Poincar'{e} cycle of a multibox Ehrenfest urn model with directed transport

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库仑碰撞截面在等离子体粒子模拟中的应用

·粒子束及加速器技术·库仑碰撞截面在等离子体粒子模拟中的应用宋萌萌1,2，周前红1，孙　强1，杨　薇1，董　烨1（1. 北京应用物理与计算数学研究所，北京 100094； 2. 中国工程物理研究院研究生院，北京 100088）摘要：在等离子体粒子模拟中，TA 模型和NanBu 模型被广泛用于处理库仑碰撞，这两种模型要求每个时间步长内全部粒子参与计算。

为了降低参与碰撞的粒子数，提高库仑碰撞的计算效率，提出了一种基于截面的库仑碰撞模拟方法，并给出了库仑碰撞概率的计算公式。

采用该方法对不同温度不同密度电子气的弛豫过程进行模拟，分别对比了电子速度分布函数、电子温度以及电子x 、y 方向上的温度与电子温度之比的模拟值与理论值，验证了该方法的准确性。

在相同的小时间步长上，该方法相比TA 模型计算效率提升可达40%以上。

对于较大的时间步长，该方法仍能得到与理论解近似的模拟结果，相比Nanbu 模型，在相同的精度下可取更大的时间步长，计算效率也有所提升。

研究表明，该方法同样适用于电子-离子碰撞。

因此在提高库仑碰撞计算效率上，该方法具有碰撞粒子数少以及适用于大时间步长的优势。

关键词：粒子模拟；库仑碰撞截面；蒙特卡罗方法； TA 模型； Nanbu 模型；等离子体中图分类号： O411.3 文献标志码： A doi : 10.11884/HPLPB202133.200179Application of coulomb collision cross-section inparticle-in-cell simulation of plasmaSong Mengmeng 1,2， Zhou Qianhong 1， Sun Qiang 1， Yang Wei 1， Dong Ye 1（1. Institute of Applied Physics and Computational Mathematics , Beijing 100094, China ;2. Graduate School of China Academy of Engineering Physics , Beijing 100088, China ）Abstract ： In particle-in-cell simulation of plasma, TA and Nanbu models have been widely used for Coulomb collision. Both models require all particles to collide. In this paper, a cross-section-based method is introduced to give a probability of Coulomb collision for each particle pair and accelerate the computation. To test this method, the relaxations of an electron gas due to e-e collisions were simulated. Comparing the simulated with the theoretical values of velocity distribution function, electron temperature, the ratio of electron temperature in x , y direction to electron temperature, the accuracy of the cross-section-based method was verified. The calculation efficiency of this method can be improved by more than 40% than the TA model at the same small time step. Furthermore, at a large time step,the simulations show agreement with the theoretical solutions, the efficiency is also improved than the Nanbu model.The simulation about the equilibration of electron and ion temperature showes that this method is also suitable for e-i collisions. Therefore in the acceleration of simulating Coulomb collision, this method has two advantages as follows:first, there is a small number of particles to collide within a step, and second, it is suitable for large time steps.Key words ： particle-in-cell simulation ； cross-section of Coulomb collision ； Monte-Carlo methods ； TA model ； Nanbu model ； plasma库仑碰撞作为带电粒子输运过程的重要组成部分，广泛存在于气体放电、聚变等离子体、空间等离子体及半导体系统中[1]。

A Chebyshev criterion for Abelian integrals

arXiv:0805.1140v2 [math.DS] 9 May 2008
A Chebyshev criterion for Abelian integrals
M. Grau, F. Man˜osas and J. VБайду номын сангаасlladelprat
Departament de Matem`atica, Universitat de Lleida, Lleida, Spain
Departament de Matem`atiques Universitat Aut`onoma de Barcelona, Barcelona, Spain Departament d’Enginyeria Inform`atica i Matem`atiques,
Universitat Rovira i Virgili, Tarragona, Spain
XH = −Hy∂x + Hx∂x and Y = P ∂x + Q∂y.
Then, see [17, 20] for details, the ﬁrst approximation in ε of the displacement function of the Poincar´e map of Xε is given by (1) with ω = P dy − Qdx. Hence the number of isolated zeros of I(h), counted with multiplicities, provides
Zeros of Abelian integrals are related to limit cycles in the following way. Consider a small deformation of a Hamiltonian vector ﬁeld Xε = XH + εY, where

W^{2,1}_p Solvability for Parabolic Poincare Problem

$W^{2,1}_p Solvability for Parabolic Poincare Problem$

1
2
L.G. SOFTOVA
coeﬃcients allowing discontinuity in t. The vector ﬁeld ℓ(x, t) generating B is deﬁned on S = ∂ Ω × (0, T ) and is tangential to it in some subset E. The kind of contact is of neutral type and we suppose that γ (x, t) = (ℓ(x, t) · ν (x)) ≥ 0 on S. It means that the boundary value problem under consideration is of Fredholm type, i.e. both the kernel and cokernel are of ﬁnite dimension. 2,1 We are interested of strong solvability of our problem in Wp (Q), p ∈ (1, ∞). Because of the loss of regularity of the solution near to the set of tangency E we impose higher regularity in E of the data. The study is based on the original Winzel’s idea to extend ℓ into Ω such that to obtain explicit representation of the solution through the integral curves of that extension. Thus the problem is reduced to obtaining of suitable a priori estimates for the solution and its derivatives on an expanding family of cylinders. Further, the solvability is proved using regularization technique which, roughly speaking, means to perturb the vector ﬁeld ℓ by adding small ε times ν, to solve the such obtained regular ODP and then pass to limit as ε → 0. The perturbed problem regards linear uniformly parabolic operator P with VMO coeﬃcients and boundary operator B with (ℓε · ν ) > 0. In this case we dispose of unique solvability result 2,1 in Wp (Q), p ∈ (1, ∞) supposing P u ∈ Lp (Q) and initial and boundary data belonging to the corresponding Besov spaces (see [15], [8]). Poincar´ e problem for linear uniformly parabolic operators with H¨ older continuous coeﬃcients is studied in [11] (see also [13]) where unique solvability in the corresponding H¨ older spaces is obtained. Moreover, the linear results were applied to the study of semilinear parabolic problem in H¨ older spaces. A tangential ODP for second-order uniformly elliptic operators with Lipschitz continuous coeﬃcients was studied in [9] (see also [8]). It is obtained strong solvability in W 2,p (Ω) but for p > n/2. In our case the parabolic structure of the equation permits to obtain an 2,1 (Q) only through the data of the a priori estimate for the solution in Wp problem. Thus we are able to prove unique solvability for all p ∈ (1, ∞) avoiding the use of maximum principle and omitting any additional conditions on the vector ﬁeld. 2. Statement of the problem and main results Let Ω ⊂ Rn , n ≥ 3 be a bounded domain with ∂ Ω ∈ C 2,1 and Q = Ω × (0, T ) be a cylinder in Rn+1 . Set ℓ(x, t) = (ℓ1 (x, t), . . . , ℓn (x, t), 0) for a unit vector ﬁeld deﬁned on the lateral boundary S = ∂ Ω×(0, T ). We consider the following oblique derivative problem ≡ ut − aij (x, t)Dij u = f (x, t) in Q, I u ≡ u(x, 0) = ψ (x) on Ω, (2.1) ∂u B u ≡ = ℓi (x, t)Di u = ϕ(x, t) on S. ∂ℓ Denote by ν (x) = (ν 1 (x), . . . , ν n (x)) the unit outward normal to ∂ Ω. Then we can write ℓ(x, t) = τ (x, t) + γ (x, t)ν (x) where τ (x, t) is tangential projection

Modular Invariance and Twisted Anomaly Cancellations of Characteristic Numbers

(12)
=
8A(T M, ∇T M )ch(TC M, ∇TC M ) − 32A(T M, ∇T M ) cosh
c (12) . 2 As an application of the general cancellation formula (1.3), when M is closed and spinc and B is an 8k + 2 dimensional oriented submanifold of M such that [B ] ∈ H8k+2 (M, Z ) is dual to w2 (T M ), the authors obtain that (sign(M ) − sign(B • B )) is divisible by 8 by using the Atiyah-Singer index theorem for spinc manifolds, where (B • B ) denotes the self-intersection of B in M . Moreover, they [9] show that −24 A(T M, ∇T M ) ec + e−c − 2 Sig(M ) − Sig(B • B ) 8
M
(1.4)
≡
A(T M, ∇T M )ch(bk (TC M + C2 − ξC , C2 )) cosh
c 2
≡ ind2 (bk (T B + R2 , R2 ) mod 2, which is the analytic version of the Ochanine congruence obtained in [18]. Formula (1.3) has interesting applications to study the Ochanine and the Finashin congruences (cf. [19], [6]). We refer interested readers to [9] for details. This shows us again how miraculous cancellation formulas imply divisibility and congruence results.

Exploring polygonal environments by simple robots with faulty combinatorial vision

Exploring Polygonal Environments by Simple Robots with Faulty Combinatorial Vision Anvesh Komuravelli1,⋆and Mat´uˇs Mihal´a k21Department of Comp.Science and Engineering,Indian Institute of TechnologyKharagpur,Indiaanvesh@cse.iitkgp.ernet.in2Institute of Theoretical Computer Science,ETH Zurich,Switzerlandmatus.mihalak@inf.ethz.chAbstract.We study robustness issues of basic exploration tasks of simplerobots inside a polygon P when sensors provide possibly faulty informationabout the unlabelled environment P.Ideally,the simple robot we consider isable to sense the number and the order of visible vertices,and can move toany such visible vertex.Additionally,the robot senses whether two visiblevertices form an edge of P.We call this sensing a combinatorial vision.Therobot can use pebbles to mark vertices.If there is a visible vertex with apebble,the robot knows(senses)the index of this vertex in the list of visiblevertices in counterclockwise order.It has been shown[1]that such a simplerobot,using one pebble,can virtually label the visible vertices with theirglobal indices,and navigate consistently in P.This allows,for example,tocompute the map or a triangulation of P.In this paper we revisit someof these computational tasks in a faulty environment,in that we modelsituations where the sensors“see”two visible vertices as one vertex.Insuch a situation,we show that a simple robot with one pebble cannot evencompute the number of vertices of P.We conjecture(and discuss)that thisis neither possible with two pebbles.We then present an algorithm thatuses three pebbles of two types,and allows the simple robot to count thevertices of ing this algorithm as a subroutine,we present algorithmsthat reconstruct the map of P,as well as the correct visibility at every vertexof P.1IntroductionNowadays one of the main research areas in microrobotics is the study of sim-ple mobile autonomous robots.The recent technological development made it possible to build small mobile robots with simple sensing and computational capabilities at a very low cost,which has launched an interest in the study of distributed robotic systems–computation with swarms of robots,not unlike the computational paradigm of wireless sensor networks(where a lot of simple,small and inexpensive devices are spread in the environment,the ⋆The work was done while the author was an internship student at ETH Zurich.devices self-deploy in a working wireless network,gather data from the en-vironment and provide simple computational tasks).Simple robots promise to bring mobile computational capabilities into areas where previous ap-proaches(usually of bulky construction)are not feasible or cost-eﬀective. The main advantages are quick and easy deployment,scalability,and cost-eﬀectiveness.This new concept raises new research problems,as the classical schemes designed for centrally operated,or overwhelmingly equipped robots are inapplicable to the lightweight and/or distributed computational models of simple robots.In this paper we consider one particular model of simple robots,the so called simple combinatorial robot.In this model the robot is modeled as a moving point inside a simple polygon P,and the sensing provides only“com-binatorial”information about the surroundings.In particular,the robot does not sense any metric information(such as angles,distances,coordinates,or direction).Also,the robot can only move to visible vertices.Study of sim-ple robots with possibly minimum requirements on the sensed information is an attractive topic both in theory and practice,as minimalistic assump-tions provide robots that are less susceptible to failures,they are robust against sensing uncertainty and can be very inexpensive to build.In theory, a minimalistic model allows a worst-case computational analysis and pro-vides insights about complexity of various tasks:the positive results identify the easy problems,while the negative results identify the diﬃcult problem for which a richer functionality and sensing is necessary.The simple combinatorial robot wasﬁrst deﬁned and studied by Suri et al.[1].The robot operates inside a polygon P.We denote the set of vertices of the polygon P by V={v0,v1,...,v n−1},where two vertices v i and v i+1,i≥0,form an edge e i=v i,v i+1of P.1The robot,initially placed at vertex v0,can only move to a visible vertex,and while moving, the robot does not sense anything about the environment.When the robot lands at a vertex of P,it senses all visible vertices,but only the presence of vertices–the vertices are unlabelled.The robot senses the vertices in a cyclic order,which is the only way the robot can distinguish the vertices from each other.Thus,a movement operation of the robot is simply of the form“move to the i-th visible vertex”.The order of visible vertices is assumed to be counterclockwise(ccw).Additionally,the robot senses whether two visible vertices form a boundary edge of P.Positioned at vertex v,this is modelled by a combinatorial visibility vector cvv(v)=(c0,...,c k),which is a binary vector that encodes,given there are k+1visible vertices,whether the i-th and(i+1)-th visible vertex,i=0,1,...,form an edge of P(c i=1)or not 1To avoid notational overhead,we assume all operations on the indices to be modulo the corresponding number(n in this case).2(c i=0).The convention is that the vertex v is visible to itself,and v is the 0-th visible vertex of v.Figure1illustrates the concept of cvv’s.The robot can use pebbles to mark vertices.If there is a visible vertex with a pebble, the robot also senses the index of this vertex in the list of visible vertices in ccw order.In case the robot uses pebbles of diﬀerent types,the robot also senses the type of the pebble.Naturally,the goal of computation with pebbles is to use few pebbles and few diﬀerent types of pebbles.Pv0v1v2v3v4v5v6Fig.1.The leftﬁgure depicts a polygon P on vertices v0,...,v6.On the rightﬁgure,a robot R is placed on vertex v0of the same polygon.The visible region of the polygon is shaded.The visible vertices(ordered ccw from the robot’s position)have only local identiﬁers0,1,2,and3(no global information)stating their position in the ccw order,and the combinatorial visibility vector of v0is cvv(v0)=(1,0,0,1),as the visible vertices0,1 form an edge,vertices1,2form a diagonal,vertices2,3form a diagonal,and vertices3,0 form an edge of PTo understand capabilities of minimalistic robots,one studies what prob-lems are solvable and which not,i.e.,one is interested in the possibility only, and does not primarily aim for the best running time of algorithms.Learning and exploring the environment is a prime problem for any robotic system. The results of Suri et al.[1]show that a simple combinatorial robot without a pebble can decide whether the polygon P is convex.On the other hand, without a pebble the robot cannot count the number of vertices,as shows the result of[2].Allowing the robot to use one pebble,the robot can virtually label the vertices of P and construct a map of P,i.e.,the visibility graph G=(V vis,E vis)of P,a graph with V vis=V and with an edge between every two vertices that are visible to each other in P.This then allows the robot to consistently navigate inside the polygon,and,for example,compute a trian-gulation of puting the visibility graph of P is essentially everything the simple combinatorial robot can do with one pebble,as was shown in[2].In this paper we study the robustness issues of the simple combinatorial robot in scenarios where the sensing does not provide accurate information.3In practice,two vertices visible from vertex v can be“seen”as being very close to each other(e.g.,they span a very tiny angle with v).If a very“sim-ple”sensory device is used,these two vertices may wrongly be recognized as a single vertex.This creates a faulty sensing for the robot.In this sec-tion we model such situations formally and study conditions in which the simple combinatorial robot can reconstruct the visibility graph of a simply-connected polygon P(the visibility graph of P is often called the map of the environment).We show in Section2that even counting the number of vertices of P is not possible with one pebble.We conjecture that this is still not possible with two pebbles.We then show that using three pebbles of two diﬀerent types allows the simple combinatorial robot to count the number of vertices of P.In Section3we present an algorithm that allows a simple robot with three pebbles of two diﬀerent types to compute the visibility graph of P,using the algorithm for counting as the main part.We conclude the paper and outline some future work in Section4.Modeling Vertex FaultsFor a given simply-connected polygon P on vertices V,a vertex fault is a set F V,|F|=2.We will sometimes refer to a vertex fault simply as a fault.A vertex that belongs to a vertex fault is called a faulty vertex.We denote by F a collection of vertex faults,i.e.,a set F={F1,F2,...,F m},where every F i,i=1,2,...,m,is a vertex fault.We assume that the vertex faults in F are mutually disjoint,i.e.,no vertex belongs to more than one vertex fault.We deﬁne and study the simple combinatorial robot with vertex faults (faulty robot for short)–a model derived from the simple combinatorial robot that reﬂects our discussion on unreliable sensing.For a given polygon P and a given set of vertex faults F,a faulty robot sitting at some vertex v∈V senses its surrounding in P via the faulty combinatorial visibility vector fcvv(v)which is deﬁned from the cvv in the following way(consult Fig.2 for illustration).Let cvv(v)be the combinatorial visibility vector of vertex v in polygon P.For any two visible vertices x and y,x,y=v,that belong to the same vertex fault F and that appear consecutively in the“vision”of vertex v(recall that the visible vertices of v are considered in ccw order), we remove from the cvv the information about x and y(i.e.,we remove the bit that encodes whether they form an edge or a diagonal in P).Doing so for any such pair of vertices deﬁnes the faulty combinatorial visibility vector fcvv of vertex v.Thus,if vertex v does not see any vertex from a vertex fault,the cvv and the fcvv are the same.Notice also that according to the deﬁnition,the robot at vertex v cannot distinguish between vertices x and y from F only4if they lie consecutively next to each other(as observed from vertex v).The reason for this is that from diﬀerent positions the vertices x and y may cause sensing problems,and from others not.Especially,if from some position the vertices x and y do not appear consecutively,i.e.,there is at least one vertex w between them,then the robot’s sensing can distinguish between x and y.The concept of the fcvv can also be seen as treating the two vertices of a vertex fault as one“virtual”vertex(as observed by a robot),and then deﬁning the fcvv as the cvv with the virtual vertices.In this understanding, the robot thus senses less vertices(than there really are).We will assume that every vertex fault F={u F,v F}is visible from a vertex of P,i.e.,there is a vertex v in P which sees both u F and u F with the correct vision(as otherwise such a vertex fault does not give any faulty vision).P vy xv′wF={x,y}cvv(v)=(1,0,1,0,1,1)fcvv(v)=(1,0,1,1,1)v′Fig.2.Illustration of a faulty combinatorial visibility vector.The leftﬁgure depicts a polygon P with one vertex fault F={x,y}.The vertices x and y appear consecutively in the ccw order as seen from v(the dotted lines are the“vision”lines)and therefore fcvv(v) diﬀers from cvv(v)–the0encoding that x and y form a diagonal in P is removed.The rightﬁgure depicts an alternative view on fcvv’s.The vertex fault{x,y}is seen by a robot at v as one virtual vertex v F,and fcvv(v)is then the cvv of this new(faulty)vision with v F in itIt remains to specify what happens if a robot decides to move to a virtual vertex v F.In our model we assume that the robot can land non-deterministically at either vertex of F.We study the worst-case behavior of algorithms,and thus assume an adversary that decides where the robot lands.Finally,if a pebble is left at a faulty vertex of a vertex fault F,a robot that sees the virtual vertex v F also sees the pebble as being placed at the virtual vertex v F.Related WorkThe concept of simple,deterministic robots that sense no metric information (distances,angles,coordinates,etc.)is a relatively new research area.The simple combinatorial robot,the model we consider in this paper,was deﬁned and studied in[1].The robot was shown be able to compute the visibility5graph of P using one pebble.A similar approach to minimalism was studied for example by Yershova et al[3].They study pursuit-evasion problems with a robot that can only sense the type of the current vertex(reﬂex or convex angle)and can only move along the boundary edges,but can continue in the same direction after reaching a vertex with reﬂex angle.In these and similar models(see e.g.[4]or[5]for other examples of similar models)the considered sensing is very simple,yet the reliability of such sensing is crucial for the solutions of the studied problems.A recent,not directly related,but well studied area of fault-tolerance with mobile robots addresses the computation issues with imprecise com-passes.In this model,a set of asynchronous autonomous robots are placed in a plane(i.e.,not in a polygon)equipped with a sense of direction(and distance)and capability to move an arbitrary distance in an arbitrary direc-tion.An imprecise compass delivers a direction that can deviate from the actual value,but the error is bounded.In this model,mainly the gather-ing problem was studied[6,7].Also for the gathering problem,the issue of not obtaining perfectly accurate sensory input,and not having a perfectly accurate movement was studied in[8]for asynchronous robots.2Counting the Number of VerticesIn this section we consider the elementary problem of inferring the number of vertices of a polygon P by a faulty robot.We shall see that this problem, being trivial in the fault-free case using one pebble,becomes non-trivial in the presence of faults even with two pebbles.We will show,however,that a robot with three pebbles of two types can compute the number of vertices of P.2.1Counting with1PebbleIt is illustrative to considerﬁrst the case when there are no vertex faults. In such a case the robot simply leaves a pebble on the current vertex and moves around the boundary,always moving to itsﬁrst visible vertex(which is its“right”neighbor),counting the number of visited vertices,until the robot comes back to a vertex with the pebble.In case of vertex faults this simple strategy does obviously not work.Consider for example a convex polygon on four vertices v0,v1,v2,v3and one vertex fault F={v1,v2}. Assume that the robot initially sits at vertex v0.The robot drops the pebble to mark v0and moves to its right neighbor,which is a virtual vertex v F.The adversary makes the robot land on v2.The robot then continues to v3and v0,visiting only three vertices in total.One could probably easily derive a6correct algorithm for this simple case,nonetheless we show that in general,using only one pebble,there is no algorithm for the problem of counting the number of vertices in the presence of vertex faults.Theorem 1.Any simple robot with one pebble cannot count the number of vertices of a polygon P with vertex faults.Proof.Let A be an arbitrary (deterministic)algorithm for the simple robot with one pebble.We will show that A cannot count the number of vertices in every polygon P .Consider polygons P 1and P 2in Fig.3with a diﬀerent number of vertices.The left polygon P 1is a square and the right polygon P 2is a convex polygon on six vertices.P 1has no vertex fault,and P 2has three vertex faults F 1={v 0,v 1},F 2={v 2,v 3}and F 3={v 4,v 5}.Thus,if we consider a robot placed at a vertex of the vertex fault {v 0,v 1}for example,it can visually distinguish between the vertices v 0and v 1,but from either of these vertices the robot sees v 2,v 3as a single virtual vertex,and v 4,v 5as another virtual vertex.Let us denote by v F 1,v F 2,and v F 3the virtual vertices that correspond tothe vertex faults F 1,F 2,and F 3,respectively.v 0v 1v 2v 3v 0v 1v 2v 3v 4v 5v F 2v F 1v F 3P 1P 2Fig.3.Polygons used for the proof of Theorem 1Observe ﬁrst that a robot has the same view in both polygons,i.e.,fcvv(v )=(1,1,1,1)for any vertex v in both polygons.Thus,if the robot does not use the pebble,it cannot count the number of vertices because if after ℓmoves and observations in P 1it determines that polygon has four vertices,then the same movements and observations can be made in the second polygon,and thus the deterministic robot has to claim P 2has four vertices,which is obviously wrong.Let us consider the situation when a robot executing A (in both poly-gons)drops a pebble.As P 1and P 2are symmetric we can,without loss of generality,assume the robot drops the pebble at vertex v 0when run on any of the two polygons.We now show that any movement of a robot execut-ing A in P 1can be mimicked in P 2as well,by appropriate choices (by the7adversary)of a vertex the robot lands at,when moving to a virtual vertex v F,i=1,2,3,such that the observed fcvv’s remain the same,together with ithe position of the pebble therein.If a robot in P1moves to itsﬁrst visible vertex(i.e.,to vertex v1in our case),then robot in P2attempts to move to v1as well,and thus the robot in P2lands at v1as well.Hence,the position of the pebble in both cases is the same–the pebble is on the vertex which is the robot’s left neighbor.Similarly,if the robot in P1moves to its last vis-ible vertex(i.e.,to vertex v3),then robot in P2attempts to move to vertex and lands at vertex v5.If the robot in P1moves to the second visible v F3vertex(vertex v2),then the robot in P2lands at vertex v3.It is easy to check that the position of the pebble is the same in both cases.Now(assuming the pebble is still at vertex v0)for any position of the robot in P1and any movement of the robot to a visible vertex,the adversary can make the robot in P2mimic the movement by an appropriate choice of landings in P2.We do not list all possible movements here,but give one more example only. Assume the robot in P1at vertex v2moves to vertex v1and then to vertex v3.If the robot in P2is at vertex v3,the algorithm A moves the robotﬁrst,and lands to vertex v2,and then attempts to move the robot to vertex v F3at vertex v5(by the choice of the adversary).If the robot picks up the pebble in P1so can the robot in P2,as we have maintained the same vision and the position of the pebble is the same for the robots in both polygons.Thus,as the adversary can force the algorithm A to produce the same vision sequence in both polygons,the algorithm cannot compute the number of vertices in both polygons.⊓⊔2.2Counting with2PebblesA natural question is to study the problem using two pebbles.While we do not know whether two pebbles suﬃce to compute the number of vertices of any polygon P,we outline the diﬃculties in designing such an algorithm.Consider a(big)polygon which consists of“triangular cells”as depicted in Fig.4.The triangular cell can be seen as a triangle whose tips were cut oﬀ.For the construction we cut oﬀjust a tiny bit so that the resulting two vertices of a loose end have distanceε(εas small as needed).Also,the two vertices of every end of the cell form a vertex fault.We can glue the triangular cells together as depicted in theﬁgure.Starting from a central triangular cell,we can grow the polygon to an arbitrary size by making the newly glued cells smaller and smaller.To make the constructionﬁnite,we just use triangular cells with no open ends.We make the construction such that the two vertices of every vertex fault F appear consecutively in ccw order as seen from any visible vertex,and thus the two vertices will be seen8by the robot as a single virtual vertex v F.For this to achieve,one has to set an appropriateεand an appropriate angle at which the new cells are glued.For brevity we omit the precise description of the construction.We note that the depiction in Fig.4is only schematic.We call the resulting polygon triangular.For the moment we assume the polygon is big enough for“anything which follows”,while the exact size will naturally become clear at the end of the section.vertexfaultFig.4.Left:A“triangular cell”is a triangle with endpoints split into open ends.The two vertices of each open end form a vertex fault.Right:The whole polygon is build from these “triangular cells”by an appropriate rotation and scalingWeﬁrst prove a useful lemma that highlights the main technique for the proof of the main result of this subsection.Lemma1.A simple robot with no pebbles can be made to stay within two neighboring cells in any triangular polygon P.Furthermore,if the initial vertex can be chosen by the adversary,the robot can be made to stay within one cell.Proof.The main trick is to choose the proper vertex v∈F where the robot lands when it attempts to move to a virtual vertex v F.We(the adversary) can choose this vertex arbitrarily(i.e.,the robot does not notice the diﬀer-ence)as long as the vision from these two vertices is the same.Observe that if the robot is not at the ending triangular cell,the vision is everywhere the same,fcvv=(1,1,1,1,1,1).Our choice of the landing vertex will depend on what the robot wants to do after landing in v F.For the following discussion, consult Fig.5.Let s denote the vertex where the robot starts.Let e be the vertex for which{s,e}is a vertex fault in P.Vertex s is a“gateway”to two neighboring triangular cells A and B,with vertices as depicted in the ﬁgure.We show how to make the robot stay in the cells A and B.Assume for example the robot wants to move to its right neighbor(which is the virtual vertex of the vertex fault{a,b}).The robot may land at a or b.We have9the freedom to choose.Depending on the robot’s next move we choose a or b such that after the next move the robot stays in A or B .The important observation is that a robot at a or b has the same sensing (the same fcvv)and thus,as the robot is deterministic,has to do the same movement,re-gardless of whether it lands at a or b .If the next move is “go to the i -th visible vertex in ccw order”,where i is 1or 2,then we make the robot land at b (as if it landed at a ,the next movement would bring the robot out of A and B ).Similarly,if the next move is “go to the i -th visible vertex in ccw order”,where i is 4or 5,then we make the robot land at a .Clearly,if the next move is “go to the 3rd visible vertex”,the robot stays within the cells A and B regardless of us choosing a or b as the landing vertex.Thus,we only choose a or b according to the robot’s ﬁrst movement that is diﬀerent from “go to the 3rd visible vertex”.After we have chosen the proper vertex a or b for the robot to land,we can similarly argue for all subsequent movements.s ba cd ef g h i AB Fig.5.A robot that does not use a pebble never leaves cells A and BFrom the aforementioned arguments it is now an easy observation that if the adversary can choose the initial vertex (i.e.,either s or e )then the robot can be made to stay within one cell (say,cell A in our case).⊓⊔Using the ideas of the previous lemma we show the following theorem Theorem 2.If a faulty robot with two pebbles can count the number of vertices of a triangular polygon P ,then at any time of the computation the two pebbles are at most two moves (of the robot)apart.Proof.Let us consider the situation where the two pebbles B 1and B 2are more than two moves apart.Thus,the pebbles are in two cells A and B which do not share a single vertex.Let us consider the moment when the robot places the second pebble B 2in cell B .We will show that the robot cannot count the number of vertices of P .We will argue that the adversary can choose landings in such a way that the robot will never come back to cell A (where the ﬁrst pebble B 1is placed).Thus,eﬀectively,this will lead10into a situation of a robot with one pebble only.In this situation,however, the robot cannot lose sight of the second pebble B2,as otherwise the robot would end up in a situation of Lemma1,according to which the adversary can make the robot stay in one cell(forever).Clearly,if the robot cannot lose the sight of the second pebble B2,it cannot visit all vertices of P(as picking up the pebble B2results into the situation of Lemma1,and thus we can make the robot to stay in one cell,never coming back to cell A),and thus it cannot count the number of vertices of P.Consider the situation in Fig.6,where B1denotes theﬁrst pebble,and B2denotes the second pebble.B1lies in cell A,B2lies in cell B,and there is at least one more cell X between the two cells(and B1and B2do not lie in X).We want to avoid the robot coming to a vertex of vertex fault F1,the“gateway”to cell A.For this,weﬁrst argue about the position of pebble B2in cell B.It is placed at a vertex of a vertex fault F4={g,h}. From the geometry of the setting and from our assumptions it follows that the robot had to came to F4from a vertex of P that did not see the pebble B1.Hence,we(the adversary)can choose whether the robot lands at g or h–the visibility will be the same,so the robot decides to place a pebble in either case.Fig.6.Pebbles B1and B2are separated by at least3moves This eﬀectively means that we(the adversary)can decide the location of the pebble B2to be g or h.Our decision depends on the next step(s)of the robot.We may assume that the next step of the robot is a movement (as collecting the right-now dropped pebble is useless and does not help the robot to navigate or compute anything).Let usﬁrst consider the case in which we let the robot land at vertex g to place the pebble B2there.If the robot never leaves the sight of B2then the robot can clearly never come to11cell A,and it also cannot count the number of vertices of P.Thus,assume the robot eventually leaves the sight of B2.Clearly,for one of the choices of landing at g or h,the“leaving”of the robot does not happen at a vertex of F2(i.e.,if for a particular choice of landing the“leaving”happens at a vertex of F2,then for the other choice of landing the“leaving”happens at a vertex of F x–the symmetrically placed vertex fault to F2;this follows because the robot will do the same sequence of movements in either case).Thus,choosing the proper landing,the robot moves from a vertex of F x to a cell with no sight of a pebble,and thus it ends up at the situation of Lemma1,which guarantees that the robot will stay in one cell(forever).⊓⊔Thus,according to the theorem,the two pebbles have to be dropped in adjacent cells,or in the same cell.This hints us that the robot should keep track of the two pebbles such that they are not very far apart.Thus,as the robot moves,it should move the pebbles too.While this may help in visiting vertices,it is not obvious it helps in counting them exactly.This provokes us to make the following conjecture.Conjecture1.A simple robot with two pebbles cannot count the number of vertices of a polygon with vertex faults.2.3Counting with3PebblesNow,we present an algorithm for counting the vertices of a polygon with any number of vertex faults using three pebbles of two diﬀerent types.Theorem3.A simple robot with three pebbles of two diﬀerent types can count the number of vertices of a polygon P with vertex faults.Proof.Our algorithm uses the distinct pebble(pebble of type2)to mark the start vertex v0,and two other identical pebbles(pebbles of type1)to traverse consistently along the boundary of P in ccw order.Starting at vertex v0,the algorithm’s goal is to be able to go to the i-th vertex on the boundary, i=1,2,3,4,...,until the pebble of type2is found again,and thus the number of vertices of P is inferred.The pebble of type2will not have any other usage in the algorithm.As we have seen in the previous sections,going to theﬁrst vertex is already impossible if no pebble is used(recall,just set{v1,v2}to be a vertex fault and let the robot land at vertex v2instead of landing at v1).Using two pebbles,traversing the boundary consistently is possible.We will show how to make one step of the traversing,i.e.,how to move to the next vertex on the boundary.The whole traversing is then just the repetition of these steps.12。

Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems

Existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systemsM.M.El-Dessoky a ,b ,⇑,M.T.Yassen b ,E.Saleh b ,E.S.Aly ba Mathematics Department,Faculty of Science,King Abdulaziz University,P.O.Box 80203,Jeddah 21589,Saudi Arabia bMathematics Department,Faculty of Science,Mansoura University,Mansoura 35516,Egypta r t i c l e i n f o Keywords:Ši’lnikov criterion Lüsystem Zhou’s systemHeteroclinic orbits Homoclinic orbits Smale horseshoesUndetermined coefﬁcients methoda b s t r a c tThis paper presents the existence of Ši’lnikov orbits in two different chaotic systems belong to the class of Lorenz systems,more exactly in the Lüsystem and in the Zhou’s system.Both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z -axis by using the undetermined coefﬁcient method.The existence of the homoclinic orbit for the Zhou’s system has been proven also by using the undetermined coefﬁcient method.As a result,the Ši’lnikov criterion along with some technical conditions guarantees that Lüand Zhou’s systems have both Smale horseshoes and horseshoe type of chaos.Moreover,the geometric structures of attractors are determined by these heteroclinic orbits.Ó2012Elsevier Inc.All rights reserved.1.IntroductionHomoclinic and heteroclinic orbits arise in the study of bifurcation and chaos phenomena,as well as their applications as in mechanics,biomathematics and chemistry [1,2].In some cases it is necessary to determine the nature or the type of cha-otic behaviors resulting from a dynamical system,one of the commonly agreeable analytic criteria for proving chaos in autonomous systems is the work of Ši’lnikov [3,4],the resulting chaos is called horseshoe type or Ši’lnikov chaos [5].Since the discovery of the famous Lorenz chaotic system [6,7],researchers think much of seeking some kind of canonical forms of all possible continuous time quadratic autonomous chaotic systems in three dimensions.The existence of hetero-clinic and homoclinic orbits of quadratic autonomous chaotic systems in three dimensions have been frequently discussed in academic researches [8–10],such as the Lorenz family system [11],the generalized Lorenz canonical form of dynamics sys-tem [12],the Chen system [13],the Liu system [14],the coupled Dufﬁng’s systems [15],the modiﬁed Lorenz system [16],the new chaotic systems [17,18],the Genesio system [19]among several others based on the Ši’lnikov criterion [3,4].The undetermined coefﬁcient method is a powerful tool to determine heteroclinic and homoclinic orbits of chaotic sys-tems and plays a great role in chaotic dynamics analysis as well as chaos control and synchronization.In this work,using the undetermined coefﬁcient method a rigorous proof is introduced to prove the existence of Ši’lnikov orbits in two different chaotic systems,more exactly in the Lüsystem and in the Zhou’s system.Both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z -axis.We prove that,the Lüsystem has only one type of Ši’lnikov orbits,i.e.,het-eroclinic orbits,but the Zhou’s system have two types of orbits,i.e.,heteroclinic and homoclinic orbits.Moreover,by apply-ing the Ši’lnikov theorem,which provides an important theoretical criterion for proving the existence of chaotic attractor,convinced that the two systems indeed are chaotic,with Smale horseshoes and the horseshoe type of chaos.0096-3003/$-see front matter Ó2012Elsevier Inc.All rights reserved./10.1016/j.amc.2012.05.048⇑Corresponding author at:Mathematics Department,Faculty of Science,Mansoura University,Mansoura 35516,Egypt.E-mail address:dessokym@.eg (M.M.El-Dessoky).This paper is organized as following:Section 2,some basic concepts and terminologies related to homoclinic and hetero-clinic orbits are reviewed.Section 3is devoted to investigation of the Lüsystem structure.In Section 4,the Ši’lnikov hetero-clinic orbits of the Lüsystem is studied in detail by using the undetermined coefﬁcient method.In this section,the algebraic expression of the heteroclinic orbit will also be derived,and the uniform convergence of its series expansion is proved.Sec-tion 5,is devoted to investigation of the Zhou’s system structure.In Section 6,the Ši’lnikov chaos of the Zhou’s system will be studied by using the undetermined coefﬁcient method.In this section,the algebraic expression of homoclinic and heteroclin-ic orbits will be derived,and the uniform convergence of its series expansion is proved.Finally,some concluding remarks will be provided in Section 7.2.Homoclinic and heteroclinic orbitConsider the third-order autonomous system:dxdt¼f ðx Þ;t 2R ;x 2R 3;ð1Þwhere the vector ﬁeld f :R 3?R 3belongs to class C r (r P 2).Let x 2R 3is the state variable of the system (1),and t 2R is the time.Suppose that f has at least an equilibrium point E .The point E is called a hyperbolic saddle focus for system (1),if the eigenvalues of the Jacobian A =Df (E )are a ,q ±i x where q a <0,x –0.A homoclinic orbit c (t )refers to a bounded trajectory of system (1)that is doubly asymptotic to an equilibrium point E of the system,i.e.lim t !þ1c ðt Þ¼lim t !À1c ðt Þ¼E .A heteroclinic orbit d (t ),is similarly deﬁned except that there are two distinct saddle-foci E 1,and E 2,being connected by the orbit,one corresponding to the forward asymptotic time,and the other,to the reverse asymptotic time limit,lim t !þ1d ðt Þ¼E 1;and lim t !À1d ðt Þ¼E 2.The heteroclinic and the homoclinic Ši’lnikov criterions for the existence of chaos,are summarized in the following two theorems [3,4].Theorem 1(The heteroclinic Ši’lnikov theorem ).Suppose that two distinct equilibrium points,denoted by v 1e and v 2e ,respectively,of system (1)are saddle foci,whose characteristic values c k and q k ±i x k (k =1,2)satisfy the following Ši’lnikov inequalities:j c k j >j q k j >0;k ¼1;2;x –0;ð2Þunder constraintq 1q 2>0or c 1c 2>0;ð3Þsuppose also that there exists a heteroclinic orbit joiningv 1e and v 2e ,then:(i)The Ši’lnikov map,deﬁned in a neighborhood of the heteroclinic orbit,has a countable number of Smale horseshoes in itsdiscrete dynamics;(ii)For any sufﬁciently small C 1-perturbation g of f,the perturbed systemdxdt¼g ðx Þ;x 2R 3ð4Þhas at least a ﬁnite number of Smale horseshoes in the discrete dynamics of the Ši’lnikov map deﬁned near the heteroclinic orbit;(iii)Both the original system (1)and the perturbed system (4)have horseshoe type of chaos.Theorem 2(The homoclinic Ši’lnikov theorem ).Suppose that one equilibrium point of system (1),denoted by v e ,is saddle focus,whose eigenvalues c and q ±i x satisfy the following Si’lnikov condition:cq <0;j c j >j q j >0;x –0;ð5Þsuppose also that there exists a homoclinic orbit connecting v e .Then:(i)The Ši’lnikov map,deﬁned in a neighborhood of the homoclinic orbit of the system,possesses a countable number of Smalehorseshoes in its discrete dynamics;(ii)For any sufﬁciently small C 1-perturbation g of f,the perturbed systemdx¼g ðx Þ;x 2R 3;ð6Þ11860M.M.El-Dessoky et al./Applied Mathematics and Computation 218(2012)11859–11870has at least aﬁnite number of Smale horseshoes in the discrete dynamics of theŠi’lnikov map deﬁned near the homoclinic orbit;(iii)Both the original system(1)and the perturbed system(6)exhibit horseshoe type of chaos.3.Structure of the LüsystemThe Lüsystem[20–22]can be described by the following differential equation:dxdt¼aðyÀxÞ;dydt¼Àxzþcy;dz¼xyÀbz;ð7Þwhere a,b2R+,c2R,this system exhibit chaotic attractor as shown in Fig.1when a=36,b=3and c=20.The Lüsystem(7) has three equilibrium:E1¼ð0;0;0Þ;E2;3¼ðÆﬃﬃﬃﬃﬃbcp;Æﬃﬃﬃﬃﬃbcp;cÞ:The characteristic equation of the system(7)at equilibrium points(x,y,z)isk3þðaþbÀcÞk2þðabþx2ÀcbÀcaþazÞkþax2Àabcþabzþaxy¼0:ð8ÞThe characteristic equation of the system(7)at E1is:ðkþaÞðkþbÞðkÀcÞ¼0:ð9ÞThe characteristic Eq.(9)has three negative real roots,therefore E1is not saddle focus.Then there is no homoclinic or het-eroclinic orbits ofŠi’lnikov type.The characteristic equation at the equilibrium points E2and E3is:k3þðaþbÀcÞk2þab kþ2abc¼0:ð10ÞDue to Descartes’rule of signs[23,24].The characteristic equation(10)has no positive real root.Thus,it has at least one negative real root.In Eq.(10).Let k¼lÀðaþbÀcÞ,then Eq.(10)becomes:l3þp lþq¼0;ð11Þwherep¼abÀðaþbÀcÞ23;q¼2ðaþbÀcÞ327ÀabðaþbÀcÞ3þ2abcandPhase portraits of the Lüsystem at a=36,b=3and c=20in the three-dimensional.M.M.El-Dessoky et al./Applied Mathematics and Computation218(2012)11859–1187011861a ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq þﬃﬃﬃﬃD p 3r þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq ÀﬃﬃﬃﬃD p 3r ;q ¼À12ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r !;x ¼ﬃﬃﬃ3p 2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r !:When D >0,therefore,the algebraic equation (10)has the following three roots:k 1¼Àða þb Àc Þþa ;k 2;3¼Àða þb Àc Þþq Æx i :ð12ÞRespectively,where k 1<0.To ensures that the real part of the complex conjugate roots is positive and it is further required that:ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r >2ða þb Àc Þ3>0:ð13ÞTherefore,the equilibrium points E 2and E 3are saddle foci at the same time.For a =36,b =3and c =20the equilibrium and their eigenvalues are given by:E 1¼ð0;0;0Þ;then ðk 1;k 2;k 3Þ¼ðÀ36;À3;25Þ;E 2¼ð7:745967;7:745967;20Þ;then ðk 1;k 2;k 3Þ¼ðÀ22:65161246;1:825806228Æ13:688730641i Þ;E 3¼ðÀ7:745967;À7:745967;20Þ;then ðk 1;k 2;k 3Þ¼ðÀ22:65161246;1:825806228Æ13:688730641i Þ:8><>:Then,when D >0and inequality (13),one can easily obtain that the two points E 2and E 3are of hyperbolic saddle foci type,but the point E 1is not of this type,then there is no homoclinic or heteroclinic orbits of Shil’nikov type that is doubly asymp-totic to the equilibrium E 1.4.The existence of heteroclinic orbits in the LüIn this part,we will investigate the undetermined coefﬁcient method to prove the existence of heteroclinic orbits of sys-tem (7).4.1.Finding heteroclinic orbits From (7),we ﬁnd that:y ¼_x aþx ;_y ¼€x a þ_x ;€y ¼x va þ€x ;z ¼ðcy À_y Þx ;_z ¼ðc _y À€y Þx À_x ðcy À_y Þx 2:8<:ð14ÞSubstituting (14)into the third equation of system (7)givesx ðx vþða þb Àc Þ€x þðab Àbc Þ_x Àabcx Þþ_x ððc Àa Þ_x À€x Àx 3Þþax 4¼0:ð15ÞIf x (t )is found,then z (t )and y (t )will also be determined.Therefore,ﬁnding the heteroclinic orbit of system (7)is now re-duced to seeking a function w (t )such that w (t )=x (t )satisfying (15)andw ðt Þ!Àﬃﬃﬃﬃﬃbc p ;as t !þ1;w ðt Þ!ﬃﬃﬃﬃﬃbc p ;as t !þ1;orw ðt Þ!ﬃﬃﬃﬃﬃbc p ;as t !À1;w ðt Þ!Àﬃﬃﬃﬃﬃbc p ;as t !À1:Without loss of generality,one may stipulate a deﬁnite direction as follows:from E 2to E 3corresponds to t ?+1,while from E 3to E 2corresponds to t ?À1.Letx ðt Þ¼w ðt Þ¼Àd þX 1k ¼1a k e k a t ;d ¼ﬃﬃﬃﬃﬃbc p ;and t >0;ð16Þwhere a <0is an undetermined constant,and a k (k P 1)are undetermined coefﬁcients.Substituting (16)into Eq.(15),we get:X1k ¼1d ðG ða k Þa ke a kt ¼H 1þH 2þH 3;ð17Þ11862M.M.El-Dessoky et al./Applied Mathematics and Computation 218(2012)11859–11870whereH1¼X1k¼2X kÀ1i¼1a3i3þðaþbÀcÞa2i2þðabÀbcÞa iþðÀa3i2þðcÀaÞa2iÞðkÀiÞþd2ð3a iþ6aÞÀabch ia i a kÀi e a kt;H2¼ÀdX1k¼3X kÀ1j¼2X jÀ1i¼14aþ3aðkÀjÞ½ a i a jÀi a kÀj e a kt;H3¼X1k¼4X kÀ1m¼3X mÀ1j¼2X jÀ1i¼1aþðkÀmÞa½ a i a jÀi a mÀj a kÀm e a ktandGða kÞ¼ða3k3þðaþbÀcÞa2k2þab a kþ2abcÞ:ð18ÞAssume that a1–0,otherwise one can inductively have a k=0for all k>1.Now,comparing the coefﬁcients of e a kt(k P1)of the same power terms,we obtain the following results.For k=1,a3þðaþbÀcÞa2þab aþ2abc¼0;ð19Þwhich is just the characteristic polynomial of the Jacobian of the linearized equation of system(7)evaluated at the equili-brium point E2or E3.Since(10)has the unique negative root for given parameters,there exist a a<0such that G(a)=0,and for k>1,Gða kÞ¼ða3k3þðaþbÀcÞa2k2þab a kþ2abcÞ–0;k>1:So,for k=2,a2¼a21ðH4Þa;ð20ÞwhereH4¼ðaþbÀcÞa2þðabÀbcÞaþððcÀaÞa2Þþd2ð3aþ6aÞÀabc: For k=3,a3¼½H5þH6a;ð21ÞwhereH5¼X2i¼1a3i3þðaþbÀcÞa2i2þðabÀbcÞa iþðÀa3i2þðcÀaÞa2iÞðkÀiÞþd2ð3a iþ6aÞÀabch ia i a3Ài;H6¼Àd½4aþ3a a31: Finally,for k P4,a k¼½H7þH8þH9a;ð22ÞwhereH7¼X kÀ1i¼1a3i3þðaþbÀcÞa2i2þðabÀbcÞa iþðÀa3i2þðcÀaÞa2iÞðkÀiÞþd2ð3a iþ6aÞÀabch ia i a kÀiH8¼ÀdX kÀ1j¼2X jÀ1i¼1½4aþ3aðkÀjÞ a i a jÀi a kÀjandH9¼X kÀ1m¼3X mÀ1j¼2X jÀ1i¼1½aþðkÀmÞa a i a jÀi a mÀj a kÀm:M.M.El-Dessoky et al./Applied Mathematics and Computation218(2012)11859–1187011863So a is completely determined by a,b and c ,and a k (k P 2)is completely determined by a ,b ,c ,a .In fact,the ﬁrst part of the heteroclinic orbit corresponding to t >0has been determined.Next,the second part corre-sponding to t <0will be constructed.Due to the symmetry of the system (7)around z -axis ((x (t ),y (t ),z (t ))is a solution,and then (Àx (t ),Ày (t ),z (t ))is also a solution for the system (7)).Thus,for t <0,we have:x ðt Þ¼d ÀX1k ¼1a k e Àa kt ;d ¼ﬃﬃﬃﬃﬃbc p and t <0:Then,we can see that the system (7)has a heteroclinic orbit see Fig.2,which connect the equilibrium points E 3and E 2,and takes the following form:x ðt Þ¼w ðt Þ¼Àd þX 1k ¼1a k e a kt ;for t >0;0;for t ¼0;d ÀX 1k ¼1a k e Àa kt ;for t <0:8>>>><>>>>:ð23ÞFrom the continuity of the solution,we have:X1k ¼1a k ¼d ;ð24Þwhich will determine the value of a 1.4.2.The uniform convergence of heteroclinic orbits series expansionThe uniform convergence of the series expansion (16)of the heteroclinic orbit is investigated.For simplicity,we only con-sider the case in which system (7)has the special parameter set that generates two-scroll attractors.For other parameter sets,the proof is similar if the heteroclinic orbit exists.For the chaotic Lüsystem,a =36,b =3,c =20and d ¼ﬃﬃﬃﬃﬃbc p ¼2ﬃﬃﬃﬃﬃﬃ15p ,the values of a and a k can be determined by (19)–(22)and (24)as j a 2j ¼0:01697769306a 21;j a 3j ¼0:002487296673a 31 ;j a 4j ¼0:00006094055197a 41 ,one can inductivity prove that j a k j <10Àðk þ1Þa k 1 ;ðk P 4Þ.We need to seek a 1with P 1k ¼1a k e a kt ¼d ¼2ﬃﬃﬃﬃﬃﬃ15p .Numerical simulation shows that a ‘‘stable’’a 1indeed exist near 7.629581737with relative error no greater than 1%.So when k P 4,a k is bounded,that is there exists an l >0,such that j a k j 6l ,k =1,2,...Consequently,P 1k ¼1a k e a kt 6l P 1k ¼1e a kt is convergent on (0,+1).So Àd þP 1k ¼1a k e a kt is convergent on (0,+1).Similarly,the convergence of d ÀP 1k ¼1a k e Àa kton (À1,0)can also be proved.Theorem 3.If c >0,D >0and the condition (13)are satisﬁed,then the system (7)has one Ši’lnikov heteroclinic orbit of which one component has the form (23),and the corresponding chaos is of horseshoe type.Obviously,the typical parameters a =36,b =3and c =20are always satisﬁed.So there exist heteroclinic orbits of Ši’lnikov type,and as a result,there exist a countable number of Smale horseshoes.Therefore,there exists an invariant set constituting the complex attractor.That is the essence of the geometric structure of the attractor.heteroclinic orbit joining E 2and E 3in the Lüsystem with 11864M.M.El-Dessoky et al./Applied Mathematics and Computation 218(2012)11859–11870gular points.The Ši’lnikov criterion guarantees that the Zhou’s system has Smale horseshoes and the horseshoe chaos.In ad-dition,there also exists one homoclinic orbit joined to the origin.The uniform convergence of the series expansions of these two types of orbits are proved in this section.It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Zhou’s system.The Zhou’s system can be described by the following differential equationdx¼a ðy Àx Þ;dy¼bx Àxz ;dzdt¼xy þcz ;ð25Þwhere a ,b and c are real parameters,this system exhibit chaotic attractor as shown in Fig.3when a =10,b =16and c =À1.When a =À0.7,b =16and c =À1,the Zhou’s system has chaotic attractors as shown in Fig.4.The Zhou’s system (25)hasthree equilibrium points E 1¼ð0;0;0Þ;E 2;3¼ðÆﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;ÆﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;b ÞThen,the characteristic equation of the system (25)the point (x ,y ,z )is:k 3þða Àc Þk 2Àðab þac Àyx Àaz Þk þx 2a þxya þacb Àcaz ¼0:ð26ÞThe characteristic equation of the system (25)at E 1is (k Àc )(k 2+a k Àab )=0.Then,the eigenvalues are c and Àa2Æﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃa 2þ4ab p 2.One can see that the equilibrium E 1=(0,0,0)will be saddle focus iff:a 2þ4ab <0and Àac <0:ð27ÞThe characteristic polynomial of the Zhou’s system (25)E 2¼ðﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;ﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;b Þwill be:k 3þða Àc Þk 2Àðbc þac Þk À2abc ¼0;ð28ÞIf c <0,due to Descartes’rule of signs,the characteristic equation (28)has no positive real root.Thus,it has at least one ne-gative real root.Let k ¼l Àða Àc Þ3,then Eq.(28)becomes:l 3þp l þq ¼0;ð29Þwherep ¼Àðbc þac ÞÀða Àc Þ2;q ¼2ða Àc Þ3þðbc þac Þða Àc ÞÀ2abc :Furthermore,denoteD ¼q 22þp 33;when D >0by Cardan formula,Eq.(29)has a unique negative real root,c ,and a conjugate pair of complex roots,q ±i x ,withthe Zhou’s system in the three-dimensional when M.M.El-Dessoky et al./Applied Mathematics and Computation 218(2012)11859–1187011865c ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq þﬃﬃﬃﬃD p 3r þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq ÀﬃﬃﬃﬃD p 3r ;q ¼À12ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r !;x ¼ﬃﬃﬃ3p 2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r !:Therefore,when D >0,the algebraic equation (28)has the following three roots:k 1¼Àða Àc Þ3þc ;k 2;3¼Àða Àc Þ3þq Æx i :ð30ÞRespectively,where k 1<0.To ensures that the real part of the complex conjugate roots is positive and it is further required that:ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2þﬃﬃﬃﬃD p 3r ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃÀq 2ÀﬃﬃﬃﬃD p 3r >2ða Àc Þ3>0:ð31ÞNote that the characteristic polynomial of the Jacobian of the linearized system of (25),evaluated at the equilibrium pointE 3,is exactly the same as (28).So,it also has three roots,identical to (30).6.Existence of Ši’lnikov-type orbits 6.1.The existence of heteroclinic orbitsIn this part,we will investigate the undetermined coefﬁcient method to prove the existence of heteroclinic orbits of sys-tem (25).From (25),we ﬁnd that:y ¼_x þx ;_y ¼€x þ_x ;€y ¼x vþ€x ;z ¼ðbx À_y Þ;_z ¼ðb _x À€y Þx À_x ðbx À_y Þ2:8<:ð32ÞSubstituting (32)into the third equation of system (25)givesx ðx vþða Àc Þ€x Àac _x þabcx ÞÀ_x ð€x þa _xþx 3Þþax 4¼0:ð33ÞIf x (t )is found,then z (t )and y (t )will also be determined.Therefore,ﬁnding the heteroclinic orbit of system (25)is nowchanged to seeking a function f (t )such that f (t )=x (t )satisfying (33)andf ðt Þ!ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;as t !þ1;f ðt Þ!ﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;as t !þ1;orf ðt Þ!ﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;as t !À1;f ðt Þ!ÀﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ;as t !À1:Without loss of generality,one may stipulate a deﬁnite direction as follows:from E 2to E 3correspond to t ?+1,while from E 3to E 2corresponds to t ?À1.Letof the Zhou’s system in the three-dimensional when a =xðtÞ¼fðtÞ¼ÀdþX1k¼1a k e a kt;d¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbcp;and t>0;ð34Þwhere a<0,is an undetermined constant and a k(k P1)are undetermined coefﬁcient.Substituting from(34)into Eq.(33),we obtain:X1k¼1ðGða kÞa k e a kt¼H1þH2þH3;ð35ÞwhereH1¼X1k¼2X kÀ1i¼1a3i3þðaÀcÞa2i2ÀðacÞa iÀða2i2þa a iÞðkÀiÞaþ6a d2À3a d2iþabch ia i a kÀi e a kt;H2¼ÀdX1k¼3X kÀ1j¼2X jÀ1i¼1ð4aþ3aðkÀjÞÞa i a jÀi a kÀj e a kt;H3¼X1k¼4X kÀ1m¼3X mÀ1j¼2X jÀ1i¼1ðaþaðkÀmÞÞa i a jÀi a mÀj a kÀm e a kt:andGða kÞ¼dða3k3þðaÀcÞa2k2ÀðacþbcÞa kÀ2abcÞ:Comparing the coefﬁcients of e a k t(k P1)of the same power terms,we obtain the following results.For k=1,a3þðaÀcÞa2ÀðacþbcÞaÀ2abc¼0;ð36Þwhich is just the characteristic polynomial of the Jacobian of the linearized equation of system(25)evaluated at the equili-brium point E2or E3.So(36)has the unique negative root for given parameters,there exist a a<0such that G(a)=0,and for k>1,Gða kÞ¼dða3k3þðaÀcÞa2k2ÀðacþbcÞa kÀ2abcÞ–0:ð37ÞThat isGða kÞ¼dða3k3þðaÀcÞa2k2ÀðacþbcÞa kÀ2abcÞ–0;k>1:So,for k=2,a2¼a21ðH4ÞGð2aÞ;ð38ÞwhereH4¼ða3þðaÀcÞa2ÀðacÞaþabcÞþða2þa aÀ3dÞaþ6d2: For k=3,a3¼½H5þH6Gð3aÞ;ð39ÞwhereH5¼X2i¼1a3i3þðaÀcÞa2i2ÀðacÞa iÀða2i2þa a iÞðkÀiÞaþ6a d2À3a d2iþabch ia i a3ÀiH6¼Àdð4aþ3aÞa31: Finally,for k P4,a k¼½H7þH8þH9Gða kÞ;ð40ÞwhereM.M.El-Dessoky et al./Applied Mathematics and Computation218(2012)11859–1187011867H 7¼X k À1i ¼1a 3i 3þða Àc Þa 2i 2Àðac Þa i Àða 2i 2þa a i Þðk Ài Þa þ6a d 2À3a d 2i þabc h ia i a k Ài ;H 8¼Àd X k À1j ¼2X j À1i ¼1ð4a þ3a ðk Àj ÞÞa i a j Ài a k Àj ;andH 9¼Xk À1m ¼3X m À1j ¼2X j À1i ¼1ða þa ðk Àm ÞÞa i a j Ài a m Àj a k Àm :So a is completely determined by a ,b and c ,and a k (k P 2)is completely determined by a ,b ,c ,a .In fact,the ﬁrst part of the heteroclinic orbit corresponding to t >0has been determined see Fig.5.Next,its second part corresponding to t <0will be constructed.Due to the symmetry of the system,one component of the heteroclinic orbit of (25)has the following form:x ðt Þ¼f ðt Þ¼Àd þX 1k ¼1a k e a kt ;for t >0;0for t ¼0;d ÀX 1k ¼1a k e Àa kt for t <0;8>>>>>>>>><>>>>>>>>>:ð41ÞFrom the continuity of the solution,we have:X1k ¼1a k ¼d ;ð42Þwhich will determine the value of a 1.6.2.The uniform convergence of heteroclinic orbits series expansionThe uniform convergence of the heteroclinic orbit series expansion is investigated.For simplicity,we only consider the case in which system (25)has the special parameter set that generates two-scroll attractors.For other parameter sets,the proof is similar if the heteroclinic orbit exists.When a =10,b =16,c =À1and d ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃÀbc p ¼4,the values of a and a k can be determined by (36)–(40)and (42)as,j a 2j ¼0:009279499509a 21;j a 3j ¼0:0001944176542a 31 ;j a 4j ¼0:0000109295777a 41,one can inductivity prove that j a k j <10Àk þ1a k 1 ;ðk P 4Þ.We need to seek a 1with P 1k ¼1a k ea kt ¼d .Numerical simulation shows that a ‘‘stable’’a 1indeed exist near 3.85125with relative error no greater than 1%.So when (k P 4)a k is bounded,that is there exists an l >0,such that j a k j 6l ,k =1,2,...Consequently,P 1k ¼1a k e a kt 6l P 1k ¼1e a ktis convergent on (0,+1).So Àd þP 1k ¼1a k e a kt is convergent on (0,+1).Similarly,the1heteroclinic orbit joining E 2and E 3in the Zhou’s system 11868M.M.El-Dessoky et al./Applied Mathematics and Computation 218(2012)11859–11870。

Multi-Pulse Continuous Phase Modulation

Multi-Pulse Continuous Phase ModulationTommy Svensson Department of Signals and Systems Communication Systems Group Chalmers University of Technology SE-41296G¨o teborg,Sweden tommy.svensson@s2.chalmers.seArne Svensson Department of Signals and Systems Communication Systems Group Chalmers University of Technology SE-41296G¨o teborg,Sweden arne.svensson@s2.chalmers.seAbstractKeywordsMulti-pulse,Continuous phase modulation,Modulation cod-ing,Spectral efﬁciency,Nonlinear systems. INTRODUCTIONThe constant envelope property of Continuous Phase Mod-ulation(CPM)makes it possible to use non-linear ampli-ﬁers,which have lower cost and signiﬁcantly higher power efﬁciency than linear ampliﬁers.It is also possible to de-ﬁne schemes with a narrow spectral main lobe and small spectral side lobes by using a long and smooth phase pulse. Such a phase pulse introduces intersymbol interference,and the optimum decoding algorithm for the so-called partial re-sponse CPM on an AWGN channel is maximum likelihood sequence detection(MLSD).For spectrally efﬁcient CPM, a non-binary symbol alphabet gives better power efﬁciency. The established performance measure for the power efﬁ-ciency of a maximum likelihood sequence detector in AWGN is the minimum Euclidean distance of the signal space,since the minimum Euclidean distance predicts the bit error rate performance at high signal to noise ratios[1].In the literature,work has been done on generalizing CPM such as multi-h[1],multi-T e.g.[4],ﬁltering plus hard-limiting[1]and non-linear CPFSK[3].In previous work [9,11],we have studied the performance of spectrally efﬁ-cient CPM(about twice as efﬁcient as in GSM).It was found that serially concatenated convolutionally encoded CPM per-formed worse than standard CPM using a numerically op-timized phase pulse and modulation index with the same spectrum requirement and information bit rate.In this paper,we propose a generalization for multi-level CPM,where each bit of the multi-level information symbol is transmitted by a separate phase pulse and a separate mod-ulation index.This generalization removes some symmetry of the CPM signal space and increases the effective dimen-sionality.The phase code becomes a non-linear function of the information symbols and potentially this increases the minimum Euclidean distance of the CPM signal space.We deﬁne multi-pulse CPM(mpCPM),show how toﬁnd the minimum Euclidean distance and how to calculate the power spectral density.Next,we investigate the performance of multi-pulse CPM compared to standard CPM by searching for optimized phase pulses.We show that the minimum Euclidean distance is increased and that the simulated er-ror probability is reduced.Finally we show by examples that multi-pulse CPM can be used to control the bit error probability of the individual bit streams in multi-level CPM. CONTINUOUS PHASE MODULATIONThe transmitted signal for CPM can be described[1]bywhere is the length of a symbol interval,is the symbol energy,the carrier frequency and is the transmitted -ary data symbol sequence,with symbols taken from the set.For symbol interval, where,the phase function is deﬁned bywhereThe function is called the phase pulse and is a continuous function with the restrictionThe parameter is called the modulation index.When re-stricting the modulation index to,where and are relative prime integers,the phase state,,takes values from aﬁnite discrete set.Hence,for the symbol interval ,the signal(1)is determined by the state,whereis called the correlative state.With a rational modulation index,the CPM signals can be described by a trellis and optimally decoded using the Viterbi decoder.The number of states in the trellis is ,but just a few states are needed for near optimum decoder performance,[8,10].MULTI-PULSE CONTINUOUS PHASE MODULATION A general formulation for the CPM phase function would be to assign arbitrary phase waveforms,with the continuous phase restriction,to each transition in a predeﬁned trellis. However,it is a very complex task to search for such wave-forms for spectrally efﬁcient CPM.We deﬁne a generalization of CPM,which we call multi-pulse CPM(mpCPM).For the phase function is deﬁned bywhere in general are symbols taken from an arbitrary set of cardinality two.Thus,bit in an information symbol is transmitted by the phase pulse.Each phase pulse follows the restriction in(4),making the accumulated phase to follow the expressionStandard CPM with a natural bit mapping rule is the spe-cial case when,for.Note that different lengths of the phase pulses are allowed by choosing to be the length of the longest phase pulse.It is also straightforward to extend the formulation to include multi-h.In this paper we restrict the analysis to the case when,hence the trellis structure of the CPM system is not changed,only the phase functions on the trellis transitions are made more general.THE EUCLIDEAN DISTANCEThe normalized minimum Euclidean distance[1]is deﬁned by For large,the asymptotic symbol error probability is controlled byfT bP o w e r s p e c t r a l d e n s i t y [d B ]Figure 1.Spectrum mask and power spectral densities for the spectrally efﬁcient case.where the time-average auto-correlation function ist/TP h a s e p u l s eFigure 3.Optimized phase pulses for the spectrally efﬁ-cient case.GMSK phase pulse is shown as a reference.Table 1.Minimum Euclidean distances.101010101010E b / N 0 [dB]S y m b o l E r r o r P r o b a b i l i t yFigure 6.Simulated symbol error probability for the less spectrally efﬁcient case with MLSD and AWGN.mapping rule (not shown).In Fig.7we demonstrate that with multi-pulse CPM it is possible to approach equal BER for all the bit streams at a target bit error probability (upper right),to let one bit stream be more reliable than the other two (lower left)and to largely separate the bit error probabilities for the individual bit streams (lower right).There are many communication system aspects that could beneﬁt from a CPM system providing unequal bit error pro-tection for the bit streams,e.g.multi-level coding and broad-cast systems with multi-resolution source encoding.CONCLUSIONIn this paper,we have deﬁned a generalization of CPM,which we call multi-pulse CPM (mpCPM),and we have shown how to ﬁnd the minimum Euclidean distance and how to calculate the power spectral density.We have investigated the performance gain for multi-pulse CPM compared to standard CPM by searching for optimized phase pulses and we have found that there is a gain in the minimum Euclidean distance.The gain is small for a spectrally efﬁcient system,and the gain increases with less spectral efﬁciency.Finally we have demonstrated that multi-pulse CPM can be used to control the bit error probability of the individual bit streams in multi-level CPM.REFERENCES [1].Plenum Press,1986.[2].John Wiley &Sons,1993.[3]Nonlinear continuous phase frequencyshift keying.,10(Oct.1991),1473–1481.Figure 7.Simulated multi-pulse CPM ,and with MLSD and AWGN.Bit error prob-ability for each bit stream using a GMSK phase pulse with is shown.The parametersare given in each subﬁgure.[4]Good multi-Tphase codes under bandwidth and complexity constraints.,5(Sept.1994),1699–1702.[5].PhD thesis,Telecommunication theory,University of Lund,May 1985.[6].McGraw-Hill,2000.[7]NLQPL:A FORTRAN-subroutinesolving constrained nonlinear programming problems.(1985),485–500.[8]Reduced state sequence detection ofpartial response continuous phase modulation.,4(Aug.1991),256–268.[9]On convolutionallyencoded partial response CPM.In(Amsterdam,TheNetherlands,Sept.1999),vol.2,pp.663–667.[10]Reduced complexitydetection of bandwidth efﬁcient partial response CPM.In(Houston,Texas,May 1999),vol.2,pp.1296–1300.[11]Maximizingminimum Euclidean distance of spectrally constrained partial response CPM.In(Rhodes,Greece,May2001),vol.2,pp.1244–1248.。

The outer derivation of a complex Poisson manifold

Revised in February, 1997 The outer derivation of a complex Poisson manifold. by J-L. Brylinski* and G. Zuckerman
arXiv:math/9802014v1 [math.DG] 3 Feb 1998
1. Hamiltonian and outesson manifold
We will work either with a C ∞ or a complex Poisson manifold M . In the ﬁrst case there is a Poisson tensor π ∈ ∧2 T M , where T M is the tangent bundle. In the second case the Poisson tensor π is a holomorphic section of ∧2 ΘM , where ΘM is the holomorphic tangent bundle. There are three interesting classes of vector ﬁelds, which we enumerate starting with the largest class: (1) the Poisson vector ﬁelds: a vector ﬁeld ξ is Poisson it it preserves the Poisson structure, that is Lξ π = {ξ, π } = 0, where { , } is the Schouten bracket. (2) the locally hamiltonian vector ﬁelds: locally ξ is of the form XH = i(dH )π , where i denotes interior product, and H is a smooth (resp. holomorphic) function deﬁned locally. (3) the hamiltonian vector ﬁelds: ξ = XH for some global H . * This research was supported in part by NSF grant DMS-9504522. 1

International Journal of Computational Intelligence and Applications c ○ World Scientific

International Journal of Computational Intelligence and Applicationsc World Scientiﬁc Publishing CompanyA Multiple-objectives Evolutionary Perspective to Interdomain TrafﬁcEngineeringSteve Uhlig∗Department of Computing Sciences and EngineeringUniversit´e catholique de Louvain,Louvain-la-neuve,1348,BelgiumURL:.ucl.ac.be/˜suhE-mail:suh@info.ucl.ac.beReceived(received date)Revised(revised date)We present an application of multiple-objectives evolutionary optimization to the problem of engi-neering the distribution of the interdomain trafﬁc in the Internet.We show that this practical problemrequires such a heuristic due to the potentially conﬂicting nature of the trafﬁc engineering objectives.Furthermore,having to work on the parameter’s space(BGP routing)of the real problem makes suchtechniques as evolutionary optimization very easy to use.We show the successful application of ouralgorithm to two practically relevant problems in interdomain trafﬁc engineering.Keywords:multiple-objectives evolutionary optimization;interdomain routing;trafﬁc engineering1.IntroductionThe Internet routing system today is divided into two views:intradomain and interdomain. The interdomain Internet is made of autonomous systems(AS).Each autonomous system uses the interdomain routing protocol(BGP)to exchange reachability information with its neighbor ASs.Autonomous systems are made of routers and links between routers that constitute the intradomain view of each AS.A router’s purpose is to forward trafﬁc toward a destination in the Internet.Routers in a given AS exchange intradomain routing infor-mation through an interior gateway protocol(IGP)that distributes the whole map of the intradomain network to all routers of the AS.Typically,IGP routers know the whole path to reach any other host inside the AS.BGP routers on the other hand only know the next hop to reach a destination in the Internet.The current interdomain routing protocol used in the Internet is BGP,that stands for border gateway protocol13.With BGP,an AS advertises to each neighbor AS all the des-tination networks(IP preﬁxes)it can reach.Among the IP preﬁxes that an AS advertises, some are internal preﬁxes that are reachable within this AS(internal to this AS)and others are preﬁxes that have been learned through its BGP neighbors.A key feature of BGP is ∗Steve Uhlig is funded by the FNRS(Fonds National de la Recherche Scientiﬁque,Belgium).This work was partially supported by the Waloon Government(DGTRE)within the TOTEM project.Fig.1.Intradomain and interdomain views of the Internetthat it allows each network operator to deﬁne its routing policies.Those policies are im-plemented by using“ﬁlters”9.A BGPﬁlter is a rule applied upon receiving a BGP route from a neighboring AS or before sending a BGP route to a neighboring AS.BGPﬁlters can prevent some routes from being accepted from or announced to peer ASs,and can also modify the attributes of the BGP routes on a per-AS basis so that some routes be preferred over others.Fig.1shows a simpliﬁed Internet made of three ASs.Each AS has a particular intrado-main topology the other ASs do not know about.Inside an AS,the intradomain routing protocol(IGP)distributes the whole map of the internal topology of the AS to the other routers of the AS so that each router of the AS knows the shortest path to reach any other router of the AS.On Fig.1,AS A is directly connected to both AS B and AS C at the interdomain level,but AS B and AS C can only reach each other by crossing AS A.With the interdomain routing,neither AS B nor AS C knows the exact path followed by its trafﬁc inside AS A.With BGP,an AS only knows the intermediate ASs crossed by its trafﬁc to reach a destination AS.Nowadays,more and more Internet Service Providers(ISP)rely on trafﬁc engineering to modify theﬂow of the trafﬁc inside their network3.Trafﬁc engineering encompasses all techniques aimed at modifying the characteristics of the trafﬁc,be it to change the load of the trafﬁc among the network elements or to inﬂuence the very characteristics of the trafﬁc. In practice,different trafﬁc classes have different engineering requirements,so that trafﬁc engineering techniques depend on the particular trafﬁc class considered.In this paper,we restrict the focus on the engineering of best-effort trafﬁc,for which no strict guarantees need to be enforced.While ISPs know their internal topology and techniques exist to tune the intradomain routing8,most of them rely on manual conﬁguration.At the interdomain level, trafﬁc engineering is even more challenging12.Operators change their routing policies andthe attributes of their BGP routes on a manual basis,without a proper understanding of the implications of such changes on theﬂow of the trafﬁc.Interdomain trafﬁc engineering is used by ISPs to automatically engineer theﬂow of their trafﬁc with neighboring ASs. Having to do it manually may often lead to router misconﬁgurations10that exacerbate the stability of interdomain routing.In this paper,we present a multiple-objectives evolutionary algorithm especially de-signed to deal with interdomain trafﬁc engineering with BGP and describe two successful applications of this algorithm.The remainder of the paper is structured as follows.Section2introduces the main ob-jectives of interdomain trafﬁc engineering.Section3discusses the choice of the optimiza-tion method.Section4describes our multiple-objectives evolutionary algorithm.Section5 discusses the practical issues of sampling a non-dominated front.Then,section6provides two applications of our algorithm to problems in interdomain trafﬁc engineering.2.Problem statementInterdomain trafﬁc engineering as considered in this paper consists in modifying theﬂow of the trafﬁc exchanged with neighboring ASs.The objectives are the following:(1)minimize the burden on the interdomain routing protocol required to implement thetrafﬁc engineering,(2)optimize one or several objectives deﬁned on the trafﬁc exchanged with other ASs oron the distribution of the trafﬁc inside the AS.Theﬁrst objective concerns interdomain routing.There are many remote networks with which an AS exchanges trafﬁc on timescales of hours to days14,16.An interdomain traf-ﬁc engineering technique should ideally minimize the number of reachable networks that need to be inﬂuenced.As the number of inﬂuenced networks corresponds to the number of the BGP routing changes that will be implemented,an interdomain trafﬁc engineering technique should try to minimize the burden placed on BGP.The second objective deals explicitly with theﬂow of the trafﬁc,as it consists of a set of objectives deﬁned on the interdomain trafﬁc.As different ASs have different engineer-ing needs,the trafﬁc engineering objectives that an AS may want to optimize will depend on its size and the type of business it focuses on.Small ASs typically pay providers for their Internet connectivity.The price of this connectivity can be high,and minimizing the cost of their trafﬁc is thus relevant especially if they have multiple connections to the rge ASs on the other hand do not have to pay providers but need to carefully distribute the load of the trafﬁc inside their network.For that purpose,one way is to tune their intradomain routing8.However,tuning the intradomain routing not only changes the distribution of theﬂow of the trafﬁc inside the AS,but also how trafﬁc enters and leaves the network1.To control how trafﬁc enters and leaves the network,large ISPs need to tweak the BGP rge providers also often rely on“hot-potato routing”,that consists in using the exit point inside the network that is closest in terms of the IGP routing metric to the ingress point where the trafﬁc has been received.Hot-potato routing however does notlead to a balanced distribution of the trafﬁc among the exit points,so that trafﬁc engineering objectives can be conﬂicting in practice.In the context of interdomain trafﬁc engineering,the problem of optimizing any trafﬁc objective is always conﬂicting with the objective of minimizing the impact on BGP,as changing theﬂow of the trafﬁc always requires to tweak BGP routing.Furthermore,trafﬁc engineering objectives that are only concerned with the trafﬁc can also be conﬂicting.A multiple-objectives algorithm is then necessary to sample the trade-offs among the possible solutions to the interdomain trafﬁc engineering problem.In the remainder of this paper,we distinguish between the trafﬁc objectives that are purely concerned with the trafﬁc and the BGP routing objective that is only concerned with the changes made to the BGP routing.BGP tweaking basics ASs can be roughly classiﬁed in two types:transit or stub.Stub ASs contain hosts that produce or consume network trafﬁc.These domains do not carry trafﬁc that is not produced by or destined to their hosts.Transit ASs interconnect different ASs together and carry trafﬁc that is produced by and/or destined to external ASs.Tweaking BGP to modify theﬂow of the outgoing trafﬁc requires knowledge of how BGP decides which route to use to reach a destination12.For the sake of simplicity,we only describe in this section how to tweak BGP routes in the case of stub ASs and for outgoing trafﬁc.For further details about BGP tweaking see12.Assume a stub AS having one link per provider.It receives a BGP routing table from each of its providers.To control theﬂow of its interdomain trafﬁc,it can rely on the in-formation found in these BGP routing tables.BGP is a path vector routing protocol.Each BGP router sends BGP advertisements to its peers.A BGP advertisement sent by an AS means that the AS that advertises the route agrees to forward IP packets to the destina-tions corresponding to this route.In addition,the AS-path attribute13contained in this route advertisement also tells through which ASs the IP packets will transit to attain their destination.This information allows the stub AS to reconstruct the AS-level topology for outgoing trafﬁc by relying on these BGP routing tables.Fig.2provides an example topology with a stub AS connected to three providers.Each provider advertises one route towards each of the three considered destinations.In this ﬁgure,an arrow from AS X to AS Y indicates that the pattern“X Y”appears in the AS path for some destination.In addition,we use three different line styles to identify each provider’s routes,so that if part of the AS path of some routes that our AS received through several providers are identical,then we shall have as many arrows as there are routes from these different providers.On Fig.2,the links between two ASs learned through routes ad-vertised by provider1are represented by continuous arrows,while the ones from provider 2are represented with medium dashed arrows and those from provider3withﬁne dashed arrows.With the BGP routes received from each provider,our AS knows through which providers it can reach a particular destination.For the example provided on Fig.2,all three destinations can be reached through any of the three providers.Assume that each of these three destinations represents one third of the outgoing trafﬁc for our AS.We couldFig.2.Example AS-level topology.choose to use one different provider for each destination so that its outgoing trafﬁc is well balanced.Let us assume for the sake of simplicity that there is only one next hop per provider although this is not always the case in practice.If the stub AS shown on Fig.2wants to achieve a good balance of its outgoing trafﬁc,it can rely on the local-pref attribute to prefer some routes over others.Recall that we assume that each destination preﬁx re-ceives one third of the trafﬁc.Suppose that the restricted BGP decision process with only local-pref and the AS path length is considered.If the routes towards the three desti-nations have a default local-pref value of100,then given the fact that the shortest AS path route would be chosen by BGP to reach a destination,the trafﬁc towards destination1 will be forwarded through provider2(AS path length of2),the trafﬁc towards destination 2will be forwarded through providers1,2or3(AS path length of3),while the trafﬁc towards destination3through providers2or3(AS path length of3).This would mean that the routes for destinations2and3are non-deterministic for this restricted BGP decision process since there are several“best routes”among which BGP can choose.In practice,the BGP decision process ensures that only one route is used but this would not automatically lead to a good balance of the trafﬁc.So a solution for evenly distributing the outbound trafﬁc of the local AS is to put a higher value of the local-pref attribute in one of the routes learned from destinations2and3to ensure that only one best route remains and that the trafﬁc balance be good among the three providers.For that purpose,our stub can attach a local-pref value of110to the route towards destination2learned from provider1 and also attach a local-pref value of110to the route towards destination3learned from provider3.This conﬁguration would ensure that each provider gets one third of theoutgoing trafﬁc if the trafﬁc is evenly distributed over the three preﬁxes.Due to the way the BGP decision process chooses the best route towards a destination, there is ample choice concerning how to tweak the BGP decision process.In this paper we decided to tweak the value of the local-pref attribute only,the effect of a BGP route change is quite simple:a“BGP route change”is a pair<preﬁx,provider>that indicates that the trafﬁc having the preﬁx preﬁx as destination will be forwarded through provider provider among the providers which advertised a BGP route towards this network.What we call a BGP route change in the remainder of this paper hence concerns a single BGP preﬁx.The actual effect of a BGP route change on the BGP routes is to force the value of the local-pref attribute of the BGP route towards preﬁx preﬁx which was learned via provider provider to be set with a value of110(higher preference).All other routes for preﬁx preﬁx(learned from other providers)have their local-pref attribute(re)set to the same value which will be strictly smaller than110.Because our focus in this paper is on tweaking BGP in the most controllable way,we rely on changing the local-pref attribute.Working on the local-pref attribute ensures that the changes performed by the trafﬁc engineering will overrule other aspects of the network conﬁguration.Note that the MED attribute of BGP routes can also be used to control the outgoing trafﬁc of stub ASs15.3.Motivations for evolutionary optimizationThe trafﬁc engineering objectives discussed in the previous section cannot be compared, i.e.an improvement in one of the objectives cannot be measured against an improvement in another objective.Optimizing a single composite objective that weights all these ob-jectives is thus useless for practical purposes as a network operator would like to have the best solution in terms of all the objectives at the same time.The interdomain trafﬁc engineering problem is thus intrinsically a multiple-objectives optimization problem.For such problems,evolutionary algorithms are a well-known technique capable ofﬁnding a non-dominated front in a single run4,5.A front is a set of solutions.A solution is said non-dominated if no other solution of the set is better in terms of all the considered objectives at the same time.Additionally,relying on the“evolutionary”paradigm allows to leverage the mechanisms of population-based search and selection among individuals.More details about Pareto-optimality can be found below.Now let us describe the main motivations for selecting the evolutionary paradigm to tackle our problem.Theﬁrst reason is interdomain routing.Our aim is to be as close as possible to the way BGP works in practice.Thus,we do not want to simplify the way BGP chooses the best route towards a particular destination as it is most critical for prac-tical interdomain trafﬁc engineering.The complexity of BGP makes it very difﬁcult to model7.The second reason is that interdomain trafﬁc engineering objectives can be non-linear,based on statistics,...Hence we consider that having to rely on strict assumptions concerning the trafﬁc objectives would be too limiting.Pareto-optimality The concept of Pareto-optimality is related to the set of solutions whose components cannot be improved in terms of one objective without getting worse in at least one of the other components.More formally,a multiobjective search space is partially ordered in the sense that two solutions are related to each other in two possible ways:either one dominates or neither dominates.Consider the multiobjective minimization problem: Minimize y=f(x)=(f1(x),...,f n(x))where x=(x1,...,x m)∈Xy=(y1,...,y n)∈Yand where x is called the decision vector,y the objective vector,X the parameter space and Y the objective space.A decision vector x1∈X is said to dominate another decision vector x2∈X(x1≺x2),iff∀i∈1,..,n:f i(x1)≤f i(x2)∧∃j∈1,..,n:f j(x1)<f j(x2).Domination is an important notion because it determines the result of the comparison of two decision vectors.A decision vector x is said Pareto-optimal iff x is non-dominated regarding X,i.e. x ∈X:x ≺x.A Pareto-optimal decision vector cannot be improved in any objective without degrad-ing at least one of the other objectives.These are global optimal points.In our context however,we are not interested in global optima but optimal points in some neighborhood of some of the objectives.More precisely,we aim atﬁnding the Pareto-optimal points with respect to the trafﬁc objectives and having a distance of at most BGP conﬁgura-tion changes compared to this default BGP routing solution.Hence we do not search for globally Pareto-optimal decision vectors but locally Pareto-optimal decision vectors: Consider a set of decision vectors X ⊆X.1.The set X’is denoted as a local Pareto-optimal set iff∀x ∈X : x∈X:x≺x ∧||x−x ||< ∧||f(x)−f(x )||<δwhere||.||denotes a distance metric, >0andδ>0.2.The set X is called a global Pareto-optimal set iff∀x ∈X : x∈X:x≺x .Note that a global Pareto-optimal set does not necessarily contain all Pareto-optimal deci-sion vectors.4.Search procedureDepending on the relationships between the trafﬁc objectives which might be conﬂicting, harmonious or neutral11,the search on the non-dominated front should have to be different. Recall that we do not know beforehand the relationship between the trafﬁc objectives.This means that our search method must be as lightly biased as possible towards any of the trafﬁc objectives to sample in the best possible manner the search space.Because sampling the whole search space would make the search space grow very large,we decided that the heuristic would iterate over the BGP routing changes by trying to add one BGP routingchange at each generation of the algorithm.Doing this puts additional pressure on the population by forcing improvements in the trafﬁc engineering objectives to have as few BGP route changes as possible early on during the optimization.1accepted=02iter=03while((accepted<MAXPOP)AND(iter==MAXITER)){4foreach individual k{5//Trying a random BGP route change6BGP change.prefix=rand uniform(1,MAXPOP)7BGP change.exit=rand uniform(1,NUM POINTS)8//If effect of BGP route change is improvement accept it 9if(improved(k,BGP change)){10accept(k,BGP change)11//update counter for accepted improved individuals12accepted++13}//end if14}//end foreach individual15//update iteration counter16iter++17}//end whileFig.3.Pseudo-code of search procedure for a single generation.Fig.3provides a pseudo-code description of the search procedure.The principle of the search is as follows.At theﬁrst generation,we start with a population of individuals initialized at the default solution found by BGP routing.Hence at generation zero all indi-viduals have the same values of the trafﬁc objectives and contain no BGP routing change. At each generation,we use a random local search aimed at improving the current popula-tion by applying an additional BGP routing change.Each individual of the population is non-dominated with respect to the other members of the population for what concerns the trafﬁc objectives.In addition,the current population is always made of individuals having the same number of BGP routing changes.At each generation,we parse the whole popu-lation and for each individual we try to apply an additional randomly chosen BGP routing change.Whenever a BGP routing change provides improvement with respect to at least one of the trafﬁc objectives,we accept this improved individual and put it in the set of accepted individuals.We iterate this procedure until weﬁnd a target number of improved individuals or stop when we have performed a target number of tries(the variable iter).Note that the pseudo-code given at Fig.3concerns only one generation.The purpose of variable iter is not to count the generations but to ensure that the search will not loop indeﬁnitely during a given generation.5.Sampling the non-dominated frontThe previous section described the procedure to search for BGP routings changes that im-prove the individuals of the previous population with respect to any of the trafﬁc objec-tives.Some of these improved individuals can be dominated since we did not check for non-domination when accepting an improved individual.Improvement was sufﬁcient to accept an individual.The next step is to check for non-domination on this population of improved individuals to obtain a non-dominated front.For that purpose,we rely on the fast non-domination check procedure6.This procedure has time complexity0(MN2)where M is the number of objectives and N the size of the population.We do not describe this pro-cedure in details but refer to the original NSGA-II paper6for the original idea and to Deb’s book5for a detailed explanation.Let us only mention the main points here.Let P denote the set of non-dominated individuals found so far at the current generation.P is initialized with anyone of the individuals among the accepted ones.Then try to add individuals from the set of accepted ones,one at a time,in the following way:•temporarily add individual k to P•compare k with all other individuals p of P:–if k dominates any individual p,delete p from P–else if k is dominated by other members of P remove k from PThis procedure ensures that only non-dominated individuals are left in P.The number of domination checks is in the order of0(N2)while for each domination check M compar-isons are necessary(one for each objective).The time complexity is thus0(MN2).Having found the non-dominated front for a given number of BGP routing changes,we are left with selecting the individuals of the population for the next generation.Actually,the number of non-dominated individuals from the set of improved ones has to be smaller than the size of the population we use during the search process(MAXPOP),unless the front is almost continuous and easy to sample.To build the population for the next generation,we have to decide how many individuals in the next population each non-dominated solution will produce.Because non-dominated individuals are not comparable among themselves, we must choose a criterion that will produce MAXPOP individuals from the set of non-dominated ones.On the one hand,we would like to include at least every non-dominated individual in the population.On the other hand,depending on the way the accepted solu-tions are spread over the non-dominated front,we must sample differently different regions of the front for a given number of BGP routing changes.This notion of sampling the non-dominated front is close to an idea of distance between neighboring individuals in the objective space.Maintaining diversity on the non-dominated front requires that individu-als whose neighbors are farther apart be preferred over non-dominated individuals whose neighbors are close.The rationale behind this is that less crowded regions should require more individuals to be correctly explored than regions having more non-dominated indi-viduals.The computation of the crowding distance for each individual is done according to the procedure described in Deb’s book5pp.248.First the non-dominated individuals are sorted with respect to each objective.Then the individuals having the smallest and largestvalue for any objective are given a crowding distance d m of∞to ensure that they will be selected in the population.For each objective m,the crowding distance of any individual i,1≤i≤(|P|−2),is given byd m i=f m i+1−f m i−1•volume-based :x $per y bytes.•destination-based :x $per Mbps for “local”trafﬁc (national for instance)and y $per Mbps for “non-local”trafﬁc (international for instance).•max-based :ﬂat rate based on the maximum available bandwidth,independent of how many bits are used.The actual billing cost of the trafﬁc hence depends both on the short-term trafﬁc dynamics on each Internet connection and the long-term trafﬁc volume exchanged with providers.We thus evaluate in this section the problem of optimizing the cost of the trafﬁc of a stub AS while balancing the short-term (10minutes intervals)load of the trafﬁc over the available providers,with as few BGP routing changes as possible.Note that in this section trafﬁc bal-ancing objectives (short-term or long-term)are measured in terms of the maximum amount of trafﬁc carried by any provider over the considered time intervals.0.50.550.6 0.65 0.70.75 0.80.85 0.9 0.9510.50.60.70.80.91Non-dominated frontNumber of BGP routing changesFig.4.Daily volume-based billing and short-term trafﬁc balancing.On Fig.4,we plot the non-dominated front found by the algorithm for a scenario of a stub AS having Internet connections with three different providers.On Fig.4,the stub AS tries to minimize the daily cost of its total trafﬁc while evenly balancing the trafﬁc over its three providers over 10minutes time intervals.The grayscale palette located at the right of Fig.4and Fig.5maps the z-value of the points to some tone to ease the interpretation of the plots.The point corresponding to the default BGP routing (upper right of Fig.4)has no BGP routing change,its values of the two trafﬁc objectives equal to 1as we normalized the trafﬁc objectives with respect to their value under no BGP routing change.Globally,two regions appear on Fig.4.The ﬁrst region concerns points for the ﬁrst few BGP routing changes (about 20).These points start at the top right of Fig.4and converge to the frontthat makes the second region of the non-dominated front (bottom left).The second region of the front indicates that the two trafﬁc objectives are conﬂicting for more than 20BGP routing changes.The conﬂicting nature of the objectives can be seen by a relatively linear (slightly convex)trade-off between the two trafﬁc objectives,for a given number of BGP routing changes.Finding a solution providing a smaller cost on the long-term for a given number of BGP routing changes requires to worsen the short-term objective value.In the same way,ﬁnding a solution providing a smaller value of the short-term objective function for a given number of BGP routing changes requires that one worsens the value of the long-term objective function.0.60.65 0.70.75 0.8 0.850.9 0.95102550751000.840.860.880.90.920.940.960.9810.60.70.80.91Non-dominated frontNumber of BGP routing changes Percentile-based billingShort-termtraffic balancing Fig.5.Daily percentile-based billing and short-term trafﬁc balancing.V olume-based billing as used above is not the most realistic trafﬁc billing scheme one can think of.Now,we use as the long-term trafﬁc objective the 95th percentile billing over 10minutes time intervals.For the short-term trafﬁc objective,we use the same trafﬁc bal-ancing objective as above.The non-dominated front for the long-term percentile-based traf-ﬁc objective is provided on Fig.5.Fig.5shows no smooth non-dominated front even for a large number of BGP routing changes,in contrast to the results of the trafﬁc cost objective above.The explanation for this phenomenon is the statistical nature of the percentile-based objective which largely depends on the short-term dynamics of the trafﬁc.Indeed,the value of the 95th percentile depends on the distribution of the values of the trafﬁc for each provider and each short-term time interval.Changing the provider used to carry the trafﬁc for some reachable network over the whole day has a non-trivial effect on the value of the percentile.A cost function like volume-based billing is insensitive to the short-term variability for some reachable network,in contrast to the percentile-based objective.A percentile-based。

Poincare-Bendixson-Theorem

23
Be trapped inside a set or not
Consider a simple closed curve defined by the equation V(x)=c, where V(x) is continuously differentiable.The vector field f(x) points inward if f(x)・∇V (x)<0; points outward if f(x)・∇V (x)>0; and it is tangent to the curve if f(x)・∇V (x)=0.
9
Proof of Lemma1
▪ It cannot return to any other point on L between A and B either, so if it ever crosses L again, it will have to be further along in the same direction on L, as in the point C indicated in the figure.
17
Proof of Poincare-Bendixson Theorem
• If the orbit through x is not closed, it must pass close enough to z that it must cross L, infinitely often in a sequence that approaches z from one side.
Furthermore, if it crosses several times, the crossing points are ordered along line in the same way as on the orbit itself.

basic elements of probability -回复

basic elements of probability -回复基础概率的基本元素概率是数学中一个非常重要的概念，它用于描述随机事件发生的可能性。

在统计学、经济学、工程学和其他许多领域中都广泛应用概率论的概念和方法。

在这篇文章中，我们将逐步介绍概率的基本元素。

第一步：样本空间和事件样本空间是指所有可能结果的集合。

例如，当我们掷一个六面的骰子时，样本空间包含所有骰子的六个面：{1, 2, 3, 4, 5, 6}。

样本空间通常用大写字母Ω表示。

在样本空间中，我们可以定义一个事件，它是样本空间的一个子集。

例如，事件"A"是骰子的结果是偶数，它的元素是{2, 4, 6}。

事件通常用大写字母表示。

第二步：概率的定义概率是衡量某个事件发生可能性的数值。

它可以是从0到1的任意实数。

如果事件是不可能发生的，概率为0；如果事件一定会发生，概率为1。

概率的定义有两种方式：频率定义和主观定义。

频率定义是指通过实际观察事件在长期重复试验中发生的次数来计算概率。

例如，在掷一个公平的骰子时，偶数的概率是1/2，因为在长期重复的掷骰子试验中，其中一半的结果将会是偶数。

另一种定义是主观定义，它基于主观判断和经验。

例如，根据经验，我们可能会认为在一个蓝色的袋子中，红色球的概率比蓝色球要大。

在这种情况下，概率的值是根据主观判断来确定的。

第三步：事件的运算概率论提供了一种对事件进行运算的方法。

这些运算包括并运算、交运算和差运算。

并运算表示两个事件同时发生的概率。

例如，事件A表示掷骰子的结果是偶数，事件B表示掷骰子的结果是大于3的数。

事件A和事件B同时发生的概率可以通过求两个事件交集的概率来计算。

交运算表示两个事件中至少有一个发生的概率。

例如，事件C表示掷骰子的结果是奇数，事件D表示掷骰子的结果是小于等于3的数。

事件C和事件D至少有一个发生的概率可以通过求两个事件并集的概率来计算。

差运算表示一个事件发生而另一个事件不发生的概率。

Matlab与通信系统仿真第2.5 章 Randon process and analog

- Non-coherent demodulation: Envelop detector.
- The SNR of the demodulator output (i.e., the SNR of the
baseband signal) versus demodulator input (i.e., the
of f=0, and small at high frequencies.
✓ Bandlimited：
If the power spectrum is zero for |f|>B.
✓ Bandpass random process
The power spectrum is larger in the band
✓ We mainly discuss the amplitude modulation,
including:
- DSB-AM
- Conventional AM(常规AM)
- SSB-AM
12
Amplitude Modulation
✓ DSB-AM
For a baseband signal m(t)，the DSB-AM signal is：
= cos 2
The spectrum is：

=
− + +
2
13
Amplitude Modulation
✓ Demodulation of DSB-AM
- The coherent demodulation of the DSB-AM signal is
receive SNR ):

=

Geodesics and almost geodesic cycles in random regular graphs

a rX iv:mat h /6189v1[mat h.MG ]2Oct26GEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALD Abstract.A geodesic in a graph G is a shortest path between two vertices of G .For a speciﬁc function e (n )of n ,we deﬁne an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C ,the distance d G (u,v )is at least d C (u,v )−e (n ).Let ω(n )be any function tending to inﬁnity with n .We consider a random d -regular graph on n vertices.We show that almost all pairs of vertices belong to an almost geodesic cycle C with e (n )=log d −1log d −1n +ω(n )and |C |=2log d −1n +O (ω(n )).Along the way,we obtain results on near-geodesic paths.We also give the limiting distribution of the number of geodesics between two random vertices in this random graph.1.Introduction A geodesic in a graph G is a shortest path between two vertices of G .Let ω(n )be any function tending to inﬁnity with n ,and put e (n )=log d −1log d −1n +ω(n ).We deﬁne an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C ,the distance d G (u,v )is at least d C (u,v )−e (n ).We investigate the existence of almost geodesic cycles through random pairs of vertices in a random graph,and related questions on geodesics and paths that are nearly geodesic,in a sense to be made precise.Our results refer to the probability space of random d -regular graphs with uniform probability distribution.This space is denoted G n,d ,and asymptotics (such as “asymptotically almost surely”,which we abbreviate to a.a.s.)are for n →∞with d ≥3ﬁxed,and n even if d is odd.Some related previous research focussed on ﬁnding (edge/internally)-disjoint paths with many sources and targets.Frieze and Zhao [4]showed that for suﬃciently large d there exist ﬁxed positive constants αand βsuch that a graph G taken from G n,d a.a.s.has the following property:for any choice of k pairs {(a i ,b i )|i =1,...,k },satisfying(i)k ≤⌈αdn/logd n ⌉,and(ii)for each vertex v :|i :a i =v |+|i :b i =v |≤βd ,there exist edge-disjoint paths in G connecting a i to b i for all i =1,2,...,k .This result is optimal up to constant factors.The paths returned by their algorithm are of length of at least 10log d n .Our focus is diﬀerent as it comes from diﬀerent motivation:studying almost geodesic cycles in G n,d .Our result on internally disjoint paths refers to one pair of vertices ﬁxed before the graph is chosen.This is a much weaker model than the model of [4],that dealt with Θ(n/log n )pairs given by an adversary after the graph is chosen.However,we show the existence of disjoint paths that approximate the optimal path (whose length is a.a.s.in [log d −1n −ω(n ),log d −1n +ω(n )])by an additive factor of log d −1log d −1n ,2ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALD whereas the result of[4]give at best a constant multiplicative factor.Additionally,our result holds for all d≥3,and that weﬁnd the maximum possible number of internally disjoint paths,d,that there can possibly be between two vertices.Theorem1.1.Take any integer d≥3and any functionω(n)withω(n)→∞.Let G∈G n,d and choose vertices u and v in V(G)independently with uniform probability. Then a.a.s.the following hold:(i)|d(u,v)−log d−1n|<ω(n),(ii)there are d paths connecting u and v such that the subgraph induced by each pair of these paths is an almost geodesic cycle.Note that the d paths in(ii)theorem are internally disjoint because each pair of them induces a cycle.In a slightly diﬀerent direction,we also investigate the distribution of the number of geodesics joining two vertices(see Theorem2.6).We may obtain the lower bound in part(i)of the theorem from an elementary observation.Note that,given G∈G n,d,the number of vertices at distance at most i from a vertex u is bounded above by1+d+d(d−1)+...+d(d−1)i−1=O (d−1)i . So,there are O n(d−1)−ω(n) vertices at distance i=log d−1n−ω(n)from any given vertex of G,whereω(n)→∞.As a consequence,if two vertices of G are chosen independently with uniform probability,then the probability that the second vertex is at distance at most i=log d−1n−ω(n)from theﬁrst is at most1GEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS3 pairs may be chosen sequentially so that the next pair is chosen uniformly at random over the remaining(unchosen)points.For more information on this model,see[5]. The numbers of geodesics are investigated in Section2.Theorem1.1is proved in Section3.Someﬁnal remarks are in Section4.2.Distribution of the number of geodesicsTheﬁrst portion of our argument is a simpliﬁed version of part of the argument of Bollob´a s and Fernandez de la Vega[3].We consider the process in which the neigh-bourhoods of u and v are exposed step by step.First,the neighbours of u and v are revealed,then the vertices at distance two,and so on.This sequential exposure of the random regular graph is analysed using the random pairing model mentioned in the Introduction.Let N i(u)denote the set of vertices at distance at most i from u.Note that,in the early stages of this process,the graphs grown from u and v tend to be trees,hence the number n i of elements in N i(u)is approximatelyn i−1+(d−1)(n i−1−n i−2).Let f i denote the number of vertices in a balanced d-regular tree,that is,f i=1+d i−1j=0(d−1)j=1+d((d−1)i−1)2log d−1n .Lemma2.1.Letω(n)be any function of n such thatω(n)→∞.For i≤i0−ω(n) a.a.s.the cardinality n i of N i(u)equals f i.Moreover,for i≤i0+ω(n)a.a.s.n i=f i−O ω(n)(d−1)3(i−i0)+ω(n) .Proof.First note that it is suﬃcient to consider the case whenω(n)=o(log n). Since f i denotes the number of vertices in a balanced tree where every non-leaf vertex has degree d,theﬁrst assertion follows if we show that a.a.s.the set of vertices at distance at most i≤i0−ω(n)of a vertex u induces a tree.In other words,if we expose, step by step,the vertices at distance1,2,...,i from u,we have to avoid,at step j,edges that induce cycles.So,we wish not toﬁnd edges between any two vertices at distance j from u or edges that join any two vertices at distance j to a same vertex at distance j+1from u.We shall refer to edges of this form as‘bad edges’.Note that the expected number of‘bad edges’at step i+1is equal to O(n2i/n)=O(f2i/n)=O((d−1)2i/n). Consider i1=⌊14ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALDIn order to prove the second assertion,notice that the expected number of‘bad edges’added between step i1+1and step i,i≤⌊i0+ω(n)⌋=⌊12log d−1n−i0,the fractionalpart of1d−2 .Proof.Denote S i the event that the neighbourhoods of u and v are separate at time i, that is,N j(u)and N j(v)did not join up to time i.We claim thatP(S i0+k |S i0+k−1)∼exp −d2(d−1)2k−2γ(n,d)−2 .This implies our result for the following reasons.If M is a positive integer,−M<k,P(S i0+k )=P(S i0−M)×kl=−M+1P(S i0+l|S i0+l−1).Furthermore,equation(1)establishes that a.a.s.d(u,v)>2i0−ω(n)for any functionω(n)with lim n→∞ω(n)=∞.In particular,givenǫ>0,we can choose M=Mǫ>0GEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS5u v `````````"!# u u ````````` "!# u u uw Figure 2.Second case –even length.suﬃciently large so that P (S i 0−M )>1−ǫ.Given such an M ,we use the previous equation to deriveP (S i 0+k )>(1−ǫ)k l =−M +1exp −d 2(d −1)2l −2γ(n,d )−2 (1−o (1))∼(1−ǫ)exp k l =−M +1−d 2(d −1)2l(d −1)2M +2γ(n,d )(d −2).The same calculations also lead us to P (S i 0+k )<k l =−M +1exp −d 2(d −1)2l(d −1)2M +2γ(n,d )(d −2) .Putting the last two equations together and letting ǫ→0,during which we may assume M ǫ→∞,we have P (S i 0+k )∼exp −d (d −1)2k −2γ(n,d )6ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALD the sets N i−1(u)and N i−1(v),which are assumed to be disjoint.Recall that,according to the pairing model(see Introduction),any vertex in U i−1and V i−1can be regarded as a cell of distinct points,where the number of points corresponds to the number of unexposed neighbours of this vertex.The probability that one given point joins another is then asymptotic to1/(dn),since any pair of unmatched points is equally likely to be paired and the whole process has,by Corollary2.2,matched at most n1/2+o(1)=o(n) pairs of points to this moment.Asymptotically,there are|U i−1||V i−1|(d−1)2pairs of points such that one is associated with a vertex in U i−1and the other with a vertex in V i−1.This is because the hypothesis i=i0+k implies,by Lemma2.1,that the number of vertices in U i−1or V i−1incident with‘bad edges’created at step i−2is a.a.s.at most O(ω(n))for anyω(n)→∞,and it is clear that each vertex in U i−1or V i−1with degree larger than1in G[N i−1(u)]∪G[N i−1(v)]is incident with a‘bad edge’. Thus,the expected number of edges of theﬁrst type joining the neighbourhoods of u and v at time i−1,that is,the number of pairs of points consisting of one point associated with a vertex in U i−1and one point associated with a vertex of V i−1exposed at time i is asymptotic to(d−1)2d2n2n|U i−1||V i−1|=(d−1)3n |U i−1||V i−1|∼d2(d−1)2i−2(d−1)2+2γ(n,d).We wish to apply the method of moments to establishP(S i0+k |S i0+k−1)∼exp −d2(d−1)2k−2γ(n,d)−2 ,so we have to verify that the j-th factorial moment of the random variable Z counting the number of joins at time i=i0+k satisﬁes E([Z]j)=E(Z)jGEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS 7The j -th factorial moment of Z is given byE ([Z ]j )=j l =0 ⋆P ((X (r m ,s m )=1,1≤m ≤l )∧(Y (r m ,s m ,t m )=1,l +1≤m ≤j )),(2)where,for any given l , ⋆denotes the sum over all distinct ordered pairs (r m ,s m ),1≤m ≤l ,and (r m ,s m ,t m ),l +1≤m ≤j .We shall prove later that the relevant terms in this sum are the ones for which all the ordered pairs are disjoint,that is,there is no repetition of vertices among the j events.Assuming this,we obtainE ([Z ]j )=j l =0 |U i −1|j |V i −1|j n −o (n )j −l jl 2l ![(j −l )!]2××(d −1)2dn 2−o (n 2)j −l .This is because there are |U i −1|j |V i −1|j n −o (n )j −l ways of choosing j vertices in each of U i and V i ,and of choosing j −l vertices in V (G )\(N i −1(u )∪N i −1(v )).Moreover,pairing l of the chosen vertices in U i with l of the chosen vertices in V i can be done in j l 2l !ways,whereas there are [(j −l )!]2ways of creating triples on the remaining chosen vertices in U i ,V i and the vertices chosen in V (G )\(N i −1(u )∪N i −1(v )).Now that we ﬁxed distinct ordered pairs (r m ,s m ),1≤m ≤l ,and (r m ,s m ,t m ),l +1≤m ≤j ,the term (d −1)2dn 2−o (n 2) j −l corresponds to the probability that all the events X (r m ,s m )=1and Y (r m ,s m ,t m )=1occur simultaneously,since there is only a ﬁnite number of them.The previous sum is asymptotic toj l =0|U i −1|jj !n j −l l ! (d −1)2dn 2 j −l =|U i −1|j |V i −1|j (d −1)2j l !(j −l )!=|U i −1|j |V i −1|j (d −1)2jn j j !=1j ! d 2(d −1)2k8ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALD terms of this form in equation (2)are bounded above byj l =0 ∗∗ j −1a |U i −1|j −a j −1b |V i −1|j −b j −l −1c n −o (n )j −l −c ×× j l 2l ![(j −l )!]2 (d −1)2dn 2−o (n 2) j −l ,where ∗∗denotes the sum over all triples (a,b,c )∈{0,...,j −1}2×{0,...,j −l −1}satisfying a +b +c ≥1.This is because there are |U i −1|j −a ways of choosing j −a vertices in U i −1and j −1a ways of building a multi-set of cardinality j with j −a given elements (and using all of them).The same is true for choosing vertices in V i −1and V (G )\(N i −1(u )∪N i −1(v )).Our last expression is smaller or equal toj l =0 ∗∗(j −1)a +b (j −l −1)c(j −a )!|V i −1|j −b (j −l −c )!××j !2dn −o (n )l (d −1)3n j j !,this is asymptotic (with respect to n )toj l =0 ∗∗K (a,b,c,j,l,d )2log d −1n .So,P (E k )<ǫfor k suﬃciently large,and our result follows.Lemma 2.5.Let k be an integer and let γ(n,d )be the fractional part of 1GEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS9 is,paths of theﬁrst case,created at step i.Let E i be the equivalent random variable for paths of even length.Then(i)Withµk=d(d−1)2k−2γ(n,d)−2,P(O i0+k =j|N i0+k−1(u)∩N i0+k−1(v)=∅)∼µjkj!exp(−νk).Proof.This can be proven by the method of moments using calculations very similar to the ones in the previous lemma,proceeding separately for joins of theﬁrst type and joins of the second type.For the former,we condition on the event that no joins occurred in previous steps of the process,and,for the latter,we further assume that no joins of theﬁrst type occurred in the current step.The details are omitted. We observe that,alternatively,the proofs of the previous lemma and of Lemma2.3 could be uniﬁed by considering joint factorial moments of random variables for joins of theﬁrst type and of the second type.We are now ready to deduce one of the main results.Theorem2.6.Fix an integer l≥1.The probability that two vertices u,v chosen independently with uniform probability in G∈G n,d are joined by exactly l distinct geodesics is asymptotic to∞k=−∞ d(d−1)2k−2γ(n,d)−2 l d−2 ×× 1+(d−1)l exp(−d(d−1)2k−2γ(n,d)−1) . Proof.Let Z l be the event that u and v are joined by exactly l geodesics,and let J i denote the event that theﬁrst join occurs at time i.Then,given a positive integer M,P(Z l)=P(Z l∧i0−Mk=1J k)+M−1 k=−M+1P(Z l∧J i0+k)+P(Z l∧ k≥i0+M J k).Theﬁrst and last element in the right-hand side can be made less thanǫ,for any givenǫ>0,by choosing M=Mǫsuﬃciently large,as ensured by Corollary2.4andby the fact that equation(1)holds a.a.s.Also,each of the terms P(Z l∧J i0+k ),for−M+1≤k≤M−1,is equal toP(N i0+k−1(u)∩N i0+k−1(v)=∅) P(O i0+k=l|N i0+k−1(u)∩N i0+k−1(v)=∅)++P(O i0+k=0|N i0+k−1(u)∩N i0+k−1(v)=∅)××P(E i0+k=l|O i0+k=0∧N i0+k−1(u)∩N i0+k−1(v)=∅) .10ITAI BENJAMINI,CARLOS HOPPEN,ERAN OFEK,PAWE L PRA L AT,AND NICK WORMALD By our previous lemmas,this is asymptotic toexp −d(d−1)2k−2γ(n,d)−2l!exp(−d(d−1)2k−2γ(n,d)−2)+ + d(d−1)2k−2γ(n,d)−1 lexp −d(d−1)2k−2γ(n,d)−1l!d−2 ×× 1+(d−1)exp(−d(d−1)2k−2γ(n,d)−1) . The probability here is a function ofγ(n,d)and oscillates asγ(n,d)varies from0to1. We include some numerical results in the table below for some values of d,where prob is the probability of a unique geodesic asγ(n,d)=0and osc is the maximum variation with respect toγ=0asγvaries from0to1.d osc0.721341.4×10−30.5444107.6×10−20.3743S(d−1)2 −γ(n,d)+log(d−1)2dd−1S(d−1)2 −γ(n,d)+log(d−1)2dd−1Setting z=c t+x givesS c(x)=1πlog c|sinh(2π2m/log c)| 1/2.(5) The term for m=0in the sum(4)is independent of x,hence it yields terms independent of n in equation(3).In the special case d=3,equation(5)leads to the following bounds on the other terms of the sum(4).For|m|=1,the bound is approximately4.32×10−3,for|m|=2,it is approximately4.94×10−6,and for larger |m|the bounds are even smaller.Similar observations explain the small oscillations when d is small.3.Almost geodesic cyclesIn the proof of Theorem1.1we shall use the following auxiliary result.Lemma3.1.Let G∈G n,d and let u,v be vertices chosen independently at random in G.Consider functionsα(n),β(n)such thatα(n),β(n)→∞,α(n)=o(log d−1n)and β(n)=o(α(n)).Then a.a.s.every vertex at distance⌊α(n)⌋from u or v lies on at most one uv-path with length less than or equal to log d−1n+β(n).Proof.We prove this result for vertices at distance⌊α(n)⌋from u,and by a similar argument the same result holds for vertices at distance⌊α(n)⌋from v.As in Section2, we consider the process of exposing the neighbourhoods of u and v based on the pairing model.Here,N(u),the neighbourhood of u,is exposed for⌊α(n)⌋steps while N(v), the neighbourhood of v,is exposed for⌊12log d−1n−α(n)+2β(n)+2.Deﬁne Y= w∈UαY i.It is clear that this lemma follows if we prove that a.a.s.Y=0. We shall do this by usingP Y≥1 ≤ w∈UαP Y w=1 ,and by showing that the right-hand side goes to zero as n tends to inﬁnity.For aﬁxed w,deﬁne the set N′w obtained by the exposure of the neighbourhood of w for1is not added to N ′w at the ﬁrst step of the process,that is,only the “new”neighboursof w are exposed.As in Lemma 2.1,we use the term “bad edges”for edges that yield cycles in N ′w .Consider the random variable X w counting the number of “bad edges”in N ′w .Then,calculations analogous to the ones in Lemma 2.1establish thatE (X w )=⌊1n=O (d −1)log d −1n −2α(n )+4β(n )2log d −1n −α(n )+2β(n )+2and with the property that,after they ﬁrst split,they do not join again.So,a bound on P Y w =1∧X w =0 may be obtained by counting the number of possible pairs of distinct paths P and Q joining u i to N (v )with lengths r and s ,where r ≤s ≤12log d −1n ⌋,P Y w =1∧X w =0 =⌊i 0−α(n )+2β(n )+2⌋ s =1⌊i 0−s ⌋ r =1r −1 j =0(d −1)⌊i 0−β(n )⌋2 n −o (n )r +s −j −2 r +s −j −2j r +s −2r −1 ×j !(r −1)!(s −1)!O(d −1)n 2 =O (d −1)2β(n )−2α(n ) .Note that the formula holds because there are at most (d −1)⌊i 0−β(n )⌋2 ways of choosing two vertices in N (v )and there are n −o (n )r +s −j −2 ways of choosing vertices in the graphto include in the two paths.Moreover,these vertices can be divided into vertices of P ∩Q ,P −Q and vertices of Q −P in r +s −j −2j r +s −2r −1 ways and can then be ordered to form the paths in j !(r −1)!(s −1)!ways.Finally,each edge on the path appears with probability at most (d −1)We conclude thatP Y w=1 =P X w≥1 P Y w=1|X w≥1 +P X w=0 P Y w=1|X w=0 ≤P X w≥1 +P Y w=1|X w=0=O (d−1)4β(n)−2α(n) +O (d−1)2β(n)−2α(n)3 from their endpoints.We prove the claim by contradiction.Suppose without loss of generality that such a vertex is closer to u on P and let w be the vertex on P at distance⌊ω(n)3a.a.s.induces a tree.But then,w lies on at least two distinct u,vpaths with length less than or equal to log d−1n+logω(n),which a.a.s.does not occur by Lemma3.1withα(n)=ω(n)¨¨¨¨¨¨r r r r r r P r r r r r r ¨¨¨¨¨¨Qr r r r r r r r r r r r r r r r r r r r r r r r r r u v v P p q v Q R Figure 3.Path RR of length at most log d −1n −ω(n )using vertices and edges on P ∪Q only.But d (u,v )≥log d −1n −ω(n )implies that R does not contain vertices at distance less than or equal to ω(n )2(log d −1n −r −s −ω(n ))steps.The probabilityofthis can be calculated as in the earlier sections,and we conclude that P(Y P,Q≥1)≤E(Y)≤⌈12(log d−1n−ω⌉)−rs=0P(X p r,q s)=4⌈12(log d−1n−ω)⌉−rs=0O (d−1)2(1By Claim1,we may assume there is no vertex in common between P and Q that is more thanω(n)/3from u and v.Forωgrowing slowly enough,there is a.a.s.no point in common that is at mostω(n)/3from u and v either,since Lemma2.1implies that a.a.s.neither u nor v is in a short cycle.Let C be the union of the paths P and Q. From the bounds on d(u,v),C has length at least2log d−1n−2K.To prove the statement about all diametrically opposite points p and q on C,we may rework the argument in Lemma3.2.The Claim proved above shows that every short path of the type we are interested in must use some edge not on P or Q.Arguing as before,we only need to eliminate the existence of A such that(6)holds.The same argument as before shows that for anyﬁxed such p and q,with Y P,Q deﬁned as before, we again have P(Y P,Q≥1)=O (d−1)−ω(n) .Now apply this inequality to the O(log d−1n)pairs of vertices p and q diametrically opposite on C.Also,putω(n)=log d−1log d−1n+γ(n).Then the probability that Y P,Q≥1for at least one of these choices of p and q is O (d−1)−γ(n) .Hence,ifγ(n)→∞,we have a.a.s.for all such p and q,d(p,q)≥f(n)−log d−1log d−1n−γ(n).Replacingγbyωgives theﬁnal statement in Claim2, with probability at least1−2ε+o(1).This statement is true for allε>0.That fact implies that theﬁnal statement in Claim2holds a.a.s.(This can be regarded as “lettingε→0suﬃciently slowly”.)Combining this with part(i)proves Claim2,since, although there may be diﬀerent functions at the diﬀerent occurrences ofω,they can be made the same.This completes the proof of Claim2.To obtain the theorem,we note that the proof of Claim2does more:it shows that the two paths can be chosen to use distinct neighbours of u in their initial step,and distinct neighbours of v in theirﬁnal step.4.Final remarksIn this article we have examined the“shape”of random regular graphs.This brings up related questions.Our proof of the main theorem can be seen to give more:a.a.s.for every pair of short (i.e.bounded length)paths,one containing u and one containing v,there is an almost geodesic cycle containing both of these paths.We also show that the paths referred to in the theorem each contain a geodesic between the two vertices at distance k from its ends,for any k tending to inﬁnity with n.A geodesic cycle C in G is a cycle in which for every two vertices u and v in C, the distance d G(u,v)is equal to d C(u,v).A signiﬁcant open problem is to determine whether in a random d-regular graph,a.a.s.almost all pairs of vertices lie in a geodesic cycle.It is not even known if at least one geodesic cycle of length asymptotic to log d−1n exists a.a.s.We may also draw conclusions on how“thin”the topological triangles are in random regular graphs.Consider the proof of Lemma2.3,which analyses the time at which two simultaneous breadth-ﬁrst reaches from u and from v join each other.The proof is concerned with an accurate estimate of the probability that there are no joins by a time near i0.It is easy to see from the ideas in the proof that for large K,the second join is quite likely to occur by time i0+K,and furthermore that theﬁrst two joinsare quite likely to be in branches that diverged,in the breadthﬁrst search from u,at time less than K,and similarly from v.Let u′and v′be the points of divergence near u and v.Then the joins give two paths P and Q from u′to v′,the shorter of which, say P,is geodesic,and we can choose another vertex,w,on Q,of distance K from u′,such that the resulting two subpaths of Q to u′and v′from w are both geodesic. Thus u′,v′,w form a geodesic triangle.By Lemma3.2(noting P and Q are2K-near geodesics from u to v),the distance between the midpoints of P and Q is a.a.s.at least log d−1(n)−ω(n),whereω(n)is any function tending to∞.Hence the midpoint p of P a.a.s.has distance at least1。

Inclusive Measure of V_ub with the Analytic Coupling Model

a r X i v :0711.0860v 1 [h e p -p h ] 6 N o v 2007ROME1/1461-07DSNA/33-2007November 2007Inclusive Measure of |V ub |with the Analytic Coupling Model U.Aglietti a , F.Di Lodovico b ,G.Ferrera c ,G.Ricciardi d a Dip.Fis.,Univ.di Roma I “La Sapienza”&INFN Roma,Roma,Italy b Queen Mary,University of London,Dep.of Phys.,London,UK c Dip.Fis.,Univ.di Firenze &INFN Firenze,Sesto Fiorentino,Firenze,Italy d Dip.Scienze Fis.,Univ.di Napoli “Federico II”&INFN Napoli,Napoli,Italy1IntroductionBy comparing various spectra in the decaysB →X u +ℓ+νℓ(1)with the predictions of a model including non–perturbative corrections to soft–gluon dynamics through an eﬀective QCD coupling [1],we obtain a value for the V ub Cabibbo–Kobayashi–Maskawa (CKM)matrix element [2]|V ub |=(3.69±0.13±0.31)×10−3,the errors are experimental and theoretical,respectively.The model basically involves the insertion,inside standard threshold resummation formulae,of an eﬀective QCD coupling ˜αS (k 2),based on an analyticity requirement and resumming absorptive eﬀects in gluon cascades [3].At the lowest order for example,it reads:˜αlo S (k 2)=12−1π ==131β0log 5(k 2/Λ2),(2)where Λ≈300MeV is the QCD scale,β0=(11−2/3nf )/(4π)is the ﬁrst QCD β-function coeﬃcient and n f is the number of active ﬂavors 1.By construction,˜αS (k 2)has no Landau pole and saturates at small scales:lim k 2→0˜αS (k 2)=11It is remarkable that all higher–order corrections to the standard coupling inside ˜αS are of log-arithmic form.That is to be contrasted to the case of the analytic coupling for space–like processes (see,f.i.,formula (25)in [1]).are proportional to|V ub|2and there are two diﬀerent methods for measuring this matrix element.In theﬁrst method,one considers a speciﬁc exclusive decay,such as for exampleB→ρℓνℓ,(5) by identifying experimentally theﬁnal–state hadrons.Dynamics are substantially non perturbative and current theoretical predictions use QCD sum rules,quark models, lattice QCD,etc.The second method involves the inclusive hadronﬁnal states X u in eq.(1).In general,given a kinematical variable p,such as for example the energy Eℓof the charged lepton,one measures the number of B’s decaying semileptonically to X u with p in some interval(a,b),divided by the total number of produced B’s(decaying into any possibleﬁnal state):B[p∈(a,b)]≡N[B→X uℓνℓ,p∈(a,b)]|V ub|2≈102,(8) b→cℓνℓdecays constitute a huge background to b→uℓνℓones as far as inclusive quantities are concerned2.To avoid(or at least substantially reduce)such backgrounds, one may consider kinematical regions where b→c transitions are kinematically forbid-den(or at least strongly disfavored).Typically,one has to consider end–point regions for the variable p.On the theoretical side,this restriction has a price,because the available phase–space to QCD partons gets strongly reduced.One ends up with the so–called threshold region,deﬁned as having parametrically3m X≪E X.(9) The perturbative expansion of spectra in the threshold region is aﬀected by large log-arithms≈αn S log2n(2E X/m X),which must be resummed to all orders inαS in order to have a reliable result[5–7].Consistent inclusion of subleading logarithms requires a prescription for the QCD coupling in the low–energy region∼Λ—in principle com-pletely arbitrary—which in our model is the analyticity condition.Furthermore Fermimotion,a genuine non–perturbative eﬀect related to a small vibration of the b quark√in the B meson,comes into play when m X becomes as small as≈2The non–vanishing charm mass reduces the b→cℓνℓrate roughly by a factor two.3This region is also called Sudakov region,large–x region and radiation–inhibited region.Let us compare virtues and short–comings of the two methods for measuring|V ub|. The exclusive determination uses a smaller event sample because it deals with a single channel,with the consequence that larger statistical errors are expected.Since therelevant hadronic matrix elements for eq.(5), ρ|J b→uµ|B ,can be computed in this case with aﬁrst–principle technique,namely lattice QCD,one expects that,with in-creasing computing resources,hadronic uncertainties can be systematically(and almost arbitrarily)reduced.On the contrary,the inclusive method suﬀers less from statistics, but needs a modelling of non–perturbative QCD eﬀects,which cannot be completely derived fromﬁrst principles in explicit form.To summarize,asymptotically in time, we expect the exclusive measure to take over the inclusive one.At present,the determinations from inclusive and exclusive decays are given with a relative precision of about7%and11%[8],respectively:|V ub|=(4.49±0.33)×10−3(inclusive),(10)|V ub|=(3.50±0.40)×10−3(exclusive).(11) There is a≈2σdiscrepancy between the above measurements,indicating some“ten-sion”between the above methods.A third independent measure of|V ub|stems from a generalﬁt of the Standard Model (SM)parameters.One assumes the validity of the Standard Model—and therefore also the unitarity of the CKM matrix—without using the direct inclusive or exclusive determinations.This indirect measure gives[9]:|V ub|=(3.44±0.16)×10−3(SMﬁt).(12) The globalﬁt of the SM therefore“prefers”the exclusive determination,while the inclusive one is in agreement at≈3σlevel only.It has been suggested that the discrepancy between the value of the experimen-tal measurement and the inclusive theoretical prediction could signal eﬀects of new physics from extra Higgs particles[10].In our opinion,the above discrepancy does not necessarily imply a signal of new physics.In other words,we believe that the“ten-sion”can be dynamically explained inside the Standard Model.Even though there are several models in literature describing non–perturbative eﬀects in inclusive B decays, which give results perfectly consistent with each other[11–16],we believe that a possi-ble interpretation of the above scenario is that the theoretical uncertainties have been under–estimated.A re–analysis of the same data with a rather diﬀerent model may therefore be useful.To re–extract|V ub|in this spirit,it is convenient to identify the diﬀerent dynamical eﬀects which come from theory and cannot be extracted from the data.As it will be explicitly shown in sec.3,one has to compute,roughly speaking, both:1.inclusive rates,(strongly)dependent on the choice of the heavy quark masses m b(and m c),as well as on the QCD coupling at a reference scale(typicallyαS(m Z)).In sec.3we present two methods which diﬀer in the treatment of the inclusive quantities;we also discuss our choices of the b and c masses;2.suppression factors,for the restriction of the kinematical variable p in some experi-mentally accessible range.These factors are aﬀected by large threshold logarithms and by the related Fermi–motion eﬀects mentioned earlier.The discrepancy of our analysis with respect to previous ones does not rely on the estimate of inclusive quantities(diﬀerent choices of quark masses,ofαS(m Z),etc.),i.e. on point1.,but on the modelling of the threshold region,i.e.on point2.Sec.4contains our results for|V ub|coming from the various distributions considered together with a discussion,while sec.5presents our conclusions.2Threshold resummation with an eﬀective couplingLet us brieﬂy describe in this section the phenomenological model used to extract|V ub|, namely threshold resummation with a time-like QCD coupling for the semi–inclusive B decays given in eq.(1)(for a more detailed discussion we refer the reader to ref.[1]). Factorization and resummation of threshold logarithms in semileptonic decays leads to an expression for the triple–diﬀerential distribution,the most general distribution,of the following form[17]:1dxdwdu=C[x,w;αS(Q)]σ[u;Q]+D[x,u,w;αS(Q)],(13)where:x=2E lm b,u=1− 1+y[(1−y)N−1−1] Q2y Q2y2dk2⊥σN(Q)is the Mellin transform ofσ(u,Q),σN(Q)= 10(1−u)N−1σ(u;Q)du.(16)The functions˜A(˜αS),˜B(˜αS)and˜D(˜αS)haveﬁxed order expansions in˜αS and are obtained from the standard ones A(αS),B(αS)and D(αS)by means of the change of renormalization scheme for the coupling constantαS→˜αS.The QCD form factor σ[u;Q]has been numerically computed for diﬀerent values ofαS(m Z)in[1].Threshold suppression—in our opinion the main theoretical ingredient for the measure of|V ub|,as discussed in the introduction—is represented by the factorΓ[B→X uℓνℓ,p∈(a,b)]W(a,b)≡d3ΓΓd3ΓΓ3|V ub|extraction:methodThe branching ratio in eq.(6)can be computed theoretically as:B[p∈(a,b)]=Γ[B→X uℓνℓ,p∈(a,b)].(20)Γ[B→(anything)]Since we do not aim to check QCD but only to extract|V ub|,we can use the experimental measure of the B lifetime,which is rather accurate4:τB=(1.584±0.007)×10−12s,(21) and limit ourself to compute the suppression factor W(a,b)deﬁned in the previous section(see eq.(17))and the inclusive B→X uℓνℓrate:G2F m5b|V ub|2Γ[B→X uℓνℓ]=.(25)m bFor example,a tiny uncertainty of±2%on the m b mass,which is compatible with the actual estimates,corresponds to about a±10%uncertainty in the total semileptonic b→u width,translating into a±5%uncertainty on|V ub|.One can eliminate the above(undesired)dependence on m5b,and the related uncer-tainty,by expressing the branching ratio as follows:B[p∈(a,b)]=B SL(27)Γ[B→(anything)]and the ratio of(b→c)/(b→u)semileptonic widths:R c/u≡Γ(B→X cℓνℓ)I(ρ)F(αS)G(αS,ρ),(30)192π3wherem2cρ≡+8ρ3−ρ4.(32)ρNote that there is an(accidental)strong dependence on the charm mass m c,because of the appearance of a large factor in the leading term inρ,namely−8.As far asinclusive quantities are concerned,the largest source of theoretical error comes indeedfrom the the uncertainty inρ.Most of the dependence is actually on the diﬀerence m b−m c,which can be estimated quite reasonably with the Heavy Quark Eﬀective Theory(HQET)—see next section.Finally,the factor G(αS,ρ)contains correctionssuppressed by powers ofαS as well by powers ofρ:G(αS,ρ)=1+∞n=1G n(ρ)αn S,(33)with G n(0)=0.Note thatG(0,ρ)=G(αS,0)=1.(34) By inserting the above expressions for the semileptonic rates,one obtains for the per-turbative expansion of R c/u:R c/u=R c/u(ρ,αS,|V ub|/|V cb|)=|V cb|26The average of inclusive and exclusive determinations of|V cb|,in good agreement with each other,is|V cb|=(41.6±0.6)×10−3[19].known that pole masses are aﬀected by the poor behavior of the perturbative series relating them to physical quantities;replacing the pole mass with theMS mass scheme.Them b(m c(1m c−m D+3m D⋆04−m D+3m D⋆04−(2.41±0.23)%B(B→X c eνe)=7We use m B0=5279.50±0.33MeV,m B⋆0=5325.1±0.5MeV,m D0=1864.84±0.17MeV and m D⋆0=2006.97±0.19MeV[19].8We neglect B→X uτντdecays which constitute at most a10%of B→XτντThe corresponding theoretical quantity strongly depends on m b and m c[23]:Γ(b→cτντ)X th(m b,m c,αS)≡4.(m X,q2):where the distribution looked at is a two dimensional distribution in theplane of the hadronic mass and the transferred squared momentum q2to the lep-ton pair.The analyses from B A B A R and Belle are described in[28]and[29],respec-tively,using the B–reconstruction analysis.Moreover,another technique,called simulated annealing,is used by Belle to select events in the(m X,q2)plane[30].Events with hadron mass lower than1.7GeV and q2larger than8GeV2are selected in all the cases.5.(Eℓ,s maxh ):where the distribution looked at is a two dimensional distribution inthe electron energy and s maxh ,the maximal m2X atﬁxed q2and Eℓ.There is aresult from B A B A R[31].The requests on the kinematic variables are Eℓ>2.0GeV and s maxh<3.5GeV2.We compute|V ub|for each of the analyses starting from the corresponding par-tial branching fractions.Then,we determine the average|V ub|value using the HFAG methodology[8].Averaged values of|V ub|are presented in the case of all the uncorrelated measure-ments,where the uncorrelation refers to the experimental errors,and in the case of each category of kinematic variables.There are large correlations among the three kinematical distributions in the B–reconstruction analysis,which are not reported in the experimental papers.Thus,we use only the m X analysis,which is the one with the smallest experimental error,when performing the average on all the available measure-ments.Table1reports the extracted values of|V ub|for all the uncorrelated analyses and their corresponding average.The errors are experimental(i.e.statistical and systematic)and theoretical,respectively.The average is:|V ub|=(3.69±0.13±0.31)×10−3,(43) consistent with the measured value of|V ub|from exclusive decays[8]and the indirect measure[9].The table shows also the criteria used for the determination of the partial branching ratio(∆B).The theoretical errors are considered completely correlated among all the experimental analyses,when performing the average.The|V ub|values and the corresponding average are plotted in Figure2.Several sources of theoretical errors have been considered:•in addition to our preferred method based on eq.(26),we also use the method based on eq.(22)to extract the value of|V ub|.Since the two methods described in the previous section basically involve diﬀerent inclusive quantities,this error allows a cross–check of their evaluations,i.e.basically of the choices of the b andc masses adopted;•we compute inclusive quantities both in theMS masses is our default,the value usingTable1:Theﬁrst column in the table shows the uncorrelated analyses,the second column shows the corresponding values of|V ub|,andﬁnally the last column shows the criteria for which∆B is available.Theﬁnal row shows the average value of|V ub|.The errors on the|V ub|values are experimental and theoretical,respectively.The experi-mental error includes both the statistical and systematic errors.Analysis∆B criteria3.44±0.14±0.283.18±0.16±0.263.47±0.20±0.293.87±0.19±0.303.73±0.25±0.283.93±0.41±0.293.87±0.26±0.29AverageFigure2:|V ub|values for the uncorrelated analyses and their average.the pole scheme masses,where the ranges deﬁned in sec.3.1are used,gives the systematic uncertainty.Since in general higher–order corrections are diﬀerent in the two schemes,that should provide an estimate of the size of unknown higher–order eﬀects.•we vary the order at which the rate is computed from the exact NLO to the approximate NNLO[32].Since the perturbative series for QCD is believed to be an asymptotic one andαS=αS(m b)≈0.22in b physics is rather large,that should provide a reasonable estimate on the truncation error;•we vary all the parameters which enter in the computation of|V ub|within their errors,as given by the PDG[19].What we cannot change is the modelling of the threshold region,which isﬁxed in our model because,as discussed in the introduction,the latter has no free parameters. That is the factor W(a,b),which appears in both methods,virtually with no error. The error on the modelling of the threshold region can only be estimated in an indirect way,by considering diﬀerent decay spectra,in which presumably threshold eﬀects enter in diﬀerent ways.Future work towards improving the determination of the b–quark mass in theMS scheme(±4.4%),eq.(38)(±4.1%)and the variation from the<0.71m B,(44)m Bbecause q2>8GeV2.The lower cut on q2therefore signiﬁcantly reduces the hard scale Q from the“natural”value Q=m B.The point is that our model has been constructed to describe B–decay spectra having the(maximal)hard scale Q=m B,and not spectra having a smaller hard scale.Indeed,the model was checked against beauty fragmentation at the Z0peak[4],where the dominant infrared eﬀects are controlled by a hard scale equal to m B.In other words,to analyze the(m X,q2)distribution,we are using the model in a region where it has not been checked and there is no surprise that it does not work so well in this case.Table2:Theﬁrst column of the table shows the diﬀerent contributions to the theoret-ical errors,the second column shows the corresponding variation,andﬁnally the third column shows the percentage contribution with respect to the|V ub|value.Theoretical ErrorsVariationαS±0.6→3.5|V cb|±1.4m b(GeV)±1.0m c(GeV)±4.4B(B→X uℓνℓ)±1.0eqs.(36),(38)+4.1pole mass(GeV)−4.6approx.NNLO rate+2.15ConclusionsWe have analyzed semileptonic B decay data in the framework of a model for QCD non–perturbative eﬀects based on an eﬀective time-like QCD coupling,free from Landau singularities.The analysis has considered the kinematical distributions in Eℓ,m X,andP+,as well as the two dimensional distributions in(m X,q2)and(Eℓ,s maxh ),taking intoaccount the experimental kinematical cuts.Our inclusive measure of the|V ub|CKM matrix element is:|V ub|=(3.69±0.13±0.31)×10−3.(45) The errors on the|V ub|values are experimental and theoretical,respectively.The experimental error includes both the statistical and systematic errors.For theﬁrst time,an inclusive value for|V ub|is obtained which is in complete agreement with the exclusive determination.Current literature presents a discrepancy among previous inclusive determinations of|V ub|on one side and the exclusive deter-minations(≈2σ)and the over–allﬁt of the Standard Model(≈3σ)on the other side[8].Let us try to identify the diﬀerences between our approach and the previous ones. Aﬁrst diﬀerence lies in the selected electron energy range.According to our model, electron spectra below≈2.3GeV measured at the B–factories suﬀer from an under–subtracted charm background.Because of that,we have limited our analysis to pretty large electron energies E e>2.3GeV.If we take a smaller cutoﬀ,we obtain a larger value of|V ub|,in order to simulate b→c events.As far as theory is concerned,our model seems to produce a smaller Sudakov suppression compared to other models constructedTable3:The table contains the|V ub|values for several analyses and the corresponding averages.The errors on the|V ub|values are experimental and theoretical,respectively. The experimental error includes both the statistical and systematic errors.|V ub|for endpoint analyses(10−3)3.44±0.14±0.283.18±0.16±0.263.47±0.20±0.29Average|V ub|for m X analyses(10−3)3.87±0.19±0.303.73±0.25±0.28Average|V ub|for(m X,q2)analyses(10−3)4.03±0.26±0.304.20±0.37±0.313.93±0.41±0.29Average|V ub|for P+analyses(10−3)3.35±0.22±0.273.62±0.31±0.29Averageon top of soft–gluon resummation,such as for example the dressed gluon exponentia-tion[14].For aﬁxed experimental rate,a smaller Sudakov suppression implies indeed larger hadronic form factors and smaller|V ub|’s.As discussed above,we are rather conﬁdent in our model for phenomenological reasons rather than theoretical ones:we have checked it in beauty fragmentation[4],where soft contributions are similar to those in b decay9.Even though our model on soft–gluon dynamics is formally without free parameters,we may say that we have constructed it,out of many possibilities,as a kind of“expert system”.Once“trained”by giving beauty fragmentation data in input, it should predict reasonable beauty decay spectra.In conclusion,the inclusive extraction of|V ub|requires the calculation of inclusive quantities,strongly dependent on b and c masses,as well as the evaluation of threshold–suppressed quantities,the latter containing large infrared logarithms and Fermi–motion (non–perturbative)eﬀects.We argue that the main diﬀerence of our model with respect to previous ones is a smaller suppression of the threshold region.AcknowledgmentsThis work is supported in part by the EU Contract No.MRTN-CT-2006-035482“FLA-VIAnet”.References[1]U.Aglietti,G.Ferrera and G.Ricciardi,Nucl.Phys.B768(2007)85[arXiv:hep-ph/0608047].[2]N.Cabibbo,Phys.Rev.Lett.10,531(1963);M.Kobayashi and T.Maskawa,Prog.Theor.Phys.49,652(1973).[3]D.V.Shirkov and I.L.Solovtsov,Phys.Rev.Lett.79(1997)1209[arXiv:hep-ph/9704333].[4]U.Aglietti,G.Corcella and G.Ferrera,Nucl.Phys.B775(2007)162[arXiv:hep-ph/0610035].[5]S.Catani,M.L.Mangano,P.Nason and L.Trentadue,Phys.Lett.B378(1996)329[arXiv:hep-ph/9602208].[6]U.Aglietti and G.Ricciardi,Phys.Rev.D70(2004)114008[arXiv:hep-ph/0407225].[7]U.Aglietti,G.Ricciardi and G.Ferrera,Phys.Rev.D74(2006)034004[arXiv:hep-ph/0507285];Phys.Rev.D74(2006)034005[arXiv:hep-ph/0509095];Phys.Rev.D74(2006)034006[arXiv:hep-ph/0509271].[8]E.Barberio et al.[Heavy Flavor Averaging Group(HFAG)Collaboration],arXiv:0704.3575[hep-ex].[9]M.Bona et al.[UTﬁt Collaboration],JHEP0610(2006)081[arXiv:hep-ph/0606167].[10]M.Bona et al.[UTﬁt Collaboration],JHEP0603(2006)080[arXiv:hep-ph/0509219];G.Isidori and P.Paradisi,Phys.Lett.B639(2006) 499[arXiv:hep-ph/0605012];W.Altmannshofer,A.J.Buras,D.Guadagnoli and M.Wick,arXiv:0706.3845[hep-ph].[11]A.K.Leibovich,I.Low and I.Z.Rothstein,Phys.Rev.D61(2000)053006[arXiv:hep-ph/9909404];R.Akhoury and I.Z.Rothstein,Phys.Rev.D54(1996) 2349[arXiv:hep-ph/9512303];A.K.Leibovich,I.Low and I.Z.Rothstein,Phys.Lett.B486(2000)86[arXiv:hep-ph/0005124].[12]M.Neubert,Phys.Lett.B513(2001)88[arXiv:hep-ph/0104280];nge,M.Neubert and G.Paz,JHEP0510(2005)084[arXiv:hep-ph/0508178]; [13]C.W.Bauer,Z.Ligeti and M. 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On the spatial mean of the Poincare cycle

a rX iv:mat h /55625v1[mat h.PR]28May25ON THE SPATIAL MEAN OF THE POINCAR ´E CYCLE LUIS B ´AEZ-DUARTE Abstract.This is an English translation of an article by the author appeared in Spanish in 1964.Let X be a measure space and T :X →X a measurable transformation.For any measurable E ⊆X and x ∈E ,the possibly inﬁnite return time is n E (x ):=inf {n >0:T n x ∈E }.If T is an ergodic tranformation of the probability space X ,and µ(E )>0,then a theorem of M.Kac states that E n E dµ=1.We generalize this to any invertible measure preserving transformation T on a ﬁnite measure space X ,by proving independently,and nearly trivially that for any measurable E ⊆X one has E n E dµ=µ(I E ),where I E is the smallest invariant set containing E .In particular this also provides a simpler proof of Poincar´e ’s recurrence theorem.The purpose of this note is to give an independent and rather elementary proof of a slight generalization of Kac’s well-known theorem [3]about the spatial mean of the Poincar´e return time.The result implies Poincar´e ’s recurrence theorem as well.Let (X,S ,µ)be a ﬁnite measure space,and T a measure preserving invertible transformation of X onto itself,that is,µ(E )=µ(T −1E )=µ(T E ),∀E ∈S .For any E ∈S deﬁne the associated function n E by n E (x ):=inf {n :n >0,T n x ∈E },which may possibly be inﬁnite;n E (x )is the ﬁrst time of return of x to E ,or the Poincar´e cycle of x .We denote by I E the smallest invariant set containing E .As usual χA is the characteristic function of the set A .Theorem 0.1.For every E ∈X (0.1) En E dµ=µ(I E ).Proof.By deﬁnition of n E we haveχE c (T x )χE c (T 2x )...χE c (T n x )=1,n <n E (x ),=0,n ≥n E (x ),2LUIS B´AEZ-DUARTE which added over n yieldsn E(x)=1+∞n=1χE c(T x)χE c(T2x),...χE c(T n x)=1+∞n=1χ( nν=1T−νE)c(x).Now we integrate the above expression for n E to obtain (0.2) E n E dµ=µ(E)+∞ n=1µ n ν=1T−νE c∩E .But T n is invertible measure preserving soµ n ν=1T−νE c∩E =µ n−1 ν=0TνE c∩T n E ,which substituted in(0.2)yieldsE n E dµ=µ(E)+∞ n=1µ n−1 ν=0TνE c∩T n E=µ ∞ ν=0TνE ,(0.3)where the last equality follows from the canonical disjoint decomposition of a countable union:∞ν=0TνE=E∪(E c∩T E)∪((E∪T E)c∩T2E)...But it is obvious that I E= n≥0T n E,which substituted in(0.3)gives (0.1). Remark0.1.The fact that E n E dµ<∞implies that n E(x)<∞for a.e. x;this is Poincar´e’s recurrence theorem in a slightly more general setting. For a probability space X and an ergodic transformation T,ifµ(E)>0 then I E=X,so we get Kac’s theorem in[3],that is E n E dµ=1.References1.L.B´a ez-Duarte,Sobre el promedio espacial del ciclo de Poincar´e,Bolet´in de laAcademia de Ciencias F´isicas,Matem´a ticas y Naturales,24No.67,(1964),64-66.2.P.R.Halmos,Ergodic Theory,Chelsea Publ.Co.,New York1956.3.M.Kac,On the Notion of Recurrence in Discrete Stochastic ProcessesBull.Am.Math.Soc.vol.53(1947),1002-1010.Departamento de Matem´a ticas,Instituto Venezolano de Investigaciones Cient´ıficas,Apartado21827,Caracas1020-A,VenezuelaE-mail address:lbaez@ccs.internet.ve。

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I. INTRODUCTION
Physical laws governing the microscopic processes are mostly reversible in time. In macroscopic world, however, people often experience time-irreversible phenomena in their daily life. To understand why the reversible microscopic processes lead to irreversible macroscopic manifestations one refers to the Poincar´ e Theorem, which states that a system having a ﬁnite energy and conﬁned to a ﬁnite volume will, after a suﬃcient long time – the so called Poincar´ e cycle, return to an arbitrarily small neighborhood of almost any given initial state [1]. The key point is to note that the typical value of a Poincar´ e cycle for even a moderate-sized system is far beyond the meaningful time scale one can measure or experience, thus the irreversiblity is realized. Usually to describe a macroscopic system one has to know only a few parameters, such as volume, pressure, and temperature. However, to describe the same system in terms of its microscopic constituents, one has to deal with a large number of parameters, such as the momenta and positions of a huge amount of particles, which are impossible to calculate in practice. Based on this reason together with the fact that the macroscopic laws are insensitive to the microscopic details (of system history), it is natural for people to adopt the probability (ensemble) description in statistical mechanics, which deals with the equilibrium state (a macroscopic state that has stationary value of state parameters) of a macroscopic system. In this kind of description the macroscopic quantities are deﬁned as the ensemble average of their microscopic correspondences. This deﬁnition connects the microscopic and macroscopic worlds. To study how a system approaches its equilibrium state one also uses probability description, where the evolution of the system is treated as a stochastic process. One fa1
mous model for simulating such a process was proposed by Ehrenfest one century ago [2], which is an N -ball, 2-urn problem. In the beginning N numbered balls are distributed arbitrarily in either urn A or urn B . At each time step one ball is picked out at random and then put into the other urn. This simple model can be exactly solved to give an explicit Poincar´ e cycle. This model was then generalized by several authors to mimic more complicated situations encountered in real physical phenomena [3–5]. An attractive feature of these urn model problems is that they are easy to formulate but not always easy to solve. The solutions obtained have, therefore, sometimes led to new mathematical techniques and insights [6–9]. Recently, some new urn models were proposed and solved analytically or numerically. Their results provide very good descriptions on granular and glass systems [10–14]. In this paper, we obtain the exact solution of a generalized urn model. Hereafter we call it “periodic urn model”. In this model, one considers N distinguishable balls which are distributed in M urns. These M urns are arranged along a circle and numbered one by one to form a cycle, that is, we deﬁne the (M + 1)th urn as the 1st urn (See Fig. 1). To begin with, the initial distribution of the N balls in the M urns is given by |m1,0 , m2,0 , · · · , mM,0 ≡ |m0 , where mi,0 is the number of balls in the ith urn at the start. At each time step one ball is picked out of the N balls such that every ball has an equal probability of being picked up. The ball is then placed into the next numbered urn. The state that the ith urn contains mi balls is represented by |m1 , m2 , · · · , mM ≡ |m , which we name it state vector. Hereafter we call a distribution string m (without knowing the numbering of the balls) a conﬁguration of the system. Otherwise, if we also know the location of each numbered ball, we call such a distribution a microstate
Poincar´ e cycle of a multibox Ehrenfest urn model with directed transport
Yee-Mou Kao
National Center for High-Performance Computing, No.21, Nan-ke 3rd.Rd., Hsin-Shi, Tainan County 744, Taiwan, Republic of China
arXiv:cond-mat/0210338v1 [cond-mat.stat-mech] 16 Oct 2002
Pi-Gang Luan
Institute of Optical Sciences, National Central University, Chung-Li 32054, Taiwan, Republic of China (February 1, 2008) We propose a generalized Ehrenfest urn model of many urns arranged periodically along a circle. The evolution of the urn model system is governed by a directed stochastic operation. Method for solving an N -ball, M -urn problem of this model is presented. The evolution of the system is studied in detail. We ﬁnd that the average number of balls in a certain urn oscillatபைடு நூலகம்s several times before it reaches a stationary value. This behavior seems to be a peculiar feature of this directed urn model. We also calculate the Poincar´ e cycle, i.e., the average time interval required for the system to return to its initial conﬁguration. The result can be easily understood by counting the total number of all possible microstates of the system. PACS numbers: 05.20.-y, 02.50.Ey, 02.50.-r,