IAC-02-Q.3.3.03 THE INTERPLANETARY INTERNET A COMMUNICATIONS INFRASTRUCTURE FOR MARS EXPLOR

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Palo Alto Networks Cortex XSOAR Threat Intelligenc

Palo Alto Networks Cortex XSOAR Threat Intelligenc

Cortex XSOAR ThreatI ntellig ence Manag ementThreat intelligence is at the core of every security operation. It applies to every security use case. Unfortunately, security teams are too overtaxed to truly take advantage of their threat intelligence, with thousands of alerts and millions of indicators coming at them daily. They require additional context, collaboration, and automation to extract true value. They need a solution that gives them the confidence to do their jobs effectively and shore up their defenses against the attacker’s next move.Cortex® X SOAR Threat Intelligence Management (TIM) takes a unique approach to native threat intelligence management, unifying aggregation, scoring, and sharing of threat intelligence with playbook-driven automation.Features and Capabilities Powerful, native centralized threat intel : Supercharge i nvestigations with instant access to the massive repository of built-in, high-fidelity Palo Alto Networks threat intelli -gence crowdsourced from the largest footprint of network, endpoint, and cloud intel sources (Tens of millions of mal -ware samples collected and firewall sessions analyzed daily).Indicator relationships : Indicator connections enable struc -tured relationships to be created between threat intelligence sources and incidents. These relationships surface importantcontext for security analysts on new threat actors and attack techniques.Hands-free automated playbooks with extensible integra-tions : Take automated action to shut down threats across more than 600 third-party products with purpose-built p laybooks based on proven SOAR capabilities.Granular indicator scoring and management : Take charge of your threat intel with playbook-based indicator lifecycle man -agement and transparent scoring that can be extended and customized with ease.Automated, multi-source feed aggregation : Eliminate manual tasks with automated playbooks to aggregate, parse, prioritize, and distribute relevant indicators in real time to security con -trols for continuous protection.Most comprehensive marketplace : The largest community of integrations with content packs that are prebuilt bundles of integrations, playbooks, dashboards, field subscription services, and all the dependencies needed to support specific security orchestration use cases. With 680+ prebuilt content packs of which 700+ are product integrations, you can buy i ntel on the go using Marketplace points.Business Value Figure 1:Control, enrich, and take actionwith playbook-driven automation Take Full Control Take complete control of your threat intelligence feeds Enrich Incident Response Make smarter incident response decisions by enriching every tool and process Actionable IntelClose the loop between intelligence and action withplaybook-driven automationFigure 2: Take control of your threat intel feedFigure 3:Make smarter decisions by enriching and prioritizing indicatorsFigure 4: Close the loop between intel and action with automationThreat Intelligence Combined with SOAR Security orchestration, automation, and response (SOAR) solutions have been developed to more seamlessly weave threat intelligence management into workflows by combin -ing TIM capabilities with incident management, orchestra -tion, and automation capabilities. Organizations looking for a threat intelligence platform often look for SOAR solutions that can weave threat intelligence into a more unified andautomated workflow—one that matches alerts both to their sources and to compiled threat intelligence data and that can automatically execute an appropriate response.As part of the extensible Cortex XSOAR platform, threat intel management unifies threat intelligence aggregation, scoring, and sharing with playbook-driven automation. It empowers security leaders with instant clarity into high-priority threats to drive the right response, in the right way, across the entire enterprise.way. Automated data enrichment of indicators provides ana-lysts with relevant threat data to make smarter decisions. Integrated case management allows for real-time collabora-tion, boosting operational efficiencies across teams, and auto-mated playbooks speed response across security use cases. Key Use CasesUse Case 1: Proactive Blocking of Known ThreatsChallengeThe security team needs to leverage threat intelligence to block or alert on known bad domains, IPs, hashes, etc. (indicators). The indicators are being collected from many different s ources, which need to be normalized, scored, and analyzed before the customer can push to security devices such as SIEM and firewall for alerting. Detection tools can only handle l imited amounts of threat intelligence data and need to constantly re-prioritize indicators.SolutionIndicator prioritization. Palo Alto Networks Threat Intelligence Management can ingest phishing alerts from email i nboxes through integrations. Once an alert is ingested, a playbook is Use Case 2: Dynamic Allow/Deny ListA dministratio nChallengeManual process for allow/deny lists. Managing a single allow list and updating across the enterprise can involve updating dozens of network devices. Security teams often have to liaise with firewall admins, IT teams, DevOps, and other teams to execute some parts of incident response.SolutionEliminate downtime by using automated playbooks to e xtract valid IP addresses and URLs to exclude from enforce-ment point EDLs, ensuring employees have access to these b usiness-critical applications at all times.Use Case 3: Cross-Functional IntelligenceS haringChallengeIntelligence sharing is unstructured. Most intelligence is still shared via unstructured formats such as email, PDF, blogs, etc. Sharing indicators of compromise is not enough. A dditional context is required for the shared intelligence to have value.Internal alerts3000 Tannery WaySanta Clara, CA 95054 Main: +1.408.753.4000 Sales: +1.866.320.4788 Support: +1.866.898.9087© 2021 Palo Alto Networks, Inc. Palo Alto Networks is a registeredt rademark of Palo Alto Networks. A list of our trademarks can be found at https:///company/trademarks.html. All other marks mentioned herein may be trademarks of their respective companies. cortex_ds_xsoar-threat-intelligence-management_062221SolutionIndicator connections enable structured relationships to be created between threat intelligence sources. These relation-ships surface important context for security analysts, threat analysts, and other incident response teams, who can collab-orate and resolve incidents via a single platform.Industry-Leading CustomerS uccessOur Customer Success team is dedicated to helping you get the best value from your Cortex XSOAR investments and giving you the utmost confidence that your business is safe. Here are our plans:• Standard Success, included with every Cortex XSOAR sub-scription, makes it easy for you to get started. You’ll have access to self-guided materials and online support tools to get you up and running quickly.• Premium Success, the recommended plan, includes every-thing in the Standard plan plus guided onboarding, custom workshops, 24/7 technical phone support, and access to the Customer Success team to give you a personalized ex-perience to help you realize optimal return on investment (ROI).Flexible DeploymentCortex XSOAR can be deployed on-premises, in a private cloud, or as a fully hosted solution. We offer the platform in multiple tiers to fit your needs.。

unit3硕士英语综合教程课文翻译

unit3硕士英语综合教程课文翻译

Unit 3(Para. 1)A recent‎simula‎t ion of a devast‎a ting cybera‎t tack on Americ‎a was crying‎for a Bruce Willis‎lead:A series‎of myster‎i ous attack‎s crippl‎e d much of the nation‎a l infras‎t ructu‎r e, includ‎i ng air traffi‎c, financ‎i al market‎s and even basic email.最近一场模拟‎美国遭受毁灭‎性网络攻击的‎演习急需布鲁‎斯·威利斯(曾在小电脑客‎马特福斯特的‎专业帮助下,打破了一个异‎国客天衣无缝‎的电脑系统入‎侵计划)这样的一个具‎有高精尖电脑‎技术的人的帮‎助:由于一系列神‎秘的攻击,国家基础设施‎陷入瘫痪,包括航空运输‎,金融市场,甚至是基本的‎电子通信。

If this was not bad enough‎, an unrela‎t ed electr‎i city outage‎took down whatev‎e r remain‎e d of the alread‎y unplug‎g ed East Coast.如果这还不够‎糟糕,那么一段与神‎秘攻击毫不相‎干的电力断供‎期的出现,让已经无电力‎供应的东部沿‎海地区停止运‎转。

(Para. 2)The simula‎t ion—funded‎by a number‎of major player‎s in networ‎k securi‎t y, organi‎z ed by the Bipart‎i san Policy‎Center‎, a Washin‎g ton-based think tank,and broadc‎a st on CNN on a Saturd‎a y night—had an unexpe‎c ted twist.这次模拟实验‎是由一群网络‎安全领域的专‎家支持的,并在两党联立‎政策中心——华盛顿智囊团‎的组织下进行‎的,于一个星期六‎的晚上在CN‎N广播公布。

高中英语必修二 unit 3 computer 小短文

高中英语必修二 unit 3 computer 小短文

The goal of early calculating machines was to simplify difficult sums. But with the help of new technology, electronic chips replaced tubes and a revolution of artificial intelligence has arisen. From then on, the appearance of computers totally changed our lives. They can not only download information from the net when connected by the network or mobile phone signals, but also solve different types of logical problems. With operators as their coaches, they can even control rockets to explore the moon and instruct androids with the human race’s characters to mop floors and watch over your naughty niece. Anyhow, computers are so helpful that they do bring happiness.However, computers are easily attacked by viruses. This reality has become a hard-to-deal with-problem. As a result computers are dangerous in a way. So, personally, I worry about their universal applications in some fields, such as finance.早期计算机器的目的是简化比较难的计算。

新核心综合学术英语教程Unit 2

新核心综合学术英语教程Unit 2

Reading 2
Language Points
2. The downside of geothermal energy is the fact that it
is expensive to build, and to ensure that no harmful by-products are going to be produced in the process, geothermal plants must be created accordingly. (Para. H, L41-43.) →There are two negative aspects of geothermal energy. One is that it is expensive to build and the other is geothermal plants must be set up to ensure no harmful by-products during the process.
翻译 地热能的不足之处就是其获取费用太高,但为了确保 : 在利用地热能过程中无有害副产品的产生,地热发电
厂的修建必不可少。
Reading 2
Language Points
5.The difference is that we use the force of the water to push the turbine which in turn powers a generator thus generating electricity.(Para. I, L48-49.) 翻译 : 那就是,现在利用水力推动涡轮机为 发电机提供动力,进而产生电能。
Reading 2
Language Points

The_Part-Time_Parliament(Paxos算法中文翻译)

The_Part-Time_Parliament(Paxos算法中文翻译)
1.1. Paxos岛 ..................................................................................................................... 2
1.2. 要求(Requirements)................................................................................................... 3
现代的议会可以雇佣秘书来记录它的每一个活动,但是在Paxos没有一个人愿意始终呆在议
会大厅里作为一个秘书从头到尾参与每一个会议。取而代之的是每一个议员都会维护一个律
簿(ledger),用来记录一系列已通过的法令,每个法令会带有一个编号。例如议员Λ(译者
注:由于古希腊字母比较难输入,原文中的希腊文姓名统一用其中的一个字母代替)的律簿
2.1. 数学结论 .................................................................................................................... 5
2.2. 初级协议(The Preliminary Protocol) .................................................................... 9
Paxos议会的所知因此比较零散。虽然基本协议是知道的,但对许多细节我们却一无所知,
而这正是我们感兴趣的地方,因此我将忝为推测Paxos人在这些具体细节上可能的做法。

OpenProblemsList

OpenProblemsList

Open Problems ListArising from MathsCSP Workshop,Oxford,March2006Version0.3,April25,20061Complexity and Tractability of CSPQuestion1.0(The Dichotomy Conjecture)Let B be a relational structure.The problem of deciding whether a given relational structure has a homomorphism to B is denoted CSP(B).For which(finite)structures is CSP(B)decidable in polynomial time?Is it true that for anyfinite structure B the problem CSP(B)is either decidable in polynomial time or NP-complete?Communicated by:Tomas Feder&Moshe Vardi(1993) Question1.1A relational structure B is called hereditarily tractable if CSP(B )is tractable for all substructures B of B.Which structures B are hereditarily tractable?Communicated by:Pavol Hell Question1.2A weak near-unanimity term is defined to be one that satisfies the following identities:f(x,...,x)=x and f(x,y,....y)=f(y,x,y,....y)=...=f(y,...,y,x).Is CSP(B)tractable for any(finite)structure B which is preserved by a weak near-unanimity term?Communicated by:Benoit Larose,Matt Valeriote Question1.3A constraint language1S is called globally tractable for a problem P,if P(S)is tractable,and it is called(locally)tractable if for everyfinite L⊆S,P(L)is tractable.These two notions of tractability do not coincide in the Abduction problem(see talk by Nadia Creignou).•For which computational problems related to the CSP do these two notions of tractability coincide?•In particular,do they coincide for the standard CSP decision problem?Communicated by:Nadia Creignou 1That is,a(possibly infinite)set of relations over somefixed set.1Question1.4(see also Question3.5)It has been shown that when a structure B has bounded pathwidth duality the corresponding problem CSP(B)is in the complexity class NL (see talk by Victor Dalmau).Is the converse also true(modulo some natural complexity-theoretic assumptions)?Communicated by:Victor Dalmau Question1.5Is there a good(numerical)parameterization for constraint satisfaction problems that makes themfixed-parameter tractable?Question1.6Further develop techniques based on delta-matroids to complete the com-plexity classification of the Boolean CSP(with constants)with at most two occurrences per variable(see talk by Tomas Feder).Communicated by:Tomas Feder Question1.7Classify the complexity of uniform Boolean CSPs(where both structure and constraint relations are specified in the input).Communicated by:Heribert Vollmer Question1.8The microstructure graph of a binary CSP has vertices for each variable/value pair,and edges that join all pairs of vertices that are compatible with the constraints.What properties of this graph are sufficient to ensure tractability?Are there properties that do not rely on the constraint language or the constraint graph individually?2Approximability and Soft ConstraintsQuestion2.1Is it true that Max CSP(L)is APX-complete whenever Max CSP(L)is NP-hard?Communicated by:Peter Jonsson Question2.2Prove or disprove that Max CSP(L)is in PO if the core of L is super-modular on some lattice,and otherwise this problem is APX-complete.The above has been proved for languages with domain size3,and for languages contain-ing all constants by a computer-assisted case analysis(see talk by Peter Jonsson).Develop techniques that allow one to prove such results without computer-assisted analysis.Communicated by:Peter Jonsson Question2.3For some constraint languages L,the problem Max CSP(L)is hard to approximate better than the random mindless algorithm on satisfiable or almost satisfiable instances.Such problems are called approximation resistant(see talk by Johan Hastad).Is a single random predicate over Boolean variables with large arity approximation resistant?What properties of predicates make a CSP approximation resistant?What transformations of predicates preserve approximation resistance?Communicated by:Johan Hastad2Question2.4Many optimisation problems involving constraints(such as Max-Sat,Max CSP,Min-Ones SAT)can be represented using soft constraints where each constraint is specified by a cost function assigning some measure of cost to each tuple of values in its scope.Are all tractable classes of soft constraints characterized by their multimorphisms?(see talk by Peter Jeavons)Communicated by:Peter Jeavons 3AlgebraQuestion3.1The Galois connection between sets of relations and sets of operations that preserve them has been used to analyse several different computational problems such as the satisfiability of the CSP,and counting the number of solutions.How can we characterise the computational goals for which we can use this Galois connection?Communicated by:Nadia Creignou Question3.2For any relational structure B=(B,R1,...,R k),let co-CSP(B)denote the class of structures which do not have a homomorphism to B.It has been shown that the question of whether co-CSP(B)is definable in Datalog is determined by P ol(B),the polymorphisms of the relations of B(see talk by Andrei Bulatov).Let B be a core,F the set of all idempotent polymorphisms of B and V the variety generated by the algebra(B,F).Is it true that co-CSP(B)is definable in Datalog if and only if V omits types1and2(that is,the local structure of anyfinite algebra in V does not contain a G-set or an affine algebra)?Communicated by:Andrei Bulatov Question3.3Does every tractable clone of polynomials over a group contain a Mal’tsev operation?Communicated by:Pascal Tesson Question3.4Classify(w.r.t.tractability of corresponding CSPs)clones of polynomials of semigroups.Communicated by:Pascal Tesson Question3.5Is it true that for any structure B which is invariant under a near-unanimity operation the problem CSP(B)is in the complexity class NL?Does every such structure have bounded pathwidth duality?(see also Question1.4)Both results are known to hold for a2-element domain(Dalmau)and for majority operations(Dalmau,Krokhin).Communicated by:Victor Dalmau,Benoit Larose3Question3.6Is it decidable whether a given structure is invariant under a near-unanimity function(of some arity)?Communicated by:Benoit Larose Question3.7Let L be afixedfinite lattice.Given an integer-valued supermodular func-tion f on L n,is there an algorithm that maximizes f in polynomial time in n if the function f is given by an oracle?The answer is yes if L is a distributive lattice(see“Supermodular Functions and the Complexity of Max-CSP”,Cohen,Cooper,Jeavons,Krokhin,Discrete Applied Mathemat-ics,2005).More generally,the answer is yes if L is obtained fromfinite distributive lattices via Mal’tsev products(Krokhin,Larose–see talk by Peter Jonsson).The smallest lattice for which the answer is not known is the3-diamond.Communicated by:Andrei Krokhin Question3.8Find the exact relationship between width and relational width.(It is known that one is bounded if and and only if the other is bounded.)Also,what types of width are preserved under natural algebraic constructions?Communicated by:Victor Dalmau 4LogicQuestion4.1The(basic)Propositional Circumscription problem is defined as fol-lows:Input:a propositional formulaφwith atomic relations from a set S,and a clause c.Question:is c satisfied in every minimal model ofφ?It is conjectured(Kirousis,Kolaitis)that there is a trichotomy for this problem,that it iseither in P,coNP-complete or inΠP2,depending on the choice of S.Does this conjecturehold?Communicated by:Nadia Creignou Question4.2The Inverse Satisfiability problem is defined as follows: Input:afinite set of relations S and a relation R.Question:is R expressible by a CNF(S)-formula without existential variables?A dichotomy theorem was obtained by Kavvadias and Sideri for the complexity of this problem with constants.Does a dichotomy hold without the constants?Are the Schaefer cases still tractable?Communicated by:Nadia Creignou4Question4.3Let LFP denote classes of structures definable infirst-order logic with a least-fixed-point operator,let HOM denote classes of structures which are closed under homomorphisms,and let co-CSP denote classes of structures defined by not having a homomorphism to somefixed target structure.•Is LFP∩HOM⊆Datalog?•Is LFP∩co-CSP⊆Datalog?(forfinite target structures)•Is LFP∩co-CSP⊆Datalog?(forω-categorical target structures)Communicated by:Albert Atserias,Manuel BodirskyQuestion4.4(see also Question3.2)Definability of co-CSP(B)in k-Datalog is a sufficient condition for tractability of CSP(B),which is sometimes referred to as having width k. There is a game-theoretic characterisation of definability in k-Datalog in terms of(∃,k)-pebble games(see talk by Phokion Kolaitis).•Is there an algorithm to decide for a given structure B whether co-CSP(B)is definable in k-Datalog(for afixed k)?•Is the width hierarchy strict?The same question when B isω-categorical,but not necessarilyfinite?Communicated by:Phokion Kolaitis,Manuel BodirskyQuestion4.5Find a good logic to capture CSP with“nice”(e.g.,ω-categorical)infinite templates.Communicated by:Iain Stewart 5Graph TheoryQuestion5.1The list homomorphism problem for a(directed)graph H is equivalent to the problem CSP(H∗)where H∗equals H together with all unary relations.•It is conjectured that the list homomorphism problem for a reflexive digraph is tractable if H has the X-underbar property(which is the same as having the bi-nary polymorphism min w.r.t.some total ordering on the set of vertices),and NP-complete otherwise.•It is conjectured that the list homomorphism problem for an irreflexive digraph is tractable if H is preserved by a majority operation,and NP-complete otherwise. Do these conjectures hold?Communicated by:Tomas Feder&Pavol Hell5Question5.2“An island of tractability?”Let A m be the class of all relational structures of the form(A,E1,...,E m)where each E i is an irreflexive symmetric binary relation and the relations E i together satisfy the following‘fullness’condition:any two distinct elements x,y are related in exactly one of the relations E i.Let B m be the single relational structure({1,...,m},E1,...,E m)where each E i is the symmetric binary relation containing all pairs xy except the pair ii.(Note that the relations E i are not irreflexive.)The problem CSP(A m,B m)is defined as:Given A∈A m,is there a homomorphism from A to B m?When m=2,this problem is solvable in polynomial time-it is the recognition problem for split graphs(see“Algorithmic Graph Theory and Perfect Graphs”,M.C.Golumbic, Academic Press,New York,1980)When m>3,this problem is NP-complete(see“Full constraint satisfaction problems”,T.Feder and P.Hell,to appear in SIAM Journal on Computing).What happens when m=3?Is this an“island of tractability”?Quasi-polynomial algorithms are known for this problem(see“Full constraint satisfaction problems”,T. Feder and P.Hell,,to appear in SIAM Journal on Computing,and“Two algorithms for list matrix partitions”,T.Feder,P.Hell,D.Kral,and J.Sgall,SODA2005).Note that a similar problem for m=3was investigated in“The list partition problem for graphs”, K.Cameron,E.E.Eschen,C.T.Hoang and R.Sritharan,SODA2004.Communicated by:Tomas Feder&Pavol Hell Question5.3Finding the generalized hypertree-width,w(H)of a hypergraph H is known to be NP-complete.However it is possible to compute a hypertree-decomposition of H in polynomial time,and the hypertree-width of H is at most3w(H)+1(see talk by Georg Gottlob).Are there other decompositions giving better approximations of the generalized hypertree-width that can be found in polynomial time?Communicated by:Georg Gottlob Question5.4It is known that a CSP whose constraint hypergraph has bounded fractional hypertree width is tractable(see talk by Daniel Marx).Is there a hypergraph property more general than bounded fractional hypertree width that makes the associated CSP polynomial-time solvable?Are there classes of CSP that are tractable due to structural restrictions and have unbounded fractional hypertree width?Communicated by:Georg Gottlob,Daniel Marx Question5.5Prove that there exist two functions f1(w),f2(w)such that,for every w, there is an algorithm that constructs in time n f1(w)a fractional hypertree decomposition of width at most f2(w)for any hypergraph of fractional hypertree width at most w(See talk by Daniel Marx).Communicated by:Daniel Marx6Question5.6Turn the connection between the Robber and Army game and fractional hypertree width into an algorithm for approximating fractional hypertree width.Communicated by:Daniel Marx Question5.7Close the complexity gap between(H,C,K)-colouring and (H,C,K)-colouring (see talk by Dimitrios Thilikos)Find a tight characterization for thefixed-parameter tractable(H,C,K)-colouring problems.•For the(H,C,K)-colouring problems,find nice properties for the non-parameterisedpart(H−C)that guaranteefixed-parameter tractability.•Clarify the role of loops in the parameterised part C forfixed-parameter hardnessresults.Communicated by:Dimitrios Thilikos6Constraint Programming and ModellingQuestion6.1In a constraint programming system there is usually a search procedure that assigns values to particular variables in some order,interspersed with a constraint propagation process which modifies the constraints in the light of these assignments.Is it possible to choose an ordering for the variables and values assigned which changes each problem instance as soon as possible into a new instance which is in a tractable class? Can this be done efficiently?Are there useful heuristics?Question6.2The time taken by a constraint programming system tofind a solution toa given instance can be dramatically altered by modelling the problem differently.Can the efficiency of different constraint models be objectively compared,or does it depend entirely on the solution algorithm?Question6.3For practical constraint solving it is important to eliminate symmetry,in order to avoid wasted search effort.Under what conditions is it tractable to detect the symmetry in a given problem in-stance?7Notes•Representations of constraints-implicit representation-effect on complexity•Unique games conjecture-structural restrictions that make it false-connectionsbetween definability and approximation•MMSNP-characterise tractable problems apart from CSP7•Migrate theoretical results to tools•What restrictions do practical problems actually satisfy?•Practical parallel algorithms-does this align with tractable classes?•Practically relevant constraint languages(”global constraints”)•For what kinds of problems do constraint algorithms/heuristics give good results?8。

Invariants of Legendrian Knots and Coherent Orientations

Invariants of Legendrian Knots and Coherent Orientations
1
2
J. ETNYRE, L. NG, AND J. SABLOFF
Chekanov’s original DGA can be recovered by setting t = 1, which will force the grading to be reduced modulo 2r, and taking the coefficients modulo 2. Part I of the paper is devoted to this goal. Concurrent with Chekanov’s work on his DGA, Eliashberg and Hofer adapted the ideas of Floer homology to the contact setting. Though we will flesh out a relative version of their “contact homology theory” in Section 7, the story goes roughly as follows: let (M, α) be a contact manifold with a Legendrian submanifold K . Let A be the free associative unital algebra generated by the Reeb chords — i.e. Reeb trajectories that begin and end on K . The generators are graded by something akin to the Maslov index. There is a differential on A that comes from counting rigid J -holomorphic disks in the symplectization (M × R, d(eτ α)) of M . Here, J is a vertically-invariant almost complex structure compatible with d(eτ α). Using Floer and Hofer’s idea of coherent orientations [8, 12], it is possible to orient all of the moduli spaces of rigid J -holomorphic disks used in the definition of the differential. As a result, we may use Z coefficients in the definition of the algebra A. The second goal of this paper, carried out in Part II, is to prove that Chekanov’s DGA, and our generalization of it, is a combinatorial translation of relative contact homology. Knowing the relation between the combinatorial and geometric versions of contact homology is quite useful. In particular, the lifting of Chekanov’s DGA from Z/2 to Z[t, t−1 ] was accomplished by studying this relationship. Moreover, explicit computations in the framework of Eliashberg and Hofer’s contact homology theory can be difficult, while computations in the combinatorial theory are more straightforward. Thus, our translation between the two theories yields many explicit computations in contact homology. The paper consists of essentially two parts. After recalling several basic ideas from contact geometry in Section 2, we proceed, in Part I, to describe the combinatorial theory. This part is self-contained apart from a few technical proofs that are relegated to an appendix of Part I. In Part II of the paper we discuss Eliashberg and Hofer’s contact homology and coherent orientations. We then prove the combinatorial theory developed in Part I is a faithful translation of this more geometric theory. 2. Basic Notions We begin by describing some basic notions in three-dimensional contact geometry. A contact structure on a 3-manifold M is a completely non-integrable 2-plane field ξ . Locally, a contact structure is the kernel of a 1-form α that satisfies the following non-degeneracy condition at every point in M : α ∧ dα = 0. In this paper, we will be interested in the standard contact structure ξ0 on R3 , which is defined to be the kernel of the 1-form α0 = dz + x dy. To each contact form α, we may associate a Reeb field Xα that satisfies dα(Xα , ·) = 0 and α(Xα ) = 1. By Darboux’s theorem, every contact manifold is locally contactomorphic to (R3 , ξ0 ). See [2, chapter 8] for an introduction to the fundamentals of contact geometry. Our primary objects of study are Legendrian knots in R3 , i.e. knots that are everywhere tangent to the standard contact structure ξ0 . In particular, we examine Legendrian isotopy classes of Legendrian knots, in which two knots are deemed equivalent if they are related by an isotopy through Legendrian knots.

Python程序设计(英语)智慧树知到课后章节答案2023年下中央财经大学

Python程序设计(英语)智慧树知到课后章节答案2023年下中央财经大学

Python程序设计(英语)智慧树知到课后章节答案2023年下中央财经大学中央财经大学第一章测试1.What is the fundamental question of computer science? ()A:How much money can a programmer make? B:What can be computed?C:What is the most effective programming language? D:How fast can acomputer compute?答案:What can be computed?2. A statement is ()A:a precise description of a problem B:a complete computer command C:a section of an algorithm D:a translation of machine language答案:a complete computer command3.The items listed in the parentheses of a function definition are called ()A:parentheticals B:both B and C are correct C:parameters D:arguments答案:both B and C are correct4.All information that a computer is currently working on is stored in mainmemory. ()A:对 B:错答案:对5. A loop is used to skip over a section of a program. ()A:错 B:对答案:错第二章测试1.Which of the following is not a legal identifier?()A:spAm B:2spam C:spam D:spam4U答案:spam2.In Python, getting user input is done with a special expression called ()A:simultaneous assignment B:input C:for D:read答案:input3.The process of describing exactly what a computer program will do to solve aproblem is called ()A:specification B:programming C:Design D:implementation答案:specification4.In Python, x = x + 1 is a legal statement. ()A:错 B:对答案:对5.The best way to write a program is to immediately type in some code andthen debug it until it works. ()A:对 B:错答案:错第三章测试1.Which of the following is not a built-in Python data type? ()A:int B:rational C:string D:float答案:rational2.The most appropriate data type for storing the value of pi is ()A:string B:int C:float D:irrational答案:float3.The pattern used to compute factorials is ()A:input, process, output B:accumulator C:counted loop D:plaid答案:accumulator4.In Python, 4+5 produces the same result type as 4.0+5.0. ()A:对 B:错答案:错5.Definite loops are loops that execute a known number of times. ()答案:对第四章测试1. A method that changes the state of an object is called a(n) ()A:function B:constructor C:mutator D:accessor答案:mutator2.Which of the following computes the horizontal distance between points p1and p2? ()A:abs(p1.getX( ) - p2.getX( )) B:abs (p1-p2) C:p2.getX( ) - p1.getX( )D:abs(p1.getY( ) - p2.getY( ))答案:abs(p1.getX( ) - p2.getX( ))3.What color is color_rgb (0,255,255)? ()A:Cyan B:Yellow C:Magenta D:Orange答案:Cyan4.The situation where two variables refer to the same object is called aliasing.()A:错 B:对答案:对5. A single point on a graphics screen is called a pixel. ()答案:对第五章测试1.Which of the following is the same as s [0:-1]? ()A:s[:] B:s[:len(s)-1] C:s[-1] D:s[0:len(s)]答案:s[:len(s)-1]2.Which of the following cannot be used to convert a string of digits into anumber? ()A:int B:eval C:str D:float答案:str3.Which string method converts all the characters of a string to upper case? ()A:capwords B:upper C:capitalize D:uppercase答案:upper4.In Python “4”+“5”is “45”. ()A:错 B:对答案:对5.The last character of a strings is at position len(s)-1. ()答案:对第六章测试1. A Python function definition begins with ()A:def B:define C:defun D:function答案:def2.Which of the following is not a reason to use functions? ()A:to demonstrate intellectual superiority B:to reduce code duplication C:tomake a program more self-documenting D:to make a program more modular 答案:to demonstrate intellectual superiority3. A function with no return statement returns ()A:its parameters B:nothing C:its variables D:None答案:None4.The scope of a variable is the area of the program where it may be referenced.()A:对 B:错答案:对5.In Python, a function can return only one value. ()答案:错第七章测试1.In Python, the body of a decision is indicated by ()A:indentation B:curly braces C:parentheses D:a colon答案:indentation2.Placing a decision inside of another decision is an example of ()A:procrastination B:spooning C:cloning D:nesting答案:nesting3.Find a correct algorithm to solve a problem and then strive for ()A:clarity B:efficiency C:scalability D:simplicity答案:clarity;efficiency;scalability;simplicity4.Some modules, which are made to be imported and used by other programs,are referred to as stand-alone programs. ()A:对 B:错答案:错5.If there was no bare except at the end of a try statement and none of theexcept clauses match, the program would still crash. ()答案:对第八章测试1. A loop pattern that asks the user whether to continue on each iteration iscalled a(n) ()A:end-of-file loop B:sentinel loop C:interactive loop D:infinite loop答案:interactive loop2. A loop that never terminates is called ()A:indefinite B:busy C:infinite D:tight答案:infinite3.Which of the following is not a valid rule of Boolean algebra? ()A:a and (b or c) == (a and b) or (a and c) B:(True or False) == True C:not(a and b)== not(a) and not(b) D:(True or x) == True答案:not(a and b)== not(a) and not(b)4.The counted loop pattern uses a definite loop. ()A:错 B:对答案:对5. A sentinel loop should not actually process the sentinel value. ()答案:对第九章测试1.()A:random() >= 66 B:random() < 0.66 C:random() < 66 D:random() >= 0.66答案:random() < 0.662.The arrows in a module hierarchy chart depict ()A:control flow B:one-way streets C:logistic flow D:information flow答案:information flow3.In the racquetball simulation, what data type is returned by the gameOverfunction? ()A:bool B:string C:float D:int答案:bool4. A pseudorandom number generator works by starting with a seed value. ()A:错 B:对答案:对5.Spiral development is an alternative to top-down design. ()A:错 B:对答案:错第十章测试1. A method definition is similar to a(n) ()A:module B:import statement C:function definition D:loop答案:function definition2.Which of the following methods is NOT part of the Button class in thischapter? ()A:activate B:clicked C:setLabel D:deactivate答案:setLabel3.Which of the following methods is part of the DieView class in this chapter?()A:setColor B:clicked C:setValue D:activate答案:setValue4.New objects are created by invoking a constructor. ()A:对 B:错答案:对5.A:错 B:对答案:错第十一章测试1.The method that adds a single item to the end of a list is ()A:plus B:add C:append D:extend答案:append2.Which of the following expressions correctly tests if x is even? ()A:not odd (x) B:x % 2 == 0 C:x % 2 == x D:even (x)答案:x % 2 == 03.Items can be removed from a list with the del operator. ()A:错 B:对答案:对4.Unlike strings, Python lists are not mutable. ()A:对 B:错答案:错5.()A:对 B:错答案:错第十二章测试1.Which of the following was NOT a class in the racquetball simulation? ()A:SimStats B:Player C:RBallGame D:Score答案:Score2.What is the data type of server in an RBallGame? ()A:SimStats B:bool C:int D:Player答案:Player3.The setValue method redefined in ColorDieView class is an example of ()A:polymorphism B:generality C:encapsulation D:inheritance答案:polymorphism;inheritance4.Hiding the details of an object in a class definition is called instantiation. ()A:对 B:错答案:错5.Typically, the design process involves considerable trial and error. ()A:错 B:对答案:对。

新标准大学英语2 Unit3译文和答案

新标准大学英语2 Unit3译文和答案

Unit 3 Ar11. deception n.the action of deceiving someoneTranslate the following sentences into Chinese:The man obtained property by deception.这人靠欺骗获得财产。

This is a range of elaborate deception.这是一个精心设计的圈套。

2. fraud n. [C, U] wrongful or criminal deception intended for result in financial or personal gainTranslate the following sentences into Chinese:He was convicted of fraud.他被判定犯有诈骗罪。

He told people he was a doctor, but he was a fraud really.他告诉人们他是医生,实际上他是个骗子。

3. fraudster n.someone obtains money by deceiving.Translate the following sentence into Chinese:New measures are needed to prevent fraudsters opening bank accounts with stolen identity.需要制定新的措施以防止诈骗犯冒用身份在银行开户。

4.forge v. forgery n. to make illegal copy of something to deceive.Translate the following sentence into Chinese:The student forged his supervisor’s signature on the dissertation.这个学生在论文上伪造他导师的签字。

民航机务英语试题及答案

民航机务英语试题及答案

民航机务英语试题及答案一、选择题(每题2分,共20分)1. The term "N1" refers to the speed of the engine in relation to:A. Ground speedB. Indicated AirspeedC. True AirspeedD. Rotational speed of the engine答案:D2. What does "MEL" stand for in aviation maintenance?A. Minimum Equipment ListB. Maximum Equipment ListC. Maintenance Equipment ListD. Manual Equipment List答案:A3. The "PBH" maintenance concept is based on:A. Parts per millionB. Parts per hourC. Power by hourD. Performance based on hours答案:B4. Which of the following is NOT a type of aircraft engine?A. Piston engineB. Jet engineC. Rocket engineD. Steam engine答案:D5. The term "VOR" refers to a type of navigation system that is:A. VisualB. Radio-basedC. Satellite-basedD. Infrared答案:B6. What does "ATC" stand for in aviation?A. Air Traffic ControlB. Automatic Temperature ControlC. Advanced Technology CenterD. Airline Transport Certificate答案:A7. The "TCAS" system is used for:A. CommunicationB. Collision avoidanceC. Takeoff and landing assistanceD. Turbine control and safety答案:B8. What is the meaning of "RVSM" in aviation?A. Reduced Vertical Separation MinimumB. Radio Visual Signaling ModeC. Radar Vertical Speed MeasurementD. Remote Visual Surveillance Module答案:A9. "EFIS" stands for:A. Electronic Flight Information SystemB. Electronic Flight Instrument SystemC. Enhanced Flight Information ServiceD. Electronic Flight Indicator System答案:B10. The "ICAO" is an organization that deals with:A. International trade regulationsB. International aviation standards and practicesC. International customs operationsD. International cargo agreements答案:B二、填空题(每空1分,共10分)11. The international distress signal "MAYDAY" is pronounced as _________.答案:Mayday12. An aircraft's "T/O" stands for _________.答案:Takeoff13. The "FAR" refers to the Federal Aviation _________.答案:Regulations14. The "RAM" in aircraft systems usually stands for_________.答案:Random Access Memory15. The term "SID" in aviation is short for _________.答案:Standard Instrument Departure16. "Landing gear" is also known as _________ gear.答案:Landing17. The "PFD" in aviation is an acronym for _________.答案:Primary Flight Display18. "CVR" is the abbreviation for the _________.答案:Cockpit Voice Recorder19. The term "GPWS" refers to a system known as the _________.答案:Ground Proximity Warning System20. "TCAS" is an acronym for Traffic _________ and Collision Avoidance System.答案:Collision Avoidance System三、简答题(每题5分,共30分)21. Explain the purpose of an aircraft's transponder.答案:An aircraft's transponder is a device that sendsout a coded signal in response to a received interrogation signal from ground-based secondary surveillance radar (SSR). This allows air traffic controllers to identify and track the aircraft's position and altitude.22. What is the significance of the "EFB" in modern aviation?答案:The Electronic Flight Bag (EFB) is a collection of electronic tools and systems that replace the traditional paper-based charts, manuals, and documents used by pilots. It enhances efficiency, reduces workload, and improves safety by providing quick access to a wide range of information.23. Describe the role of an "AOG" situation in aviation.答案:Aircraft on Ground (AOG) refers to a situation where an aircraft is grounded due to a technical issue that cannot be resolved quickly. It is a critical event for airlines as it impacts operations, schedules, and can incur significant costs.24. What does "ETOPS" stand for and what does it regulate?答案:ETOPS stands for Extended-range Twin-engine Operational Performance Standards. It is a set of regulations established by the International Civil Aviation Organization (ICAO) that specifies the operational requirements for。

On the power structure over the Grothendieck ring of varieties and its applications

On the power structure over the Grothendieck ring of varieties and its applications
k ∈Zr ≥0 kr 1 where u = (u1 , . . . , ur ), k = (k1 , . . . , kr ), u k = uk 1 · . . . · ur , (see [9]).
(1 − u k t)−pk ,
2
4. (A more geometric example.) Let R be the K -functor K (X ) of the space X , i.e. the Grothendieck ring of (say, real or complex) vector bundles over X . For a vector bundle E over X , let Λk E be the k -th exterior power of the bundle E . The series λE (t) = 1 + [E ]t + [Λ2 E ]t2 + [Λ3 E ]t3 + . . . defines a pre-λ structure on the ring K (X ). To a pre-λ structure on a ring R one can associate a power structure over R: a notion introduced in [8]. Definition: A power structure over a (semi)ring R with a unit is a map (1 + t · R[[t]]) × R → 1 + t · R[[t]]: (A(t), m) → (A(t))m , which possesses the following properties: 1) (A(t))0 = 1, 2) (A(t))1 = A(t), 3) (A(t) · B (t))m = (A(t))m · (B (t))m , 4) (A(t))m+n = (A(t))m · (A(t))n , 5) (A(t))mn = ((A(t))n ) , 6) (1 + t)m = 1 + mt+ terms of higher degree, 7) A(tk )

2022_2023学年新教材高中英语Unit3TheInternetSec学生用书新人教版必修第二册

2022_2023学年新教材高中英语Unit3TheInternetSec学生用书新人教版必修第二册

Section ⅢDiscovering Useful Structures 每/日/金/句:Over the Internet, we can learn more knowledge and broaden our horizons.通过互联网,我们可以学习更多的知识并开阔我们的视野。

语言基础集释1(教材P30)Have you confirmed the Wi-Fi password?你确认Wi-Fi密码了吗?◎confirm vt.确认;使确信confirm sb. in sth. 使某人确信某事confirm sth./that... 证实……It has been confirmed that... 已经证实/确定……[佳句] Planning to take a group to pay a visit to your school next month, I'm writing to confirm some information.计划下个月组团到贵校参观,我写信确认一些信息。

[练通]——单句语法填空①My teacher's encouragement confirms me ________ the belief that I will surely win the championship of Chinese Poetry Competition.②There was frost on the ground, ________ (confirm) that fall had arrived in Canada.[写美]——应用文佳句③已经确认我们将要爬山,并在那里野餐。

________________________________ we will go to climb a mountain,where we can have a picnic.2(教材P30)Have you pressed the button yet to copy the file?你已经按下按钮复制文件了吗?◎press vt.按,压;敦促(1)press...against/to... 把……压/挤/贴在……press sb. to do sth. 极力劝说/敦促某人做某事(2)pressure n. 压(力),压强;强制put pressure on sb. 给某人施加压力under pressure 承受压力[佳句] Although under great pressure, encouragement from my parents comforts me greatly.尽管压力很大,但父母的鼓励让我感到非常安慰。

新编剑桥商务英语课件中级Unit-2a精选全文

新编剑桥商务英语课件中级Unit-2a精选全文

• reasonable prices 合理的价格
Vocabulary 2-2
• interior design
室内装修
• group dining
团体用餐
• group menu
团体用餐菜单
• catering requirement 用餐要求
• tour groups
旅游团
• birthday party
the event? • How much do we want to spend – budget?
• what are the advantages and disadvantages of mixing business with pleasure?
Vocabulary1-2
• customer satisfaction form 顾客满意调查问卷
• 阅读下列顾客满意调查问卷表,其中 那三个标准对你来说最为重要?
Text discussion
• 在阅读问卷的时候,我们要思考这样一个问题 Which three of the criteria are most important for you? 那三个标准对你来说最为重要。这个 句子中有一个单词要引起我们的特别注意: criteria 标准。这个单词的复数形式是不用加s 的。 在这个题目中“酒店”一词是复数形式, Restaurants.而在第一段的第一个句子中“a Signet House Restaurant”,这里的复数指 各家连锁店,单数是指连锁店中的一家。所以 文章中的这个酒店的复数形式在翻译的时候, 我们增加了“连锁”这个词,以便区别出来
地位的
• Facilitate one-to-one conversation 有利于一对一 交流

新标准大学英语第二版综合教程2 Unit 3 A篇练习答案及课文翻译

新标准大学英语第二版综合教程2 Unit 3 A篇练习答案及课文翻译
新标准大学英语第二版综合教程2 Unit 3 A篇练习答案及课文翻译
Coning Up
1. What’s your first impression of American football game?
2. What are the differences between American football and soccer?
Text
1 Work in pairs and discuss the questions.
1 What are the most popular sports in China? 2 Do you think foreigners can understand Chinese sports
such as t’ai chi and kung fu? 3 Do you understand foreign sports, such as football and
Text
3 Do you understand foreign sports, such as football and tennis?
• Well, I know the sports, but I don’t know the rules and hardly watch them.
• Yes. I understand a bit. Football is a ball game in which two teams of 11 players, using any part of their bodies except their hands and arms, try to manoeuver the ball into the opposing team’s goal. Tennis is played on a court by two players (or four for doubles). Players on opposite sides of a net hit a ball over the net with a racket so that it lands on the other side of the court.

《Unit7TheInternetGroupWork》作业设计方案-中职英语高教版21基础模块1

《Unit7TheInternetGroupWork》作业设计方案-中职英语高教版21基础模块1

《The Internet Group Work》作业设计方案(第一课时)一、作业目标本作业旨在通过互联网小组合作学习的方式,提高学生的英语实际应用能力,增强团队协作精神,同时巩固第一课时所学的互联网相关英语词汇和句型。

通过作业的完成,学生应能够熟练使用英语进行网络交流,并能够在小组中有效沟通、协作完成任务。

二、作业内容1. 词汇积累:学生需掌握与互联网相关的英语词汇,如“email”、“website”、“online chat”等,并能够运用这些词汇进行简单的造句。

2. 小组任务:学生将被分成若干小组,每个小组需选择一个网络热点话题(如“网络安全”、“社交媒体利弊”等),并用英语进行讨论和资料收集。

要求每组制作一份简短的英文报告,包括至少三名组员的姓名和他们对所选话题的观点陈述。

3. 交流与合作:小组内部成员需通过网络平台(如微信、QQ 或邮件)进行至少三次有效的交流和讨论,并确保每个成员都能够在报告中有发言权,对各自的任务和观点进行充分的阐述。

4. 成果展示:每个小组需将英文报告以PPT或Word文档的形式提交至教师邮箱或在线平台上,报告应包含清晰的结构和逻辑,语言准确、流畅。

三、作业要求1. 每位学生需积极参与小组讨论和资料收集工作,确保自己的任务能够按时完成。

2. 交流过程中应使用英语进行沟通,以提高英语口语和书面表达能力。

3. 报告内容应紧扣所选话题,观点明确,语言简洁明了。

4. 提交的成果需符合格式要求,并注明每位组员的贡献。

四、作业评价1. 教师将对每份报告的内容、结构、语言表达及团队协作情况进行评价。

2. 组内成员之间的合作程度和贡献也是评价的重要依据。

3. 教师将选取部分优秀作品进行课堂展示和表扬,激励学生积极参与到英语学习中来。

五、作业反馈1. 教师将对每位学生的作业进行批改和点评,指出存在的不足和需要改进的地方。

2. 教师将根据作业情况为学生提供有针对性的学习建议和方法指导,帮助学生提高英语学习和团队协作能力。

The analytic structure of an algebraic quantum group

The analytic structure of an algebraic quantum group

a rXiv:funct-a n /9771v128J u l1997The analytic structure of an algebraic quantum group J.Kustermans 1Institut for Matematik og Datalogi Odense Universitet Campusvej 555230Odense M Denmark July 1997Abstract In [14],Van Daele introduced the notion of an algebraic quantum group.We proved in [5]and [9]that such algebraic quantum groups give rise to C ∗-algebraic quantum groups according to Masuda,Nakagami &Woronowicz.In this paper,we will pull down the analytic structure of these C ∗-algebraic quantum groups to the algebraic quantum group.Introduction Van Daele introduced the notion of an algebraic quantum group in [14].It is essentially a Multiplier Hopf-∗-algebra with a non-zero left invariant functional on it.He proved that these algebaic quantum groups form a well-behaved category :•The left and right invariant functionals are unique up to a scalar.•The left and right invariant functionals are faithful.•Each algebraic quantum group gives rise to a dual algebraic quantum group.•The dual of the dual is isomorphic to the original algebraic quantum group.This category of algebraic quantum groups contains the compact and discrete quantum groups.It is also closed under the double construction of Drinfel’d so it contains also non-compact non-discrete quantum groups.Most of the groups and quantum groups will however not belong to this category.It is nevertheless worth while to study these algebraic quantum groups :•The category contains interesting examples.•The theoretical framework is not technically complicated.In a sense,you have only to worry about essential quantum group problems.Not over C ∗-algebraic complications.At the moment,Masuda,Nakagami&Woronowicz are working on a possible definition of a C∗-algebraic quantum group.To get an idea of this definition,we refer to[10].You can also get aflavour of it in[5] and[9].This definition is technically rather involved and C∗-algebraic quantum groups in this scheme have a rich analytical sructure.We proved in[5]and[9]that an algebraic quantum group with a positive left invariant functional gives rise to such C∗-algebraic quantum groups.In this paper,we will pull down this analytic structure to the algebra level.It shows that such algebraic quantum groups are truely algebraic versions of C∗-algebraic quantum groups according to Masuda,Nakagami&Woronowicz.In thefirst section,we give an overview of the theory of algebraic quantum groups as can be found in[14]. The second and third section are essentially only intended to assure that an obvious theory of analytic one-parameter groups worksfine on this algebra level.The most important results can be found in section4where we prove that the analytic objects of the C∗-algebraic quantum groups are of an algebraic nature.In the last section,we connect the analytic objects of the dual to the analytic objects of the original algebraic quantum group.We end this section with some conventions and notations.Every algebra in this paper is an associative algebra over the complex numbers(not necessarily unital).A homomorphism between algebras is by definition a linear multiplicative mapping.A∗-homomorphism between∗-algebras is a homomorphism which preserves the∗-operation.If V is a vector space,then L(V)denotes the set of linear mappings from V into V.If V,W are two vector spaces,V⊙W denotes the algebraic tensor product.Theflip from V⊙W to W⊙V will be denoted byχ.We will also use the symbol⊙to denote the algebraic tensor product of two linear mappings.We will always use the minimal tensor product between two C∗-algebras and we will use the symbol⊗for it.This symbol will also be used to denote the completed tensor product of two mappings which are sufficiently continuous.If z is a complex number,then S(z)will denote the following horizontal strip in the complex plane: S(z)={y∈will be denoted by the same symbol as the original mapping.Of course,we have similar definitions and results for antimultiplicative mappings.If we work in an algebraic setting,we will always use this form of non-degeneracy as opposed to the non degeneracy of∗-homomorphisms between C∗-algebras!For a linear functionalωon a non-degenerate∗-algebra A and any a∈M(A)we define the linear functionalsωa and aωon A such that(aω)(x)=ω(xa)and(ωa)(x)=ω(ax)for every x∈A.You canfind some more information about non-degenerate algebras in the appendix of[17].Letωbe a linear functional on a∗-algebra A,then:1.ωis called positive if and only ifω(a∗a)is positive for every a∈A.2.Ifωis positive,thenωis called faithful if and only if for every a∈A,we have thatω(a∗a)=0⇒a=0.Consider a positive linear functionalωon a∗-algebra A.Let H be a Hilbert-space andΛa linear mapping from A into H such that•Λ(A)is dense in H.•We have for all a,b∈A thatω(a∗a)= Λ(a),Λ(b) .Then we call(H,Λ)a GNS-pair forω.Such a GNS-pair always exist and it is unique up to a unitary.We have now gathered the necessary information to understand the following definitionDefinition1.1Consider a non-degenerate∗-algebra A and a non-degenerate∗-homomorphism∆fromA into M(A⊙A)such that1.(∆⊙ι)∆=(ι⊙∆)∆.2.The linear mappings T1,T2from A⊙A into M(A⊙A)such thatT1(a⊗b)=∆(a)(b⊗1)and T2(a⊗b)=∆(a)(1⊗b) for all a,b∈A,are bijections from A⊙A to A⊙A.Then we call(A,∆)a Multiplier Hopf∗-algebra.In[17],A.Van Daele proves the existence of a unique non-zero∗-homomorphismεfrom A to•(ω⊙ι)(∆(a))b=(ω⊙ι)(∆(a)(1⊗b))•b(ω⊙ι)(∆(a))=(ω⊙ι)((1⊗b)∆(a))for every b∈A.In a similar way,the multiplier(ι⊙ω)∆(a)is defined.Letωbe a linear functional on A.We callωleft invariant(with respect to(A,∆)),if and only if (ι⊙ω)∆(a)=ω(a)1for every a∈A.Right invariance is defined in a similar way.Definition1.2Consider a Multiplier Hopf∗-algebra(A,∆)such that there exists a non-zero positive linear functionalϕon A which is left invariant.Then we call(A,∆)an algebraic quantum group.For the rest of this paper,we willfix an algebraic quantum group(A,∆)together with a non-zero left invariant positive linear functionalϕon it.An important feature of such an algebraic quantum group is the faithfulness and uniqueness of left invariant functionals:1.Consider a left invariant linear functionalωon A,then there exists a unique element c∈It is also possible to introduce the modular function of our algebraic quantum group.This is an invertible elementδin M(A)such that(ϕ⊙ι)(∆(a)(1⊗b))=ϕ(a)δbfor every a,b∈A.Concerning the right invariant functional,we have that(ι⊙ψ)(∆(a)(b⊗1))=ψ(a)δ−1bfor every a,b∈A.This modular function is,like in the classical group case,a one dimensional(generally unbounded) corepresentation of our algebraic quantum group:∆(δ)=δ⊙δε(δ)=1S(δ)=δ−1.As in the classical case,we can relate the left invariant functional to our right invariant functional via the modular function:we have for every a∈A thatϕ(S(a))=ϕ(aδ)=µϕ(δa).If we apply this equality two times and use the fact that S(δ)=δ−1,we get thatϕ(S2(a))=ϕ(δ−1aδ) for every a∈A.Not surprisingly,we have also thatρ(δ)=ρ′(δ)=µ−1δ.Another connection betweenρandρ′is given by the equalityρ′(a)=δρ(a)δ−1for all a∈A.We have also a property which says,loosely speaking,that every element of A has compact support(see e.g.[6]for a proof):Consider a1,...,a n∈A.Then there exists an element c in A such that c a i=a i c=a i for every i∈{1,...,n}.In a last part,we are going to say something about duality.We define the subspaceˆA of A′as follows:ˆA={ϕa|a∈A}={aϕ|a∈A}.Like in the theory of Hopf∗-algebras,we turnˆA into a non-degenerate∗-algebra:1.For everyω1,ω2∈ˆA and a∈A,we have that(ω1ω2)(a)=(ω1⊙ω2)(∆(a)).2.For everyω∈ˆA and a∈A,we have thatω∗(a)=2.We have for everyθ∈M(ˆA)and a∈A thatθ∗(a)=C into the set of homomorphisms from A into A such that the following properties hold:1.We have for every t∈I R thatαt is a∗-automorphism on A.2.We have for every s,t∈I R thatαs+t=αsαt.3.We have for every t∈I R thatαt is relatively invariant underϕ64.Consider a∈A andω∈ˆA.Then the function C:z→ω(αz(a))is analytic.Then we callαan analytic one-parameter group on(A,∆).Except for the last proposition in this section,we willfix an analytic one-parameter groupαon(A,∆). We will prove the basic properties ofα.Result2.2Consider z∈z(a∗)for a∈A.Proof:Choose a∈A.Take b∈A.Then we have two analytic functionsC:u→u(a∗)b∗)and C:u→ϕ(bαu(a))These functions are equal on the real axis:we have for all t∈I R that(a∗)b∗)=tϕ(αz(a∗)∗)=ϕ(bαz(a)).So the faithfulness ofϕimplies thatαIn the next result,we extend the group character ofαto the whole complex plane.Result2.3Consider y,z∈C→C→C→C→Corollary2.4Consider z∈We want to useαto define a positive injective operator in the GNS-space H ofϕ.In order to do so,we will need the following results.Result2.5There exists a unique strictly positive numberλsuch thatϕαt=λtϕfor t∈I R.Proof:By assumption,there exists for every t∈I R a strictly positive elementλt such thatϕαt=λtϕ. It is then clear thatλ0=1and thatλsλt=λs+t for every s,t∈I R.Now there exist a,b∈A such thatϕ(ba)=0.We have that the function I R→.So we see that the function[−t0,t0]→I R+0:t→λt is continuous.ϕ(α−t(b)a)Therefore the function I R→I R+0:t→λt is continuous.We get from this all the existence of a strictly positive numberλsuch thatλt=λt for t∈I R.So ϕαt=λtϕfor t∈I R.C.Proof:Choose a,b∈A.Then we have two analytic functionsC:u→λuϕ(aα−u(b))and C:u→ϕ(αu(a)b)These functions are equal on the real line:we have for all t∈I R thatλtϕ(aα−t(b))=ϕ(αt(aα−t(b)))=ϕ(αt(a)b)So they must be equal on the whole complex plane.We have in particular thatϕ(αz(a)b)=λzϕ(aα−z(b)),henceϕ αz(aα−z(b)) =λzϕ(aα−z(b)). Because A=Aα−z(A),we infer from this thatϕαz=λzϕ.2Λ(αt(a))for a∈A and t∈I R.Proof:We can define a unitary group representation u from I R on H such that u tΛ(a)=λ−t2 Λ(αt(a)),Λ(b)=λ−tC: u tΛ(a),Λ(b) is continuous.Because u is bounded andΛ(A)is dense in H,we conclude that the function I R→We want to show that every P iz is defined onΛ(A)and that the formula in the above definition has its obvious generalization to P iz.First we will need a lemma for this.8Lemma2.8Consider a∈A.Then the function C:z→λ−Re zϕ(αz(a)∗αz(a))is bounded on horizontal strips.Proof:Take r∈I R.We will prove that the function above is bounded on the horizontal strip S(ri). We have for every t∈I R thatϕ(αti(a)∗αti(a))=ϕ(α−ti(a∗)αti(a))=λ−itϕ(a∗α2ti(a))which implies that the function I R→I R+:t→ϕ(αti(a)∗αti(a))is continuous.So there exists M∈I R+such thatϕ(αti(a)∗αti(a))≤M for t∈[0,r].Hence we get for every z∈Result2.9Consider z∈2Λ(αz(a))for a∈A.Proof:Take v∈H and define the function f from C such that f(u)= λ−u C.Define also for every b∈A the function f b:C such that f b(u)= λ−u C. Then f b(u)=λ−u C which implies that f b is analytic.Choose y∈C around y.By the previous lemma,we get the existence of a positive number M such thatλ−Re uϕ(αu(a)∗αu(a))≤M2for u∈B.This implies easily that λ−u2Λ(αu(a)),Λ(b n) − λ−uC.It is also clear that f(t)= P itΛ(a),v for t∈R.This implies thatΛ(a)∈D(P iz)and that P izΛ(a)=λ−zResult2.10Consider z∈Proof:It is clear that alsoα−i=β−i.Now there exist strictly positive numbersλandµsuch thatϕαt=λtϕandϕβt=µtϕfor t∈I R. Then there exist also positive injective operators P and Q in H such that P itΛ(a)=λ−t2Λ(βt(a))for t∈I R and a∈A.We know thatΛ(A)is a core for both P and Q.We have moreover for every a∈A thatµi2λi2µi2QΛ(a)So we get thatµi2Q.It is clear thatµiλiC thatλ−z2Λ(βz(a))which by the faitfulness ofϕimplies thatλ−z2βz(a).Now we see thatµλα2is multiplicative.Because alsoα2is multiplicative,thisimplies thatµ3Analytic unitary representationsThis section contains the same basic ideas as the previous one.We only apply them in a differentsituation.We will use the results in this section to introduce complex powers of the modular function of an algebraic quantum group in the next section.We will againfix an algebraic quantum group(A,∆)with a positive left Haar functionalϕon it.Let (H,∆)be a GNS-pair forϕ.Let us start of with a definition.Definition3.1Consider a function u fromC→zfor every z∈C→ϕ(u−C→ϕ(u−ϕ(u−t a∗)=ϕ(a u∗−t)=ϕ(a u t)for all t∈I R.So they must be equal on the whole complex plane.We have in particular that z a∗)=ϕ(a u z)which implies thatϕ(a u z)=ϕ(a u∗−z=u z.Result3.3Consider y,z∈C→C→C→C→Corollary3.4Consider z∈C: v tΛ(a),Λ(b) is continuous.Because v is bounded andΛ(A)is dense in H,we conclude that the function I R→We want to show that every P iz is defined onΛ(A)and that the formula in the above definition has its obvious generalization to P iz.First we will need a lemma for this.Lemma3.6Consider a∈A.Then function C:z→ϕ((u z a)∗(u z a))is bounded on horizontal strips.Proof:Take r∈I R.We will prove that the function above is bounded on the horizontal strip S(ri). We have for every t∈I R thatϕ((u ti a)∗(u ti a))=ϕ(a∗u ti u ti a)=ϕ(a∗u2ti a)which implies that the function I R→I R+:t→ϕ((u ti a)∗(u ti a))is continuous.So there exists M∈I R+such thatϕ((u ti a)∗(u ti a))≤M for t∈[0,r].11Hence we get for every z∈Result3.7Consider z∈C into C. Define also for every b∈A the function f b:C such that f b(c)= Λ(u c a),Λ(b) for c∈C which implies that f b is analytic.Choose y∈C around y.By the previous lemma,we get the existence of a positive number M such thatϕ((u c a)∗(u c a))≤M2 for c∈B.This implies easily that Λ(u c a) ≤M for c∈BNow there exists a sequence(b n)∞n=1in A such that(Λ(b n))∞n=1converges to v.We have for every c∈B and n∈I N that|f(c)−f b n(c)|=| Λ(u c a),Λ(b n) − Λ(u c a),v |≤M Λ(b n)−vThis implies that(f b n)∞n=1converges uniformly to f on B,so f is analytic on B.This implies that f is analytic in y.So we see that f is analytic onResult3.8Consider z∈C thatΛ(u z a)=Λ(v z a)which by the faitfulness ofϕimplies that u z a=v z a.4The analytic structure of an algebraic quantum groupIn[5]and[9],we proved that every algebraic quantum group gives rise to a reduced and universal C∗-algebraic quantum group in the sense of Masuda,Nakagami and Woronowicz.These C∗-algebraic quantum groups have a very rich(complicated?)analytic structure.We show in this section that this analytic structure can be completely pulled down to the algebraic level.Consider an algebraic quantum group(A,∆)and letϕbe a positive left Haar functional of(A,∆).Let (H,Λ)be a GNS-pair forϕ.In[9],we constructed a universal C∗-algebraic quantum group(A u,∆u)out of(A,∆).We denote the canonical embedding of A into A u byπu.Soπu is an injective∗-homomorphism into A u such thatπu(A)is dense in A u.We have also for every a∈A and x∈A⊙A that∆u(πu(a))(πu⊙πu)(x)=(πu⊙πu)(∆(a)x)and(πu⊙πu)(x)∆u(πu(a))=(πu⊙πu)(x∆(a)) As a rule,we will give objects associated with A u a subscript‘u’.So the left Haar weight on(A u,∆u) will be denoted byϕu and its modular group byσu.The scaling group of(A u,∆u)will be denoted byτu,the anti-unitary antipode by R u.The modular group of the right Haar weightϕu R u will be denoted byσ′u.We will also need the co-unit on the universal C∗-algebraic quantum group(A u,∆u).Recall that we have a co-unitεon the algebraic quantum group(A,∆).This gives rise to a non-zero∗-homomorphism εu from A u intoC thatπu(a)belongs to D(δiz u)and that δiz uπu(a)belongs again toπu(A).We will use this fact to define any complex power ofδon the algebra level.The definition that we introduce in this paper is much better than the ad hoc definitions in[5]and [9]because we work here within the framework of the algebraic quantum group(A,∆).Proposition4.1There exist a unique analytic unitary representation w on A such that w−i=δ.We defineδz=w−iz for z∈Proof:We know already thatδiy uπu(A)⊆πu(A)for y∈C into M(A)such thatπu(w y a)=δiy uπu(a)for y∈y for y∈C→A u:y→δiy uπu(a)is analytic.This gives us that the mapping C:y→ϕu(πu(c)∗(δiy uπu(a))πu(b))is analytic. But we have for every y∈C→Notice that we have also proven the following characterization for powers ofδ.Proposition4.2We have for every z∈C thatδy+z=δyδz3.We have for z∈z4.Consider z∈2δt2is self adjoint.We have also the following analyticity property.Result4.5We have for everyω∈ˆA that the function C:z→ω(δz)is analytic.By proposition12.2of[9],we know thatπ(δu)=δr.Hence we get thatπ(δz u)=δz r for z∈Proposition4.6We have for every z∈We want now to prove a generalization of this result for complex parameters.The idea behind this proof is completely the same but we have to be a little bit careful in this case.Recall from[9]that every element ofπu(A)is analytic with respect toσu,σ′u andτu.First we prove a little lemma.15Lemma4.8Consider z∈C into C. We have by the previous proposition for every t∈I R that(σu)t(x)belongs toπu(A),which implies that ω (σu)z((σu)t(x)) =0.Hence f(z+t)=ω((σu)z+t(x))=ω (σu)z((σu)t(x)) =0for t∈I R.This implies that f=0.In particular,ω(x)=f(0)=0.So the density ofπu(A)in A u implies thatω=0.Therefore Hahn-Banach implies that(σu)z(πu(A))is dense in A u.C and a∈A.Then(σ′u)z(πu(a))=(σu)z(πu(δiz aδ−iz)).Proof:We introduce the one-parameter representations L,R,θon A u such thatL t(x)=δit u x R t(x)=xδ−itu θt(x)=δit u xδ−itufor x∈A u and t∈I R.Then L and R commute andθt=L t R t for t∈I R.This implies that L z R z⊆θz(see e.g.proposition3.9 of[8]).We know thatπ(a)∈D(R z)and that R z(πu(a))=πu(aδ−izu).This implies that R z(πu(a))∈D(L z)andthat L z(R z(πu(a)))=πu(δiz u aδ−izu ).So we get thatπu(a)∈D(θz)and thatθz(πu(a))=πu(δiz u aδ−izu).We have moreover thatσu andθcommute and that(σ′u)t=(σu)tθt for all t∈I R.This implies again that(σu)zθz⊆(σ′u)z.Therefore(σ′u)z(πu(a))=(σu)z θz(πu(a)) =(σu)z(πu(δiz aδ−iz)).C.Then(σu)z(πu(A))=(σ′u)z(πu(A)).Now we are in a position to prove a generalization of proposition4.7.Proposition4.11Consider z∈C.Consider elements a1,...,a m,b1,...,b m∈A and p1,...,p n,q1,...,q n∈A such that the equalitym i=1∆(a i)(1⊗b i)= n j=1p j⊗q j holds.Thenmi=1∆u(πu(a i))(1⊗πu(b i))=nj=1πu(p j)⊗πu(q j)We have now the following two analytic functions:andCorollary4.12We have that R u(πu(A))=πu(A).Proof:We know by theorem9.18of[9]that R u((τu)−iThese results imply that al the objects associated to the C∗-algebraic quantum group(A u,∆u)can be pulled down to the algebraic level.Thefirst(and typical)result is contained in the next proposition. Proposition4.13There exists a unique analytic one-parameter groupσon A such thatσ−i=ρ.We have moreover thatπu(σz(a))=(σu)z(πu(a))for every a∈A and z∈Proof:By proposition4.11,we know that(σu)z(πu(A))=πu(A)for every z∈C into the set of mappings from A into A such that πu(σz(a))=(σu)z(πu(a))for every a∈A and z∈C thatαz is a homomorphism on A.2.We have for every t∈I R thatαt is a∗-automorphism on A.3.We have for every s,t∈I R thatαs+t=αsαt.Choose z∈A.Thenπu(a)∈D((σu)z)∩Mϕu and(σu)z(πu(a))=πu(σz(a))∈Mϕu.Becauseϕu is invariant underσu,this implies by proposition2.14of[7]thatϕu(πu(a))=ϕu (σu)z(πu(a)) =ϕu πu(σz(a))which implies thatϕ(a)=ϕ(σz(a)).Take a∈A andω∈ˆA.Then there exist b,c∈A such thatω=cϕb∗.We know by proposition8.8of[9]thatπu(a)is analytic with respect toσu.This implies that the functionC→C thatϕu(πu(b)∗(σu)z(πu(a))πu(c))=ϕu(πu(b)∗πu(σz(a))πu(c))=ϕu(πu(b∗σz(a)c))=ϕ(b∗σz(a)c)=ω(σz(a))So the function C:z→ω(σz(a))is analytic.Hence we get by definition thatσis an analytic one-parameter group on A.Using proposition8.8of[9],we see thatσ−i=ρ.C thatϕσz=ϕ.There exists a canonical GNS-construction(H,Λu,π)for the weightϕu such thatπu(A)is a core forΛu andΛu(πu(a))=Λ(a)for a∈A(see definition10.2of[9]and theorem10.6of[9]).Denote the modular operator ofϕu by∇and the modular conjugation ofϕu by J(both with respect to this GNS-construction).Then these objects are completely characterized by the following two results.Result4.15Consider z∈Corollary4.16We have for every a∈A that JΛ(a)=Λ(σiBy the remarks after definition8.4of[9],we know thatπ(σu)t=(σr)tπfor t∈I R.Consequently,π(σu)z⊆(σr)zπfor z∈C and a∈A that(σr)z(πr(a))=πr(σz(a)).Completely similar to proposition4.13,we have the next results.Remember that we have proven in corollary7.3of[5]the existence of a unique strictly positive numberνsuch thatϕr(τr)t=νtϕr for t∈I R. Proposition9.19of[5],proposition8.17of[5]and proposition8.20of[7]imply thatϕr(σ′r)t=ν−tϕr for t∈I R.This gives us also thatϕr(τu)t=νtϕu and thatϕu(σu)z=ν−tϕu for every t∈I R.Proposition4.18There exists a unique analytic one-parameter groupσ′on A such thatσ′−i=ρ′. We have moreover thatπu(σ′z(a))=(σ′u)z(πu(a))for every a∈A and z∈C thatϕσ′z=ν−zϕ.Proposition4.20We have for every a∈A and z∈C.We callτthe scaling group of(A,∆).Result4.22We have for every z∈C thatπr(τz(a))=(τr)z(πr(a)).Now it is the turn of the anti-unitary antipode R u to be pulled down.This is possible thanks to corollary 4.12Definition4.24We define the mapping R from A into A such thatπu(R(a))=R u(πu(a))for a∈A. Then R is a∗-anti-automorphism on A such that R2=ι.We call R the anti-unitary antipode of(A,∆).We know by[9]that R rπ=πR u which gives us,as usual,the next result.Proposition4.25We have for every a∈A thatπr(R(a))=R r(πr(a)).19In the rest of this section,we will prove the most basic relations between the objects introduced in this section.In most cases,the proofs consist of pulling down the corresponding relations on the C∗-algebra level.We will make use of the results on the reduced C∗-algebra level because they were proved before the results on the universal level(in fact,most of the latter results depend on the results on the reduced level).First we describe the polar decompositionResult4.26We have for every z∈.2These are immediate consequences of corollary5.4of[5]and theorem5.6of[5].Corollary4.28We have the following commutation relations:1.We have for every z∈C thatτyσz=σzτy.Proof:The remarks before this result imply easily thatσtτs=τsσt for s,t∈I R.Take a,b∈A.Fix t∈I R for the moment.Then we have two analytic functionsC:u→ϕ(σ−t(b)τu(a))and C:u→νuϕ τ−u(b)σt(a)These functions are equal on the real line:we have for every s∈I R thatϕ(σ−t(b)τs(a))=ϕ bσt(τs(a)) =ϕ bτs(σt(a)) =νsϕ(τ−s(b)σt(a))So they must be equal onC→C→Now we prove some relations in connection with the comultiplication.Proposition4.30Consider z∈•(τz⊙τz)∆=∆τz•(τz⊙σz)∆=∆σz•(σ′z⊙τ−z)∆=∆σ′z•(σz⊙σ′−z)∆=∆τzProof:We only prove thefirst equality.The others are proven in the same way.Choose a∈A.Take b∈A.By proposition5.7of[5],we have for every t∈I R that(πr⊙πr)(∆(τt(a))(τt(b)⊗1))=∆ πr(τt(a)) (πr(τt(b))⊗1)=∆ (τr)t(πr(a)) ((τr)t(πr(b))⊗1)=((τr)t⊗(τr)t) ∆r(πr(a)) ((τr)t(πr(b))⊗1)=((τr)t⊗(τr)t)(∆r(πr(a))(πr(b)⊗1))=((τr)t⊗(τr)t) (πr⊙πr)(∆(a)(b⊗1)) =(πr⊙πr) (τt⊙τt)(∆(a)(b⊗1))which implies that∆(τt(a))(τt(b)⊗1)=(τt⊙τt)(∆(a)(b⊗1))(*)Choose p,q∈A.Then we have two functionsC:y→(ϕ⊙ϕ)((1⊗p)∆(q)∆(τy(a))(τy(b)⊗1))andC:y→(ϕ⊙ϕ) (1⊗p)∆(q)(τt⊙τt)(∆(a)(b⊗1))We see immediately that the second function is analytic.Because(ϕ⊙ϕ)((1⊗p)∆(q)∆(τy(a))(τy(b)⊗1))=ϕ(pτy(b))ϕ(qτy(a))for y∈A,also thefirst function is analytic.We know by(*)that both functions are equal on the real axis,so they must be equal on the whole complex plane.In particular,(ϕ⊙ϕ)((1⊗p)∆(q)∆(τz(a))(τz(b)⊗1))=(ϕ⊙ϕ) (1⊗p)∆(q)(τz⊙τz)(∆(a)(b⊗1)) . Hence the faithfulness ofϕimplies that∆(τz(a))(τz(b)⊗1)=(τz⊙τz)(∆(a)(b⊗1))=(τz⊙τz)(∆(a))(τz(b)⊗1)So we see that∆(τz(a))=(τz⊙τz)(∆(a)).C.Thenετz=ε.Corollary4.33We have thatεR=ε.In the next part,we look at some formulas involving the modular functionδ.Thefirst one says that everyδz is a one-dimensional corepresentations of(A,∆).Proposition4.34We have for every z∈Proof:Take a,b∈A.Then we have by proposition8.6of[5]for every t∈I R that(πr⊙πr)(∆(δit)∆(a)(b⊗1))=(πr⊙πr)(∆(δit a)(b⊗1))=∆r(πr(∆(δit a))(πr(b)⊗1) =∆r(δit rπr(a))(πr(b)⊗1)=(δit r⊗δit r)(πr⊙πr)(∆(a)(b⊗1))=(πr⊙πr)((δit⊗δit)∆(a)(b⊗1))which implies that∆(δit)∆(a)(b⊗1)=(δit⊗δit)∆(a)(b⊗1)(∗)We have now two functionsC:u→(ϕ⊙ϕ)(∆(δiu)∆(a)(b⊗1))andC:u→(ϕ⊙ϕ)((δiu⊗δiu)∆(a)(b⊗1))The second function is clearly analytic.Because(ϕ⊙ϕ)(∆(δiu)∆(a)(b⊗1))=ϕ(δiu a)ϕ(b)for u∈A, also the second is analytic.Furthermore,(*)implies that both functions are equal on the real line. So they must be equal on the whole complex plane.In particular,(ϕ⊙ϕ)(∆(δz)∆(a)(b⊗1))= (ϕ⊙ϕ)((δz⊗δz)∆(a)(b⊗1))Hence the faithfulness ofϕimplies that∆(δz)∆(a)(b⊗1)=(δz⊗δz)∆(a)(b⊗1)So we get that∆(δz)=δz⊗δz.C thatε(δz)=1and S(δz)=δ−z.Result4.36Consider y,z∈C→C→Combining the result concerningτwith corollary4.35and proposition4.27,we get the following one.Result4.37Consider z∈C that(σ′r)z=R r(σr)−z R r.So we get the next result.Result4.38Consider z∈C and a∈A thatσ′z(a)=δizσz(a)δ−iz.We end this section with some remarks concerning the right Haar functional on(A,∆).Recall that we have a right Haar functionalϕS on(A,∆)but we do not know(yet)whetherϕS is positive.By the formulaχ(R⊙R)∆=∆R,we have however the following proposition.Theorem4.40The functionalϕR is a positive Haar functional on(A,∆).Because S=τ−i2ϕR.BecauseϕS=δϕ,we have also thatϕ(R(a))=ϕ(δ12)for a∈A.5The analytic structure of the dualIn this section,we will connect the alytic objects associated to the dual of an algebraic quantum groups to the analytic objects of this algebraic quantum groups.So consider an algebraic quantum group(A,∆)with a positive left Haar functionalϕon it.As in the previous section,we will use the notationsσfor the modular group of the left Haar functional,σ′for the modular group of the right Haar functional,τfor the scaling group and R for the anti-unitary antipode. The corresponding objects on the dual quantum group(ˆA,ˆ∆)will get a hat on them,e.g.ˆσwill denote the modular group of the left Haar functional on(ˆA,ˆ∆).First we start with the modular group of the dual quantum group(ˆA,ˆ∆).(Similar results are also considered in[10]).We will introducefirst a temporary notation.Considerω∈M(ˆA).Then we defineωz∈A′such that ωz(a)=ω(τz(a)δ−iz)for a∈A.Lemma5.1Consider z∈where we used result4.36in the second last equality.Consequently,(ωz⊙ι)∆(a)=τ−z [(ω⊙ι)∆(τz(a)δ−iz)]δiz .The other equality is proven in the same way.C.Then we have the following properties.1.We have for everyω∈M(ˆA)thatωz∈M(ˆA).2.Considerω,θ∈M(ˆA).Then(ωθ)z=ωzθz.Lemma5.3Consider z∈Lemma5.4Considerω∈ˆA and a∈A.Then the fuction C:z→ωz(a)is analytic.Proof:There exist b,c,d∈A such thatω=bcϕd∗.We know thatπr(a)is analytic with respect toτr and thatπr(b)is analytic with respect toδr.This implies that the function C→A r:z→πr(b)πr(c))is analytic.ϕr(πr(d)∗(τr)z(πr(a))δ−izrBut we have for every z∈C→Proposition5.5We have for everyω∈ˆA,a∈A and z∈C into L(ˆA)such thatβz(ω)=ωz for z∈2.Choose s,t∈I R.Takeω∈ˆA.Then we have for every a∈A that[βs(βt(ω))](a)=[βt(ω)](τs(a)δ−it)=ω(τt(τs(a)δ−it)δ−is)=ω(τt(τs(a))δ−itδ−is)=ω(τs+t(a)δ−i(s+t)))=[βs+t(ω)](a), implying thatβs(βt(ω))=βs+t(ω).So we have proven thatβsβt=βs+t.3.Lemma5.2implies thatβt is multiplicative for every t∈I R.4.Choose t∈I R.Takeω∈ˆA.Then we have for every a∈A that[βt(ω)]∗(a)=ω(τt(S(a)∗)δ−it)=ω((δit S(τt(a)))∗)=C that b(βz(ω))=βz(ω)(a)=ωz(a))which implies that the function C:z→b(βz(ω))is analytic by lemma5.4So we can conclude from this al thatβz is an analytic one-parameter group onˆA.Choose a∈A.Lemma2.8of[6]implies thatˆσi(ψa)=ψδS2(a).We know by the proof of lemma5.3 that(ψa)i=ψδS2(a).So we see thatˆσi(ψa)=βi(ψa).Henceˆσi=βi which by proposition2.11implies thatˆσ=β.C thatˆσz(ω)(a)=ω(τz(a)δ−iz).The proof of the next result is completely similar(an easier)to the proof of the previous proposition.It is a consequence of the fact thatˆS2(ω)=ωS2forω∈ˆA.Proposition5.7We have for everyω∈M(ˆA),a∈A and z∈,we get easily the following on.2Corollary5.8We have for everyω∈M(ˆA)thatˆR(ω)=ωR.Remembering thatˆσ′z=ˆRˆσ−zˆR,it is now easy to check the next equality.Corollary5.9We have for every a∈A,ω∈M(ˆA)and z∈。

立体几何法解决平面中的共线共点问题

立体几何法解决平面中的共线共点问题

立体几何法解决平面中的共线共点问题参赛队员:鲁海昊 臧佳玮指导老师:严贤付学校:青岛二中PrefaceGenerally speaking, when we deal with the three-dimensional geometry problems, we used to transfer them into the plane geometry first. However, I cannot agree with the idea that we should always transfer the three-dimensional geometry problems into the plane geometry mode.In the following article, we solve the problem in an opposite way, which means solving the problems of a total of three points or three planes by using the three-dimensional geometry. The main idea is simple: if three points both are on two planes, they are in line; if three lines are the interception lines of three different planes which intercept each other, they have a common point. The key to this point is how to construct the two or three planes. It follows that we provide three methods to construct the planes, and independently prove some basic theories and properties.(1)transferring plane graphs into three-dimensional graphs(2)using proportional relationship and similarity(3)symestrically finding out two planes which do not parallel to the base Therefore, let’s walk into the three-dimensional world, and research the common problems in a different vision.前言:平时我们解决立体几何问题一般都要转化为平面几何来处理。

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IAC-02-Q.3.3.03THE INTERPLANETARY INTERNET:A COMMUNICATIONS INFRASTRUCTURE FOR MARS EXPLORATIONS. Burleigh, Jet Propulsion Laboratory, Pasadena, CA, USAV. Cerf, WorldCom Inc., Loudon, VA, USAR. Durst, MITRE Corporation, Reston, VA, USAK. Fall, Intel Corporation, Berkeley, CA, USAA. Hooke, Jet Propulsion Laboratory, Pasadena, CA, USAK. Scott, MITRE Corporation, Reston, VA, USAH. Weiss, Sparta Inc., Columbia, MD, USA53rd International Astronautical Congress The World Space Congress - 200210-19 Oct 2002/Houston, TexasFor permission to copy or republish, contact the International Astronautical Federation3-5 Rue Mario-Nikis, 75015 Paris, FranceIAC-02-Q.3.3.03THE INTERPLANETARY INTERNET:A COMMUNICATIONS INFRASTRUCTURE FOR MARS EXPLORATIONScott Burleigh, Jet Propulsion Laboratory, Pasadena, CA, USAScott.Burleigh@Vinton Cerf, WorldCom Inc., Loudon, VA, USAvinton.g.cerf@Robert Durst, MITRE Corporation, Reston, VA, USAdurst@Kevin Fall, Intel Corporation, Berkeley, CA, USAkfall@Adrian Hooke, Jet Propulsion Laboratory, Pasadena, CA, USAadrian.hooke@Keith Scott, MITRE Corporation, Reston, VA, USAkscott@Howard Weiss, Sparta Inc., Columbia, MD, USAhsw@ABSTRACT∗A strategy is being developed whereby the current set of internationally standardized space data communications protocols can be incrementally evolved so that a first version of an operational “Interplanetary Internet” is feasible by the end of the decade. This paper describes its architectural concepts, discusses the current set of standard space data communications capabilities that exist to support Mars exploration and reviews proposed new developments. We also speculate that these current capabilities can grow to support future scenarios where human intelligence is widely distributed across the Solar System and day-to-day communications dialog between planets is routine.∗ This work was performed in part at and under the direction of the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.THE FUTURE:AN INTERPLANETARY INTERNETIn 1998 the US Defense Advanced Research Projects Agency (DARPA), as part of its “Next Generation Internet” initiative, began funding asmall group at NASA's Jet Propulsion Laboratory in Pasadena, California to study the technical architecture of an “Interplanetary Internet”. The idea was to blend ongoing workin standardized space communications capabilities with state of the art techniques being developed within the terrestrial Internet community, with a goal of achieving end-to-end communication in a multi-network interplanetary environment. The “Interplanetary Internet” name was deliberately coined to suggest a far-future integration of space and terrestrial communications infrastructure to support the migration of human intelligence throughout the Solar System.Joining the JPL team in this work wasone of the original designers of the Internet andco-inventor of the “Transmission Control Protocol/Internet Protocol” (TCP/IP) protocol suite. Support for the work has recently transitioned from DARPA to NASA.The future Interplanetary Internet architectural concept is deceptively simple:1. Use Internet or Internet-related protocols toform local networks in low delay, relativelylow noise environments such as aroundEarth, within a free flying spacecraft, onand around another planet, etc.2. A specialized deep space backbone networkof long-haul wireless links interconnectingthese local internets. This interplanetarybackbone is expected to evolve to includemultiple space-based data relay satellites.3. The resulting interplanetary internet thusconsists of a "network of internets". Just asthe familiar TCP/IP suite unites the Earth's"network of networks" into the Internet, theInterplanetary Internet will employ a newoverlay protocol concept called bundling totie together a set of heterogeneous internets.A routing function will direct bundles (messages) through a concatenated series of Internets, just as the Earth’s current Internet protocol (IP) routes data through a series of independent networks on Earth. To guarantee reliability of the end-to-end transfer, the bundles will also contain retransmission mechanisms functionally similar to those provided by the terrestrial Internet’s Transmission Control Protocol (TCP). However, the similarity of bundling with TCP/IP ends there.Unlike the Earth’s backbone environment of continuous connectivity, negligible delay and clean data channels, the hallmarks of the interplanetary backbone are therefore intermittent connectivity, huge propagation delays and noisy data channels.While the Earth’s backbone network is wired – large numbers of fiber or copper circuits interconnecting fixed hubs – the interplanetary backbone is dependent on fragile wireless links. In addition, the hubs on the interplanetary backbone (relay spacecraft or gateways into remote local Internets) are all moving with respect to each other. Planets travel in fixed orbits and sometimes bodies like the Sun cause line of sight occultations that last for days on end. Landed vehicles on remote planetary surfaces will move out of sight of Earth as the body rotates, and may have to communicate through local relay satellites that only provide data transmission contacts for a few minutes ata time.The bundling protocol handles this environment in two ways:• It operates in a “store and forward” mode, very similar to e-mail, where bundles areheld at routers along the way until suchtime as a forward path is established.• It avoids the need for a sender to store data until an acknowledgement is received fromthe other end by operating in a "custodial"mode. In this mode, intermediate nodes inthe network can assume responsibility forensuring that bundles reach their destinations, allowing senders (and previous custodians) to reassign resourcesto new observations.• In the presence of high error rate links, the hop-by-hop store-and-forward bundling model with per-hop error control increasesthe probability of successful end to endtransmission.One key problem in the design of an Interplanetary Internet is identifying the communicating endpoints. If a common, solar system-wide IP address convention were adopted, then every component of the system would need to be upgraded on a common timescale to preserve the commonality of the address space. The current concept is that rather than have a single address space across the entire Solar System end point identifiers comprise a two-part name. One part of the name (the routing part) gets the bundledelivered to a remote destination “region” of the Interplanetary internet. The second part of the name (the administrative part) contains the information required to deliver to one or more local destinations. Thus for Mars operations the routing part of the name will be used to move the bundle across the Deep Space backbone to the entry gateway on the appropriate region on Mars, where the administrative part of the name comes into play and identifies the local recipient(s) on the Martian internet.THE PRESENT: MARSCOMMUNICATIONSIN THE COMING DECADEA successful program of Mars exploration will need a robust, dependable and high capacity space communications infrastructure. In the terrestrial environment, the g TCP/IP suite provides these features. Programs of Mars exploration will need an analogous set of standard capabilities to support automated communications over the vast distances, heterogeneous and stressed environments that make up the Earth-Mars communications system. Although space exploration is unlikely to experience the sort of commercially driven growth that has been seen in the terrestrial Internet in the last decade, some clear “market forces” are already emerging in the space community:a. Hard requirements exist for interoperabilityamong the many agencies and organizationsthat will be cooperatively executing theintegrated program of Mars missions.b. Each of those missions needs to avoid thecost and risk of developing unique systems,so there is the inevitable allure of deployingsystems in space that have a pedigreerooted in their robust Earth-based commercial counterparts.c. In order to feed an intense interest by thegeneral public to participate in the excitement of exploration, large quantitiesof data must be rapidly acquired from thevicinity of Mars and processed fordissemination via the Web.Since the early 1980s, the Consultative Committee for Space Data Systems1 (CCSDS)– an international organization currently supported by thirty-four space agencies - has been incrementally developing a basic set of standardized space communications techniques that are now in ubiquitous use within the world space community. In fact, approximately two hundred and twenty space missions are currently committed to use the CCSDS capabilities.In a typical space mission configuration, as shown in Figure 1, programmed intelligence is widely distributed across both the space and ground segments and large numbers of computers need to exchange information either locally (within the space system or within the ground system) or end-to-end between space and ground.Figure 1: Space Mission Data InterfacesThe CCSDS is organized into technical panelsto develop the standards that cluster into five major categories where international interoperability is needed:1. Data handling interfaces within or betweenspacecraft, such as the mechanismswhereby a payload may connect to theonboard data system or whereby a landedvehicle may talk to an orbiter via a spacelink.2. Long-haul data links that connect aspacecraft with its ground system.3. End-to-end data paths that utilize thosespace links to support networked data flowbetween ground and space.These first three categories form the “space/ground communications system” that provide bi-directional data exchange in support of users, who access:4. Mission management services (such asnavigation, flight operations and facilitycontrol) that are exposed by one organization to another.5. Mechanisms for describing, sharing andarchiving the scientific information products derived from the mission.CURRENT SPACE/GROUNDCOMMUNICATIONS STANDARDS CCSDS protocol standards are “layered” so that they stack together in a modular fashion.Figure 2: Space Protocol StackThe main logical components of the protocol stack are shown in Figure 2. At the top of the stack are the user “applications” (typically sensor and control systems) that run on computers located in space or on the ground. When two applications need to exchange information, several underlying layers of standard data communications protocol support them. Most of these standard layers contain multiple options that can be selected to meet mission needs, and most of the layers can be bypassed if not required. Working from the bottom the stack upwards, the layers are as follows.1. Wireless standards. These standards specifythe frequencies and efficient modulationtypes to be used to create the channelconnecting the spacecraft to its groundstations or other spacecraft.2. Coding standards. These “clean up” errorson those wireless channels and make themmore suitable for automated data transfer.The CCSDS coding standards include avariety of high performance technologiesincluding Convolutional, Reed-Solomonand Turbo Codes.3. Link standards. The “frames” that carryhigher layer data across the space link arespecified here. The CCSDS “PacketTelemetry” and “Packet Telecommand”standards handle virtually all the long-haullinks. Packet Telecommand supports Link-layer reliability by providing a ‘go-back-n’frame retransmission protocol, known asthe “Command Operation Procedure”(COP), that works best in a short-delayenvironment. A more general retransmission protocol for long delaycommunications is planned but not yetavailable. CCSDS “Advanced OrbitingSystems” is an adaptation of PacketTelemetry to handle high rate data transfer,and is used by the International SpaceStation and many Earth-observing missions.A new CCSDS “Proximity-1” protocolhandles short-range communications, suchas between landed vehicles and planetaryorbiters, or between multiple spacecraftflying in a constellation. It is derived fromCCSDS Telecommand and provides bi-directional Link layer reliability via aderivative of the COP retransmissionscheme.4. Networking Standards. The Space link isjust one component of the end-to-end datapath between a spacecraft instrument and auser. In order to traverse the whole path,“routing’ information needs to beassociated with each chunk of user data.The CCSDS Packet (the “packet” part ofPacket Telemetry and Telecommand) hasbeen in use as a “CCSDS Path” (connectionoriented) networking protocol for well overa decade. It exploits the fact that for mostcurrent missions there is a highlypredictable data routing path between aninstrument and a user, so there is little needfor adaptive packet routing (and theconcomitant extra communications overhead associated with carrying largesource and destination addresses in everypacket).More recently, CCSDS has added thecapability to allow onboard systems to havetheir own Internet Protocol (IP) addresses.This is accomplished by either direct use ofIP, or an abbreviated form of IP that is theNetwork Protocol (NP) component of afour-part stack of protocols known as theSpace Communication Protocol Standards(SCPS)2. Both of these capabilities allowpackets to be dynamically routed throughdifferent paths in a connectionless manner.5. Security Standards. As missions becomemore Internet-accessible, they become morevulnerable to attack. Basic authenticationand encryption can be accomplished withinthe CCSDS Link standards but morepowerful end-to-end techniques can protectthe entire flow of user data. Two standardprotocol choices exist: Internet ProtocolSecurity (IPSec) and a SCPS SecurityProtocol (SP). Both provide multiple levelsof data protection:Access Control – prevention of unauthorized users from sending data.Authentication – guarantee of the identity of the sender.Integrity – protection against the intentional or accidental modification ofthe user data during transit.Confidentiality – protection from disclosure of the contents of the userdata.6. End-to-End Reliability Standards. All ofthe standards up to this layer have beenprimarily associated with getting a singlepacket of data delivered between two endsystems. By concatenating powerful channel coding with Link layer retransmission over the space link, andassuming no loss on Earth or in thespacecraft or local Mars networks, there is ahigh probability that the packet will bedelivered.However, if the packet gets lost due tobuffer overflows somewhere in the end-to-end path, or damaged by bit errors duringtransit, there will be a gap in the user data.Absent any other hop-by-hop remedies, theonly way to fill such gaps is via end-to-endretransmission. This retransmission can beperformed three ways: manually by humans; by custom code running in each ofthe applications that are sending andreceiving data; or by invoking a general-purpose communications protocol that isdedicated to that job.For short delay communications, the CCSDS recommends a protocol solutionand has adopted the Internet “TransmissionControl Protocol” (TCP) and SCPS extensions to TCP know as “TCP Tranquility”. For those applications notneeding TCP’s services, the Internet UserDatagram Protocol (UDP) can be used tosegment and encapsulate user data.7. Space File Transfer Standards. This layerof protocol – the first of several so-called“Application Services” that will probablybe developed in the near future – directlysupports the user applications that arerunning end-to-end. In recent years therehas been a rapid shift towards organizingspace data transfer into standalone andautonomous files that may be assigneddifferent priorities and individually accounted-for. This is particularly important as ground networks such as theDSN become heavily subscribed, so that alarge amount of two way traffic betweenthe spacecraft and the ground can beconducted and verified in a short intervaland the tracking assets can then be releasedto service another spacecraft.The CCSDS currently supports two file-based standard capabilities:The Internet File Transfer Protocol (FTP), and SCPS space-adaptedextensions to FTP. These are primarilyfor use in short-delay Internet-likeenvironments, and assume anunderlying layer of TCP.The CCSDS File Delivery Protocol (CFDP). This is a delay tolerantprotocol whose model of operations isfundamentally store-and-forward, muchlike e-mail that conveys files asattachments. The protocol as currentlydesigned contains its own reliabilitymechanisms and does not assume anunderlying retransmission capability. Itpresently operates point-to-point acrossa single link and contains three parts:file manipulation commands that allowfiles to be created and exchanged;filestore commands used to manageremote file systems; and a reliabilityprotocol that ensures that all of thepieces of the file are properly deliveredacross the link, with any missing piecesbeing automatically retransmitted.The long-term CFDP concept has its own notion of “custodial transfer” where a sender can transmit a file to a receiver over a single link and, upon receipt of the entire file, the receiver can notify the sender that it will take care of any successive forward transmission hops. This allows the sender to release local processing and storage resources and to deploy them on new data acquisition – a very important feature for transmission of data to and from the surface of Mars. For missions not wanting to use file transfers, applications can bypass the file process and access most of the underlying layers directly.COMMUNICATIONS SCENARIOFOR MARS OPERATIONSData communications in support of Mars exploration has already moved into the “networked” era as we have started to deploy communications relay orbiters around the planet.Figure 3: Mars Networking ScenarioA typical scenario (Figure 3) reveals that space-based data communications cluster into three main groups:a. Local networking among surface vehiclesthat are stationary, roving and in theatmosphere.b. Short-haul relay communications betweenlanded vehicles and Mars-orbiting spacecraft.c. Long haul data transfer directly between thesurface and the Earth, or from the relayspacecraft to Earth.For the most part, the international standards that are now coming into widespread use can satisfy these communications problems. A typical mapping of these existing standard capabilities across these various environmentsis shown in Figure 4.Figure 4: Mars Communications Protocol Stack 1. The CCSDS File Delivery Protocol (CFDP)is emerging as the leading candidate for theubiquitous “end-to-end” protocol for mostnear-term Mars operations. It operates bi-directionally and allows users to exchangefiles between assets on and around Marsand facilities on the ground.2. Within the Earth’s Internet, CFDP will betransported using standard Internet protocols. On the Deep Space backbone,CFDP is transferred using the CCSDS Networking protocols, running over the CCSDS Long-Haul space link and codingprotocols, which run over the S, X or Ka-band wireless channels.3. For those missions with direct linksbetween the Martian surface and Earth, theDeep Space backbone long-haul protocolswill run all the way down to the surface andnear-surface vehicles.4. For those missions using communicationsrelay spacecraft, the long haul link protocols will be terminated at the orbiterand the CCSDS Proximity Link and Codingprotocols will be used to communicate between the orbiters and the landed assets.The orbiters will bridge the CCSDS Networking protocol from the long-haul link protocol to the proximate protocol.STANDARDS EVOLUTIONIN THE COMING DECADEThe current CCSDS File Delivery Protocol, CFDP, is by design a prototypical form of the bundling protocol that will be required for the future Interplanetary Internet.Figure 5: Current CFDP ArchitectureThe current CFDP architecture (Figure 5) consists of three parts:a. file handling mechanisms, plus;b. point to point reliability mechanisms,which;c. draw upon underlying space link datatransfer services.Extensions to CFDP are currently under development that will allow it to support multi-hop custodial file data transfers of the sort envisioned by the “Mars Network” concept3. This set of capabilities should therefore be sufficient to satisfy immediate Mars mission needs.The current bundling protocol architecture (Figure 6) improves on CFDP in several key respects:Figure 6: Current bundling Architecturea. It is not confined to supporting just filetransfer, but it can handle virtually any end-to-end space application. Eventually, CFDPwill simply “move up the stack” to becomeone of those applications. b. Its internal functions are more clearlymodular than CFDP, so that it should beeasier to evolve over time.c. It will provide a more flexible custodialtransfer capability than is achievable withCFDP.The detailed specification and prototyping of bundling is already underway and it is hoped by the middle of this decade to be in a position for CFDP to make a smooth transition from running directly over the space link to running over bundling. At that point, we will be ready to begin Interplanetary Internet operations.INTERPLANETARY INTERNET:DEPLOYMENT STRATEGYThe terrestrial Internet did not just “happen overnight”; it has evolved over a period of thirty years. Similarly, standardized space communications have been evolving for the last twenty years. In the coming decades, intense interest in Mars exploration and the correspondingly large number of missions that will be flown provides a good opportunity to accelerate that evolution.Individual mission organizations are rarely altruistic. They typically focus their scarce resources on solving immediate problems, and let subsequent missions solve their own. Coordinated Mars exploration, however, provides a rare sense of community because many space vehicles from many different organizations are all focused on a common goal. In particular, Mars exploration offers a unique communications opportunity because of the presence of multiple orbiting spacecraft that can act as communications relays.Anything landed on the surface of Mars, or deployed in its atmosphere, faces stringent communications challenges, of which power availability is currently the most constraining. Instead of having to transmit and receive overthe vast interplanetary distances to and from Earth, surface vehicles can now exploit proximate relay spacecraft to reduce their communications burdens. In order to expedite these relays, payloads such as the Electra Mars Network Transceiver4 are already being developed that may be shared by multiple missions. Even if large numbers of dedicated communications relay satellites are not affordable, communications payloads can be added to scientific orbiters that increase their utility and inherently make them part of a larger community.We believe that the Interplanetary Internet will follow a similar path, where “primary” scientific missions will be given “secondary” relay capabilities that will facilitate the slow accretion of communications network infrastructure throughout the Solar System. Missions will help each other out by supporting each other, because they themselves may be the beneficiaries of such cooperation.CONCLUSIONAs we enter this exciting period of Mars exploration, we already have a rich and proven set of international standards to support our needs for communicating between Mars and Earth. Those standards will continue to evolve and grow in capability, and by the 2005-2007 timeframe we should have sufficient capabilities on hand to make Mars one of the first true extensions of the Internet: the Interplanetary Internet is about to be launched. To track its progress, refer to the Interplanetary Internet Special Interest Group5 within the Internet Society.REFERENCES1. Consultative Committee for Space DataSystems: 2. Space Communications Protocol Standards:3. Mars Network 4. “The Electra Mars Network Transceiver”;Bell, Edwards, Graf, Komarek, Lehman.Paper IAC-02-Q.3.3.01, International Astronautical Congress, Houston, October2002.5. Interplanetary Internet Special InterestGroup (IPNSIG): 。

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