibility双数Even number罗马数字Roman numeral数学 mathematics, maths(BrE), math(AmE) 公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis, hypotheses(pl.)命题 proposition算术 arithmetic加 plus(prep.), add(v.), addition(n.)被加数 augend, summand加数 addend和 sum减 minus(prep.), subtract(v.), subtraction(n.) 被减数 minuend减数 subtrahend差 remainder乘 times(prep.), multiply(v.), multiplication(n .)被乘数 multiplicand, faciend乘数 multiplicator积 product除 divided by(prep.), divide(v.), division(n.) 被除数 dividend除数 divisor商 quotient等于 equals, is equal to, is equivalent to大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than运算符 operator平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n 次方根倒数（reciprocal） x的倒数为1/x有理数 rational number无理数 irrational number实数 real number虚数 imaginary number数字 digit数 number自然数 natural number整数 integer小数 decimal小数点 decimal point分数 fraction分子 numerator分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制 hexadecimal system权 weight, significance进位 carry截尾 truncation四舍五入 round下舍入 round down上舍入 round up有效数字 significant digit无效数字 insignificant digit代数 algebra公式 formula, formulae(pl.)单项式 monomial多项式 polynomial, multinomial系数 coefficient未知数 unknown, x-factor, y-factor, z-factor等式，方程式 equation一次方程 simple equation二次方程 quadratic equation三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数，幂 exponent乘方 power二次方，平方 square三次方，立方 cube四次方 the power of four, the fourth powern次方 the power of n, the nth power开方 evolution, extraction二次方根，平方根 square root三次方根，立方根 cube root四次方根 the root of four, the fourth rootn次方根 the root of n, the nth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量 constant变量 variable坐标系 coordinates坐标轴 x-axis, y-axis, z-axis横坐标 x-coordinate纵坐标 y-coordinate原点 origin象限quadrant截距(有正负之分)intercede（方程的）解solution几何geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 right angle钝角 obtuse angle平角 straight angle周角 perigon底 base边 side高 height三角形 triangle锐角三角形 acute triangle直角三角形 right triangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem钝角三角形 obtuse triangle不等边三角形 scalene triangle等腰三角形 isosceles triangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram矩形 rectangle长 length宽 width周长 perimeter面积 area相似 similar全等 congruent三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine反余弦 arc cosine反正切 arc tangent反余切 arc cotangent反正割 arc secant反余割 arc cosecant补充：集合aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field of definition 值域 range单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列，级数 series微积分 calculus微分 differential导数 derivative极限 limit无穷大 infinite(a.) infinity(n.)无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefinite integral复数 complex number矩阵 matrix行列式 determinant圆 circle 圆心 centre(BrE), center(AmE)半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference轨迹 locus, loca(pl.)平行六面体 parallelepiped立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron旋转 rotation轴 axis球 sphere半球 hemisphere底面 undersurface表面积 surface area体积 volume空间 space双曲线 hyperbola抛物线 parabola四面体 tetrahedron五面体 pentahedron六面体 hexahedron菱形 rhomb, rhombus, rhombi(pl.), diamond正方形 square梯形 trapezoid直角梯形 right trapezoid等腰梯形 isosceles trapezoid五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateral polygon相位 phase周期 period振幅 amplitude内心 incentre(BrE), incenter(AmE)外心 excentre(BrE), excenter(AmE)旁心 escentre(BrE), escenter(AmE)垂心 orthocentre(BrE), orthocenter(AmE)重心 barycentre(BrE), barycenter(AmE)内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average方差 variance标准差 root-mean-square deviation, standard deviation比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率，或然率 probability分布 distribution正态分布 normal distribution非正态分布 abnormal distribution图表 graph条形统计图 bar graph柱形统计图 histogram折线统计图 broken line graph曲线统计图 curve diagram扇形统计图 pie diagramEnglish Chineseabbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线；分角线 angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西 - 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲 congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲 ;圆锥曲conic section 二次曲 ;圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer's rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre's theorem 棣美弗定理De Morgan's law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法)determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的)discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns (variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler's formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of…以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat's last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron's formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder's Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke's law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射。

核专业英语背诵材料2.0

Unit 1Helium---------------------氦uranium------------铀Gaseous state-----------气态的artificially------------人工的The perfect gas law------理想气体定律Boltzmann constant--- 玻尔兹曼常数neutrons --------------中子electrostatic -------静电的，静电学的Specific heat capacity--- 比热容Plank constant---------普朗克常量Fission----------------裂变fusion-----------------聚变Maxwellian distribution--麦克斯韦分布microscopic------------微观的Macroscopic-----------宏观的quantum number-------量子数Laser-----------------激光deuterium--------------氘Tritium----------------氚deuteron---------------氘核Trition----------------氚核atomic mass unit------原子质量单位Avogadro’s number----阿伏伽德罗常数binding energy----------结合能Substance-------------物质internal-----------------内部的Spontaneously --------自发地circular-----------------循环的Electronic ------------电子的neutral-----------------中性的Qualitative -----------定性的dissociation-------------分解分离Disrupt--------------使分裂A complete understanding of the microscopic structure of matter and the exact nature of the forces acting (作用力的准确性质) is yet to be realized. However, excellent models have been developed to predict behavior to an adequate degree of accuracy for most practical purposes. These models are descriptive or mathematical often based on analogy with large-scale process, on experimental data, or on advanced theory.一个完整的理解物质的微观结构和力的确切性质(作用力的准确性质)尚未实现。

Radial basis functions

ξ ∈Ξ
λξ φ( x − ξ ),
x ∈ Rn ,
(1.1)
suitable adjustments being made when x is not from the whole space, and the coeﬃcient vector λ = (λξ )ξ∈Ξ is an element of RΞ . In many instances, particularly those that will interest us in Section 3, the interpolation requirements s |Ξ = f |Ξ (1.2) for given data f |Ξ lead to a positive deﬁnite interpolation matrix A = {φ( ξ − ζ )}ξ,ζ ∈Ξ . In that case, we call the radial basis function ‘positive deﬁnite’ as well. If it is, the linear system of equations that comes from (1.1) and (1.2) and uses precisely that matrix A yields a unique coeﬃcient vector λ ∈ RΞ for the interpolant (1.1). All radial basis functions of Section 3 have this property of positive deﬁn-
1. Introduction
There is a multitude of ways to approximate a function of many variables: multivariate polynomials, splines, tensor product methods, local methods and global methods. All of these approaches have many advantages and some disadvantages, but if the dimensionality of the problem (the number of variables) is large, which is often the case in many applications from statistics to neural networks, our choice of methods is greatly reduced, unless

The Alexander polynomial of a plane curve singularity and the ring of functions on it

a rX iv:mat h /252v1[mat h.AG ]7Feb2The Alexander polynomial of a plane curve singularity and the ring of functions on it A.Campillo F.Delgado ∗S.M.Gusein–Zade †Abstract We give two formulae which express the Alexander polynomial ∆C of several variables of a plane curve singularity C in terms of the ring O C of germs of analytic functions on the curve.One of them expresses ∆C in terms of dimensions of some factorspaces corresponding to a (multi-indexed)ﬁltration on the ring O C .The other one gives the coeﬃcients of the Alexander polynomial ∆C as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).A version of this text has been published in Russian Mathematical Surveys,v.54(1999),N 3(327),p.157–158.The ring O X of germs of holomorphic functions on a germ X of an ana-lytic set determines X itself (up to analytic equivalence).Thus all invariants of X ,in particular,topological ones,can “be read”from O X .There arises a general problem to ﬁnd expressions for invariants of X in terms of the ring O X .The Alexander polynomial ∆C of several variables is a complete topo-logical invariant of a plane curve singularity C ⊂(C 2,0)([Y]).A formula ofD.Eisenbud and W.Neumann ([EN])expresses the Alexander polynomial in terms of an embedded resolution of the curve C .In this note we give two for-mulae for the Alexander polynomial directly in terms of the ring of germs ofanalytic functions on the curve C.One of them expresses the Alexander poly-nomial∆C in terms of dimensions of some factorspaces corresponding to a (multi-indexed)ﬁltration on the ring O C.The other one gives the coeﬃcients of the Alexander polynomial∆C as Euler characteristics of some explicitly described spaces(complements to arrangements of projective hyperplanes). It seems to be theﬁrst result which describes the coeﬃcients of the Alexander polynomial(and thus of the zeta–function of the monodromy)as Euler char-acteristics of some spaces.Another formula which expresses the Lefschetz numbers of iterates of the monodromy(and therefore the zeta–function of it) for a hypersurface singularity of any dimension in terms of Euler character-istics of some subspaces of the space of(truncated)arcs is given in a paper of J.Denef and F.Loeser(xxx-Preprint series,math.AG/0001105).Let C be a germ of a reduced plane curve at the origin in C2and let C=r i=1C i be its representation as the union of irreducible components(with aﬁxed numbering).Let O C2,0be the ring of germs of holomorphic functions at the origin in C2and let{f=0}(f∈O C2,0)be an equation of the curve C.Let O C be the ring of germs of analytic functions on C(∼=O C2,0/(f)), and let∆C(t1,...,t r)be the Alexander polynomial of the link C∩S3ε⊂S3εforε>0small enough(see,e.g.,[EN]).Remarks. 1.According to the deﬁnition,the Alexander polynomial ∆C(t1,...,t r)is well deﬁned only up to multiplication by monomials±t=±t m11·...·t m r r(t=(m1,...,m r)∈Z r).Weﬁx the Alexander polynomial assuming that it is really a polynomial(i.e.,it does not contain variables with negative powers)and∆C(0,...,0)=1.2.There is some diﬀerence in deﬁnitions(or rather in descriptions)of the Alexander polynomial for a curve with one branch(r=1)or with many branches(r>1)(see,e.g.,[EN]).In order to have all the results(Theorems 1and2below)valid for r=1as well,for an irreducible curve C,∆C(t) should be not the Alexander polynomial,but rather the zeta-functionζC(t) of the monodromy,equal to the Alexander polynomial divided by(1−t).In this case∆C(t)is not a polynomial,but an inﬁnite power series.However for uniformity of the statements we shall use the name”Alexander polynomial”for this∆C(t)as well.Letϕi:(C i,0)→(C2,0)be parametrizations(uniformizations)of the components C i of the curve C,i.e.,germs of analytic maps such that Imϕi= C i andϕi is an isomorphism between C i and C i outside of the origin.For2a germ g∈O C2,0,let v i=v i(g)and a i=a i(g)be the power of the leading term and the coeﬃcient at it in the power series decomposition of the germ g◦ϕi:(C i,0)→C:g◦ϕi(t i)=a i·t v i i+terms of higher degree(a i=0). If g◦ϕi(t)≡0,v i(g)is assumed to be equal to∞and a i(g)is not deﬁned. The numbers v i(g)and a i(g)are deﬁned for elements g of the ring O C of functions on the curve C as well.The semigroup S=S C of the plane curve singularity C is the subsemi-group of Z r≥0which consists of elements of the form v(g);av from the semigroup S C of the curve C.For the case of an irreducible curve C(r=1)the corresponding connection has been described in[CDG2].In this caseζC(t)= i∈S C t i(S C⊂Z≥0).Letπ:ˆS C→Z r be the natural projection:(v)→v∈Z r,let F v)⊂{v}×C r be the correspondingﬁbre of the extended semigroup([CDG1]).Theﬁbre F v∈S C.For v)={g∈O C:v i(g)≥v i;i= 1,...,r}be an ideal in O C.One has a natural linear map j v)→C r, which sends g∈J(v)⊂C r be the image of the map j v)=dim C(v)∼=J(v+1=(1,...,1),and that F v)∩(C∗)r(under the natural identiﬁcation of{v(vhyperplanes in the vector space C(v∈Z rk(v v )∈Z,generally speaking,inﬁnite in all directions.L is not a ring,but a Z[t1,...,t r]–(or even Z[t1,...,t r,t−11,...,t−1r]–)module.The polynomial ring Z[t1,...,t r]can be in a natural way considered as being embedded into L.Let L C(t1,...,t r)= v)·t∈L,P′C(t1,...,t r)=(t1−1)·...·(t r−1)·L C(t1,...,t r).One can easily see that P′C(t1,...,t r)is in fact a polynomial,i.e.,P′C(t1,...,t r)∈Z[t1,...,t r].This follows from the fact that, if v′i and v′′i are negative,then c(v1,...,v′i,...,v r)=c(v1,...,v′′i,...,v r).Let P C(t1,...,t r)=P′C(t1,...,t r)/(t1·...·t r−1)∈Z[[t1,...,t r]].Proposition.For r>1,the polynomial P′C (t1,...,t r)is divisible by(t1·...·t r−1),i.e.,P C(t1,...,t r)∈Z[t1,...,t r].For r=1,P C(t)=L C(t).Theorem1P C(t1,...,t r)=∆C(t1,...,t r).Theﬁbre F v)be the projectivization of theﬁbre F v)=F v)of theﬁbre F v≥δis the conductor of the semigroup S C of the curve C,then theﬁbre F v))of its projectivization is equal to1for r=1and to0for r>1.Let χ(PˆS C):= v))·t.Theorem2∆C(t1,...,t r)=χ(PˆS C).(∗) LetζC(t)(=∆C(t,t,...,t))be the zeta–function of the monodromy of the germ f(the equation of the curve C).Let|v:|v) ·t.4Remark.For an irreducible plane curve singularity all coeﬃcients of the zeta–function of the monodromy are equal to0or1.In terms of the equation (∗),0=χ(∅),1=χ(point).The proof consists of calculation of the polynomialχ(PˆS C)in terms of a suitable(not minimal one)embedded resolution of the curve C⊂(C2,0) and comparing it with the formula for the Alexander polynomial from[EN]. These calculations involve a detailed knowledge about the structure of the semigroup and its relation with the resolution of a singularity.In fact the polynomials P C(t1,...,t r)andχ(PˆS C)coincide for any(not necesser-aly plane)curve.The proof will be published elsewhere.A global version of the result from[CDG2]for a plane algebraic curve with one place at inﬁnity was obtained in[CDG3].References[CDG1]Campillo A.,Delgado F.,Gusein–Zade S.M.The extended semi-group of a plane curve singularity.Proceedings of the Steklov Institute of Mathematics,v.221,139–156(1998).[CDG2]Campillo A.,Delgado F.,Gusein–Zade S.M.On the monodromy ofa plane curve singularity and the Poincare series of its ring of functions.Functional Analysis and its Applications,v.33,N1,66-68(1999). [CDG3]Campillo A.,Delgado F.,Gusein–Zade S.M.On the monodromy at inﬁnity of a plane curve and the Poincare series of its coordinate ring.To be published in Proceedings of the Pontryagin Memorial Conference, Moscow(2000).[EN]Eisenbud D.,Neumann W.Three-dimensional link theory and invari-ants of plane curve singularities.Ann.of Math.Studies110,Princeton Univ.Press,Princeton,NJ,1985.[W]Waldi R.,Wertehalbgruppe und Singularit¨a t einer ebenen algebraischen Kurve.Dissertation.Regensburg(1972).[Y]Yamamoto M.Classiﬁcation of isolated algebraic singularities by their Alexander polynomials.Topology,v.21,N3,277–287(1982).5。

6阶多项式劳斯–赫尔维茨判据

6阶多项式劳斯–赫尔维茨判据英文版6th Degree Polynomial Routh-Hurwitz CriterionIn the field of control theory, stability analysis is crucial for determining the behavior of dynamic systems. The Routh-Hurwitz criterion is a mathematical tool used to assess the stability of linear time-invariant systems. Specifically, it helps determine whether the roots of a system's characteristic equation—which govern its dynamic response—lie in the left half of the complex plane, indicating stability.For polynomial equations of lower degrees, the application of the Routh-Hurwitz criterion is relatively straightforward. However, as the degree of the polynomial increases, the complexity and computational requirements also increase. This article focuses on the application of the Routh-Hurwitz criterion to 6th-degree polynomials, which represent a significant challenge in control system design.The Routh-Hurwitz criterion involves constructing a tabular array known as the Routh array. This array is constructed by arranging the coefficients of the characteristic polynomial in a specific manner. The criterion states that if all the elements in the first column of the Routh array are positive, then the system is stable.For 6th-degree polynomials, the construction of the Routh array becomes more complex. The array now has six rows, with each row representing a specific coefficient of the polynomial. The elements of the array are obtained by calculating ratios and differences of the coefficients.Once the Routh array is constructed, it can be used to determine the stability of the system. If all the elements in the first column are positive, the system is stable. If any element is zero or negative, the system's stability cannot be determined solely based on the Routh-Hurwitz criterion. In such cases, further analysis, such as examining the signs of the remainingelements in the array or performing a root locus analysis, may be necessary.The application of the Routh-Hurwitz criterion to 6th-degree polynomials is crucial in control system design, especially when dealing with complex systems that exhibit higher-order dynamics. By carefully constructing the Routh array and analyzing its elements, engineers can gain valuable insights into the stability of these systems and make informed decisions about their design and operation.中文版6阶多项式劳斯–赫尔维茨判据在控制理论领域，稳定性分析对于确定动态系统的行为至关重要。

局部单叶对数调和映射的Bohr半径

91
所以结合 (3), (4) 可以得到上界
|f (z)| = z|z|2βh(z)g(z) ≤ |z|2|β|+1|h(z)||g(z)| = |z|2|β|+1 · exp
4|z| 1 − |z|
.
(5)
对于左边下界的估计, 结合文献 [13, 定理 2],
log |h(z)| ≥ inf
0≤θ≤
g(z) = exp
z 0
ψ (t) ψ(t)
·
1
ω(t) − ω(t)
dt
,
所以产生
|g(z)| ≤ exp
r 0
1 1
+ −
t t
·
1
1 −
t dt
= (1 − r) · exp
2r 1−r
,
(3)
和
|h(z)| =
ψ(z)
g(z) z
≤
1
1 −
r
·
exp
2r 1−r
.
(4)
No.1
江清华等: 局部单叶对数调和映射的 Bohr 半径
的对数调和映射就是一类非线性偏微分方程
fz¯ f¯
=
ω
fz f
,
的解, 其第二复特征 ω ∈ B(D). 因此 Jacobian 行列式
Jf = |fz|2(1 − |ω|2) > 0. 当 f 是一单位圆盘上非退化的对数调和映射, 有下列表达式
f (z) = h(z)g(z),
这里的 h, g 均属于 A(D). 在文献 [7], 作者 Mao 等人对于非退化的对数调和映射引进了 Schwarz 导数概念, 研究 Schwarz 引理并获得了两种 Landau 型定理.

招聘英语数学老师的英语作文初一

招聘英语数学老师的英语作文初一Wanted: The Best English Math Teacher Ever!Hi there! My name is Lily and I'm a student at Greenfield Middle School. Our math department is in desperate need of a new English math teacher, so I've been put in charge of writing this job ad. I have to find someone really amazing at both English and math, because those are probably the two most important subjects we learn in school.First, let me tell you about the kinds of math we're learning right now. In my pre-algebra class, we're studying things like fractions, decimals, percents, ratios, exponents, and algebraic expressions. I have to admit, I find some of those topics pretty confusing! We spend a lot of time solving word problems too, which can be really tricky when you have to take all the information and set it up as an equation. I could definitely use some extra help understanding all that.Then in algebra, from what I've heard from my older friends, you start learning about linear equations, polynomials, factoring, and all sorts of crazy stuff with variables and solving for x. It sounds super hard! But I know if I have an awesome teacher who can make it fun and interesting, I'll be able to get the hang of it.Apart from just being a total math whiz, the teacher we hire is going to have to be incredible at English too. We write a ton of essays, book reports, research papers, and other assignments in all our classes. And not just in English class - in history, science, you name it! So it's really important that our new math teacher can coach us on things like:• Proper grammar a nd how to structure clear, well-organized paragraphs• How to craft a strong thesis statement• Effective writing techniques like using vivid vocabulary and showing examples instead of just telling• Citing sources properly to avoid plagiarism• Editing and revising our drafts to make them polished and free of errorsYou'd be amazed at how many kids struggle with those kinds of writing skills! We need someone who can patiently guide us through the whole writing process for any kind of assignment.And of course, since this is an English math teacher position, it would be awesome if you could find creative ways to blend the two subjects. Like using word problems to practice writing clearexplanations. Or having us write essays analyzing the genius mathematical patterns found in nature. Making cross-curricular connections like that really helps make the lessons stick in our minds.On top of being a math master and an English expert, we're looking for a teacher who has a great personality too. Someone who makes learning fun, but still pushes us to work hard and do our best. You've got to be able to keep the class engaged and not let kids get bored or disruptive. Cracking jokes, playing games, and getting a little silly are all allowed to keep things interesting! As long as we're still learning, of course.It would also be great if our new teacher could start up an after-school math club or tutoring sessions. I know I'd go for sure, and I'll bet tons of other students would too. We can never get enough practice or bonus instruction to really get this complicated content down pat.Lastly, you've got to be a really caring, supportive teacher who wants to see us all succeed. Don't judge kids who are struggling - just be patient and find new ways to explain things until we understand. Celebrate our accomplishments and give plenty of high fives and encouragement. Building our confidence is just as important as building our skills.So that's the kind of remarkable, one-of-a-kind English math teacher we need at Greenfield. Someone with a profound mastery of both subjects, who can engage us and make learning exciting. A teacher who truly goes above and beyond in every way. If you've got what it takes for this position, we'd be so thrilled to have you!Well, I think I've covered all the key requirements for our dream candidate. Thanks for taking the time to read my job ad. I really hope we can find an extraordinary English math teacher to join our school soon. We're all counting on you!。

University of Liverpool,

1
Introduction
This paper has been inspired by recent observations of Ochiai and Jun Murakami about the Homﬂy skein theory of m-parallels of certain symmetrical 2-tangles. In [8] Ochiai remarks that the 3-parallels of the tangle AB in ﬁgure 1 and its mirror image AB = BA are equal in the Homﬂy skein of 6-tangles, in other words, in the Hecke algebra H6 , [1].
K
=
F
G
A new knot K ′ can be formed by replacing the tangle F with the tangle F = τi (F ) given by rotating F through π in one of three ways,
′
τ1 (F )
=
F
, τ2 (F )
A
=
B
Figure 1: As a consequence, the 3-parallels of any mutant pair of knots given by composing the 2-tangles AB and BA with any other 2-tangle C and then closing will share the same Homﬂy polynomial. This is in contrast with the known fact that 3-parallels of mutant knots in general can have diﬀerent Homﬂy polynomials, [7, 4]. There is interest in the extent to which the Homﬂy polynomial of m-parallels or other m-string satellites can distinguish mutants which are closures of ABC and BAC with A and B as above. Ochiai has found that the 4-parallels of AB and BA are diﬀerent in the skein H8 .

On H.Weyl and J.Steiner polynomials

n +p Voln+p (TRM (t))
n+p TR : dist(x, M) ≤ t}, M (t) = {x ∈ R
n +p
(1.4)
[n 2]
= ωp t
p l=0
u2l,p (M) · t2l ,
(1.5)
π p/2 Γ( p 2 + 1) is the p-dimensional volume of the unit p - dimensional ball. ωp = II. The coeﬃcients u2l,p (M) depend on p as follows: u2l,p (M) = 2−l Γ( p 2 + 1) · w2l (M) , + l + 1) Γ( p 2 0≤l≤
2
Victor Katsnelson
Table of contents.
1. H. Weyl and J. Steiner polynomials. 2. Formulation of main results. 3. The explicit expressions for the Steiner and Weyl polynomials associated with the ‘regular’ convex sets. 4. Weyl and Steiner polynomials of ‘regular’ convex sets as renormalized Jensen polynomials. 5. Entire functions of the Hurwitz and of the Laguerre-P´ olya class. Multipliers preserving location of roots. 6. Properties of entire functions generating Steiner and Weyl polynomials of ‘regular’ convex sets and their surfaces. 7. The Hermite-Biehler Theorem and its application. 8. Properties of Steiner polynomials 9. The Routh-Hurwitz Criterion. 10. The case of low dimension: proofs of Theorems 2.5 and 2.6. 11. Extending the ambient space. 12. The Steiner polynomial of the Cartesian product of convex sets. 13. Properties of entire functions generating the Steiner and Weyl Polynomials for the degenerate convex sets B n+1 × 0q . 14. Concluding remarks. References.