k-order Slant Toeplitz Operators on the Bergman Space
Distributed Order Calculus an Operator-Theoretic Interpretation
(7)
2
where κ (t) is the inverse Laplace transform of the function z →
γ +i∞
1 , z K (z ) (8)
d 1 κ (t) = dt 2πi
γ −i∞
ezt 1 · dz, z z K (z )
γ > 0.
It was proved in [11] that κ ∈ C ∞ (0, ∞), and κ is completely monotone; for small values of t, 1 κ (t) ≤ C log , t 1 |κ ′ (t)| ≤ Ct−1 log , t (9)
td D(µ) ϕ (t) = Nhomakorabeadt0
k (t − τ )ϕ(τ ) dτ − k (t)ϕ(0).
(4)
The right-hand side of (4) makes sense for a continuous function ϕ, for which the derivative d t k (t − τ )ϕ(τ ) dτ exists. dt 0 If a function ϕ is absolutely continuous, then
1. Introduction and Preliminaries. In the distributed order calculus [11], used in physics for modeling ultraslow diffusion and relaxation phenomena, we consider derivatives and integrals of distributed order. The definitions are as follows. Let µ be a continuous non-negative function on [0, 1]. The distributed order derivative D(µ) of weight µ for a function ϕ on [0, T ] is
On the parabolic kernel of the Schrodinger operator
(0.2)
where q is a function on M alone. We shall point out that for the heat equation (q=0), upper estimates for the heat kernel were obtained in [7] and [5]. However the estimate which we obtain is so far the sharpest, especially for large time. When the Ricci curvature is nonnegative the sharpness is apparent, since a comparable lower bound is also obtained in w4. A lower bound for the fundamental solution of (0.2) is also derived for some special situations. Applications of these estimates for the heat kernel are discussed in w5. A generalization of Widder's [25] uniqueness theorem for positive solutions of the heat equation is proved (2) (Theorem 5.1). In fact, the condition on the curvature is best possible due to the counter-example of Azencott [2]. We also point out that generalizations of Widder's theorem to general elliptic operators in R n were derived in [21], [1 l] and [l]. When M has nonnegative Ricci curvature, sharp upper and lower bounds of Green's function are derived. This can also be viewed as a necessary and sufficient condition for the existence of Green's function which was studied in [23]. In fact, in [24], our estimates on Green's function were proved for nonnegatively Ricci curved manifolds with pole and with nonnegative radial sectional curvatures. In [13], Gromov proved lower bounds for all the eigenvalues of the Laplacian on a compact nonnegatively Ricci curved manifold without boundary. We generalized these estimates to allow the manifold to have convex boundaries with either Dirichlet or Neumann boundary conditions. These lower bounds can also be viewed as a generalization of the lower bound for the first eigenvalue obtained in [16]. Another application is to derive an upper bound of the first Betti number, bi, on a compact manifold in terms of its dimension, a lower bound of the Ricci curvature, and an upper bound of the diameter. The manifold is allowed to have convex boundaries, in which case bl can be taken to be the dimension of either HI(M) of HI(M, aM). It was proved in [14] that if M has no boundary, then bk can be estimated from above by the
exactly-once
Exactly-once semantics in a replicated messaging systemAbstractA distributed message delivery system can use repli-cation to improve performance and availability.How-ever,without safeguards,replicated messages may be delivered to a mobile device more than once,making the device’s user repeat actions(e.g.,making unnec-essary phone calls,firing weapons repeatedly).In this paper we address the problem of exactly-once delivery to mobile clients when messages are replicated.We define exactly-once semantics and propose algorithms to guarantee it.We also propose and define a relaxed version of exactly-once semantics which is appropri-ate for limited capability mobile devices.We study the relative performance of our algorithms compared to weaker at-least-once semantics,andfind that the performance overhead of exactly-once can be mini-mized in most cases by careful design of the system. 1IntroductionIn a replicated messaging system,messages are repli-cated at various servers,pending delivery to mobile clients.When the client makes a connection to the network,it contacts any of the servers,and downloads its pending messages.If messages are not replicated, the client could not get its messages when its only server was down.Furthermore,the connection to its server may be slow,making it impossible to down-load all the messages in a reasonable time.Both of these problem are solved with replication:the client has a choice of servers,and can contact one that is “nearby”at the time of connection and that offers good response time.Message replication is useful in critical applications where it is important to deliver messages as soon as a client makes any connection.For instance,in a mili-tary scenario,it is critical to get messages to units in thefield,no matter at what point they connect,and no matter how short their connection is.Even for non-critical applications,replication can be very use-ful.For example,when an American traveler visits Europe,it would be much more convenient to access his or her emails through a local server,rather than relying on a slow trans-Atlantic connection.Replication,of course,introduces the problem of duplicate delivery.A client can connect to one server and download message M1,and later connect to a different server and receive M1a second time.Dupli-cate delivery can range from“disastrous”to simply “annoying.”For example,a message“fire missile”or ”buy stock”may cause an action to be unintention-ally performed twice with clearly undesirable conse-quences.In other cases,a duplicate message to“call Fred back”can be simply confusing,since the recip-ient does not know whether Fred wants to talk to him/her a second time.In this paper we study mechanisms for ensuring that messages are delivered exactly once(no delivery and duplicate delivery must be ruled out).While this problem has been extensively studied in the past(see our related work section),we believe it is important to revisit the problem in the context of a replicated message system for mobile clients.In particular,we will•Study how servers with replicas can coordinate to eliminate or reduce the likelihood of duplicate delivery.•Explore duplicate elimination mechanisms that may be appropriate for clients with no or limited stable storage.•Study“weaker”exactly-once semantics that may be useful in a mobile environment.•Present an evaluation metric for comparing exactly-once mechanisms,useful for highlighting their weaknesses or strengths in a mobile envi-ronment.We stress that we are not discovering“new”mech-anisms for exactly-once delivery.All of the ideas we present(e.g.,sequence numbers to identify dupli-cates)are well known.Our goal is instead to adapt these well known“building blocks”into mechanisms that are well suited for a mobile environment,and to evaluate the tradeoffs.We also stress that not all applications require strict exactly-once semantics.However,if we canfind a mechanism that guarantees exactly-once delivery at a reasonable performance and complexity cost,thenwe may still want to use it.Thus,we believe that regardless of the application,it is important to un-derstand how exactly-once delivery can be achieved with replicated messages,so that an informed deci-sion can be made regarding message delivery options. Finally,note that some applications may not re-quire exactly-once delivery because they implement their own safeguards.For example,the message to fire a missile may specify which missile tofire,so that a duplicate message will not cause a secondfir-ing.However,these applications mechanisms(e.g., missile numbers,application sequence numbers)are identical to the mechanisms we describe here.Thus, whether exactly-once is guaranteed by the messaging system or by the application,the issues are the same, and the results we present here are relevant.The structure of our paper is as follows.We start in Section2by defining more precisely our framework and the notion of exactly-once delivery.The algo-rithms for guaranteeing exactly-once semantics are given in Section3.Section4describes our perfor-mance model,while Section5presents selected com-parisons of the strategies.We discuss related work in Section6and conclude in Section7.2Delivery semanticsFigure1illustrates a replicated messaging system. Servers are located at various locations around the globe.The servers are connected by afixed wide area network,e.g.,the Internet.These servers are respon-sible for storing and delivering messages to clients. We call these messages notes to distinguish them from other network traffic that does not require delivery guarantees(e.g.,exactly-once semantics).When a new note is generated,it is replicated to all servers for delivery to the client.The replication happens under the guidance of a separate replication proto-col,which we do not consider in detail here.A client,which is typically a wireless information access device like a PDA,stays disconnected from the server network most of the time.It occasionally es-tablishes a wireless connection into a network access point,which then lets it communicate with one of the servers to get its notes.Because client connections can be short lived,or may not provide good connec-tivity to all servers,it is important to replicate notes at multiple servers.This way,when a connection is made,note delivery can be made from the server that offers best service.However,without any safeguards, a note can be delivered more than once to a client by different servers.The semantics of a note specify the requireddeliv-Figure1:Replicated message delivery system. ery properties,e.g.,whether duplicates are allowed, or whether the note needs to be delivered by a dead-line.In this paper,we focus on the following seman-tics:•Exactly-once:a note should eventually be deliv-ered to its target once,and only once.As mentioned earlier,notes that lead to irrevoca-ble changes in the physical world often need exactly-once guarantees.We focus on notes destined for a mobile,wireless device,for instance,asking someone on the shippingfloor to send some goods,instructing a soldier in thefield to initiate some action,asking a traveling manager to change the asking price for an acquisition,or telling a nurse in the emergency department to administer some drugs.The mobile device may have limited or no storage capacity to re-member what notes have been received in the past, making it more challenging to guarantee exactly-once delivery.Also,as we will see,some mechanisms for duplicate elimination work better than others when the client is mobile and is only connected for short periods.There are other possible delivery semantics,some of which are limited notions of exactly-once.For ex-ample,with at-least-once semantics,a note should be delivered at least once to the target,and dupli-cate delivery is not a concern.(For instance,a note stating that“the building is onfire”may require at-least-once semantics.)As another example,limited lifetime semantics may indicate that duplicate deliv-eries are not allowed up to a maximum note lifetime(expiration time),but that beyond that there is no guarantee(presumably because the note is of no sig-nificance after the expiration time).Although we do not study these limited semantics here,we believe that solutions for those semantics can be obtained by“relaxing”the exactly-once solutions.Thus,the exactly-once algorithms we will present here can form the basis for many other algorithms.3Exactly-once algorithmsIn this section we describe algorithms that guaran-tee exactly-once delivery semantics.Additional im-plementation details(pseudo-code)for selected algo-rithms are given in Appendix A.The algorithms are divided into two major groups:those(Section3.2) that guarantee strict exactly-once semantics,and those that guarantee a“relaxed”version which will be defined in Section3.3.The former group assumes that the clients have some stable storage,while the latter group does not have such a restriction.Our description of the algorithms assumes N servers and one client for simplicity.The algorithms can easily be extended to service multiple clients. 3.1Delivery proceduresAlgorithms that guarantee exactly-once delivery use a procedure Deliver(T,m)to send a message m to target node T(server or client).Deliver(T,m)is guaranteed to return by a certain predefined time. If the call is successful,then T executed a match-ing Receive call and successfully received m,and the sender received an acknowledgment confirming receipt of m.(Note that T could have received m more than once.)If Deliver(T,m)fails,the sender was unable to receive the acknowledgment from T before timing out.Note that it is possible for a call to Deliver(T,m)to timeout,while the match-ing Receive(m)completes successfully,because the acknowledgment was not properly received by the sender.3.2Strict exactly-once algorithms The strict exactly-once algorithms must cope with the different failures modes of Deliver(T,m).The key in all algorithms is to assign some form of iden-tifier(id)to each note,so that duplicate or missing notes can be handled.The algorithms vary in how they identify notes,how note ids can be recycled, and how long used note ids have to be remembered by servers and clients.These“small details”have significant impact on the potential performance and the usability of the algorithms.Strategy1(Sequenced Streams)Client remem-bers latest sequence number from each server.Se-quence numbers are taken from an infinite domain. Ourfirst algorithm resembles the well-known slid-ing window protocol[5]with unbounded sequence numbers.However,our algorithm is a“multiple-stream”version of that protocol with a sliding win-dow width of one.When a new note m is generated,it isfirst stored in any one of the servers,for example,server B.Each server keeps a sequence number counter,which is in-cremented for each new note generated at that server. The globally unique id of note m is the unique node id of server B prepended to m’s sequence number at B,e.g.,B.5.After this unique id is assigned,m will be replicated to all the other servers,where it will have the same id.In this algorithm,servers deliver(using procedure Deliver)all notes with the same server id sequen-tially.In other words,the notes are divided into N separate“streams,”by server id.To enforce sequen-tiality,the client c keeps an array R of sequence num-bers,one for each server.Array R is kept in stable storage so it survives client failures.The client will only accept note B.n if R[B]+1=n.If the note is accepted,R[B]is incremented;otherwise note B.n is discarded.Notice that a note is not considered“deliv-ered”(as far as exactly-once semantics is concerned) until it is accepted and processed by the client.Also note that the acceptance of a note and the corre-sponding counter increment should be an atomic op-eration at the client.Each server A maintains a list K of to-be-delivered notes,sorted by increasing sequence number for each “stream.”A note m can be removed from K after it is successfully delivered to the client.Furthermore, the servers periodically synchronize with each other to purge notes that have been delivered.(See Ap-pendix A for more details.)Example1The system consists of two servers,A and B,plus a client c.Note m,with id A.3,is waiting to be delivered on both A and B.In other words, K A=K B={A.3}.Client cfirst connects to server A and gets m.At this point R[A]=3.If A’s call to Deliver(c,m)returns success,A will remove A.3 from K A.A may also synchronize with B to inform it of m’s delivery.This synchronization can be done immediately after m’s delivery,or periodically. Later when c connects to either A or B,the server may attempt to send note m again,perhaps becauseA’s call to Deliver failed,or because server synchro-nization has not been performed yet between A andB.But c will not process m again because m’s se-quence number(3)is not what is currently expected(R[A]+1=3+1=4).To avoid problems,we cannot reuse id x.s untilall servers have removed x.s.Furthermore,we mustalso ensure that the following id,x.(s+1)is not inuse either!To see why,suppose in our example thatA.5is reused when B has removed A.5but not somemessage m3with id A.6.When the client receivesthe new m2with A.5there is no problem,but R[A]isadvanced to6.This makes it possible for c to receivethe old m3again,instead of a new A.6that followsm2.Thus,the following condition is the one thatguarantees the correct reuse of sequence numbers inour replicated delivery system.Condition1An id x.s can only be reused(i.e.,anew note generated on server x with s as its sequencenumber)T e time after server x has received confir-mation from all other servers that note x.(s+1)hasbeen removed from their respective K lists.To implement this condition,when server x syn-chronizes with server y,y will reply with the currentstate of its K list.If the reply says that y has alreadyremoved note x.(s+1)from K,x has effectively re-ceived a promise from y that it will not attempt todeliver note x.(s+1)in the future.It is easy to see how this synchronization avoids theproblem illustrated in Example2.If server A does notknow that m1with id A.5has been removed from B,then it will not be able to reuse number5.Only afterA knows that m1does not exist at any server,will itstart counting the required T e seconds to make surem1is also gone from the network.Although Strategy2guarantees exactly-once se-mantics,it still has the following potential shortcom-ings:1.The client needs a minimum amount of stablestorage(N times the size of a sequence numbercounter)to store its R array.The strategy willnot work if,for example,the client can only re-member one sequence number in its limited sta-ble storage space.2.It requires that notes with the same server id bedelivered in strict sequential order.The problemcan be alleviated if the client is willing to re-member k sequence numbers per server id.Thisallows up to k consecutive notes in a stream to be delivered out-of-order.Unfortunately,the so-lution further increases the storage requirement on the client k-fold.3.Since the size of R is proportional to the totalnumber of servers(N),the strategy does not scale well if there are potentially many servers in the system.As an example,assume a16-bit integer is used for server ids,but not all possible ids are in use at the same time.The client either needs to remember216sequence numbers,or it must be informed whenever servers join or leave the system,which significantly complicates the dynamic server reconfiguration protocols.4.The strategy also does not scale well with manyclients.Because not every note is destined for all the clients,a note must have one sequence number per client in order to keep the sequence numbers consecutive.To cope,a server must now have as many sequence number counters as there are clients,and the protocol is significantly more expensive as a result.The following strategy is designed to address the above problems,although it has its own drawbacks, as we will see in the performance section.Strategy3(Id List)Client remembers individual note ids it has received.For this strategy we simply assume that notes have globally unique ids,but the ids need not form se-quenced streams.For example,each note may be assigned a server/date/time id.The client keeps in stable storage an explicit id list R of individual notes it has received.For each new note received,the client first checks to see if its note id appears in R.If so,the client simply ignores the note;otherwise,it processes the note and inserts the note id into R.As before, processing the note and updating the data structure that records the processing must be an atomic oper-ation.If the client had infinite storage space for R,such a strategy would trivially guard against duplicate de-livery,since the client would be able to remember the ids of all the notes it has ever received.If the client has only afixed number of slots,q,to store R,we need to purge entries from R,while still guarantee-ing exactly-once semantics.The client can purge a note id m only after it is certain that no server will ever try to deliver m in the future,and only when it is sure that m is not lingering in the network itself.For the latter,we assume a maximum network message lifetime,and use it as in Strategy2(Sequenced Wrap-around Streams).Here we only focus on ensuring that m is not re-delivered by a server.A note m can be in one of three possible delivery states on any server:1.“Undelivered”:server thinks that m has notbeen delivered to its target yet;2.“Delivered”:server knows that m has been de-livered,but it cannot yet forget about m com-pletely because other servers might not know about this delivery yet;3.“Purged”:server knows that all other servers areaware that m has been delivered.To keep track,each server keeps the following two data structures:•K:list of notes that have not been purged,in-cluding Undelivered and Delivered notes;•D:the subset of notes in K that are“Delivered.”When a note m is created,it is entered into list K on every server.When a server B delivers m(success-ful completion of Deliver(c,m)),m is put into list D on that server.Periodically or immediately upon m’s delivery,server B will attempt to synchronize its state with the other servers,by exchanging the contents of their K and D lists.During the synchro-nization,other servers will realize that m has been delivered,and thus will put m into their respective D as well.Moreover,once B confirms that m has been put into every server’s D,it can authorize the client to purge m from its R list.The servers can also purge m by removing m from both K and D at the same time.Example3We have two servers A and B plus one client c,as before.Suppose that two notes,m1and m2,are waiting to be delivered.Client cfirst connects to server A and gets note m1.At this point,the data structures have the following values:K A=K B={m1,m2}D A={m1}D B=∅R c={m1}At this point(or periodically),A tries to synchro-nize with B.However,let us assume that c initiates a connection to server B before A could successfully in-form B about m1’s delivery.In this case,B attemptsto deliver note m1again,but c will simply ignore itbecause m1is already in R c.Assume that B proceeds to deliver m2.The datastructures are now as follow:K A=K B={m1,m2}D A={m1}D B={m1,m2}R c={m1,m2}When B synchronizes with A,A will also add m2to D A.Moreover,after B gets A’s state information,B will realize that D A=D B={m1,m2}.Henceit is safe to purge both m1and m2.In particular,Bwill authorize c to remove these two note ids from R c.They will also be purged from K B and D B,and fromK A and D A as well during the next synchronization.1Incidentally,notice that even though the device on whichthe client application runs does not rely on stable storage,exactly-once delivery is still desired.Even though the deviceitself will not keep a record of an accepted note,the note willaffect the real-world where the device runs,e.g.,firing a mis-sile.Accepting the same note again is unacceptable,even if thedevice does not know that the missile has beenfired already.Functionρ(A,m)represents the number of times that node A has received and processed the same message m2.Assume that before the sender S calls the pro-cedure aDeliver(T,m),we haveρ(T,m)=0.Then an atomic delivery procedure guarantees the following:when aDeliver(T,m)returns success,ρ(T,m)=1;and when aDeliver(T,m)returns er-ror,ρ(T,m)=0.Moreover,in the former case,ρ(T,m)remains1unless aDeliver(T,m)is called again,in which case the outcome will be undefined. In other words,it is up to our exactly-once algorithms to guarantee that aDeliver is never called again if thefirst time it was successful.In the case where aDeliver(T,m)returns error,ρ(T,m)will remain0 until the next time aDeliver(T,m)is called.We fur-ther assume that the return value of aDeliver(T,m) is always logged in permanent storage.Consequently, even if the server crashes right at the moment when the function returns,it is still able tofigure out later the outcome of the call from the log.In summary,a server-centric algorithm has to per-form“risky”operations when it delivers a note to a client.The aDeliver procedure encapsulates thisrisky operation.By saying that aDeliver exists,we are conceptually saying that there is no risk,simply so we can focus on other potentially dangerous op-erations,ones that can be avoided with a good al-gorithm.In other words,our algorithms should not perform any more“risky”operations than absolutely necessary.We will say that an algorithm guaran-tees relaxed exactly-once semantics when it guaran-tees exactly-once delivery under the assumption that aDeliver exists.When we compare an algorithm that guarantees strict exactly-once semantics to one that offers re-laxed semantics,we must keep in mind that the latter has some probability of failure.(This probability can be reduced by increasing the timeout period a server waits before declaring a delivery failure.)Of course, the relaxed algorithm has an important advantage in that it does not require stable storage.3.3.2AlgorithmsThe strategies presented in this section differ in the amount of participation required of the client.Notes can have any type of globally unique identifier.Strategy4(Server Synchronization)Client has no state.In this strategy the client is not required to re-member anything when it is turned off.For a client Strategy5(Delay Reconnect)Client does not reconnect within time T r.In the two algorithms that follow,we use time to prevent duplicate deliveries.We assume a clock syn-chronization protocol,such as NTP[13].We also as-sume a bounded maximum clock drift,which,if not zero,can be added to the appropriate parameters in our algorithms.Suppose that the client promises to refrain from connecting to any server until time T r has passed since its last connection to any server3(How the selec-tion of parameter T r affects the overall performance of the strategy will be investigated in Section5).The reconnect delay can be achieved by the client remem-bering the timestamp of its last connection,or the client can use a timer that starts ticking at the end of a connection.Alternatively the human user of the client device may be trusted to discipline him or her-self.When T r has passed,and the client connects again, the new server checks to make sure that,for each of the other servers s,the most recent synchronization message received from s is within T r time.If not, the server will attempt to synchronize with the prob-lematic server(s)while letting the client wait.If all servers have synchronized within T r time,then it issafe to deliver notes to the client,because the notes delivered to the client during its most recent connec-tions are known to the current server.To be more precise,each server maintains an array T S[s],containing the timestamp of the most recently received synchronization message from each server s. The servers periodically send these synchronization messages to each other,which include the originatingserver’s list D,as in Strategy4(Server Synchroniza-tion).Example5Client c connects to server A and re-ceives note m1.Now D A includes m1.After time T r has passed,c makes another connection with serverB.Thefirst thing B does is to check when was the last synchronization message it received from A.•Case(i):if the synchronization was sent within T r time,it must have been generated at serverA when D A contained m14.BecauseB incor-porates D A into its own list D,m1must be inD when B starts delivery.•Case(ii):if the A synchronization was more than T r seconds old,B is not allowed to start delivery until it has contacted A and received an up-to-date D A,which should contain m1.Thus,because client connections must be separatedby T r time units,the union of all servers’synchroniza-tion messages less than T r old will include all notes the client has received.4Or,if D A didn’t include m1,that means m1must have been purged from A,which in turn implies that m1’s delivery was known to all servers.Each server maintains a table T S of the most re-cent synchronization timestamps from other servers just as in Strategy5(Delay Reconnect).However, the client is no longer subject to the T r reconnec-tion time limit.Moreover,the servers are no longer required to synchronize with each other periodically, but only when there is new information.The client now needs to remember the last time it connected to each server.This is recorded in a history array H,where H(A)is the time when the last connection to server A ended.When the client makes a new connection,itfirst uploads to the server its H array.The server must ensure that it has received synchronization messages from each other server that are fresher than the client’s last connect time. Example6Note m1is replicated at servers A and B.Server A delivers m1to c at time t1.At a later time t2,c connects to B,giving it the value H(A)= t1.Server B makes sure it has synchronized with A after time t1,thus ensuring that B knows about m1’s delivery.4EvaluationThe relative merits and shortcomings of our algo-rithms can be evaluated along several dimensions:•Semantic guarantees.In particular,server-centric strategies provide a less strict delivery guarantee than the client-centric ones.•Reliability.For instance,how likely is it thata client connection may be broken by networkproblems?•Availability.For example,what is the likelihood that the client will be able to connect to a server and get notes when it wants to.•Performance.For example,what is the overall network traffic incurred,or how fast can notes be delivered to the client?We have studied several metrics that quantify the reliability,availability or performance of the algo-rithms.Due to space limitations,in this paper we only present one metric,which we have found to be especially useful.Our metric is the expected through-put,defined as the expected number of notes that a client can successfully receive during a connection that lasts T c seconds.We have chosen this metric because it captures a number of factors.The metric captures the syn-chronization overhead of the algorithms,because al-gorithms that require more synchronization before or while notes are delivered will have less over-all throughput.The metric also captures the ben-efits of replicating notes at servers:As we add more servers,clients willfind closer or better con-nected servers for download,achieving better deliv-ery throughput.The expected throughput is also a good indirect measure of system reliability,because a higher throughput means that the client can get the same number of notes faster,and is therefore less prone to disconnects.Moreover,our metric focuses on the connection-time performance as observed by the client,which we feel is the most critical compo-nent of the overall system performance in a wireless communication scenario.Having selected a metric,the biggest challenge is finding a good system model.The model should be rich enough to capture the main tradeoffs we want to study,while not being so detailed that we are over-whelmed with parameters and components.Our goal is not to predict exact performance of one algorithm versus another in a particular scenario.Rather,our goal is to clearly demonstrate and quantify the im-portant features of the various strategies,so we strive for the simplest model that allows us to doso.Figure2:System model.The global network is represented by a circle,as shown in Figure2.Each node(e.g.,a server or a mobile access point)of the network resides on the circle.The network distance between any two nodes, measured in network hops,is proportional to their distance on the circumference of the circle.For in-stance,if the circumference is30network hops,then the distance between two nodes directly across from each other is15hops.In Figure2,S i is the i’th server. The mobile client is represented by c,and the Net-work Access Point(NAP)is its entry into the global network.A wireless line(dotted line)connects c and NAP.We choose a circular representation of the network because it allows us to easily model the fundamen-tal characteristics of a server network.For example, the effect of placing more servers around the globe to increase access locality is reflected by the reduced distance to the nearest server from a random point on the circle.(Of course,a sphere may have been a more accurate model for the Earth’s networks,but we do not believe we can gain more insights from such a substantially more complex model.)We assume that the servers are uniformly dis-tributed on the circle.Consequently,the more servers there are,the shorter the distance between neighbor-ing servers.For example,with3servers and30total hops,neighboring servers are separated by3010=3hops.When a client c wants to get its notes,itfirst con-nects to a random access point(NAP)on the circle. In real life,this access point could be an ISP for mo-bile units,or a LAN segment which takes in the trav-eling client as a guest.From this access point,the clientfigures out where the nearest server is on the。
Parton Distributions in the Virtual Photon Target and Factorization Scheme Dependence
a r X i v :h e p -p h /0006025v 1 2 J u n 20001Parton Distributions in the Virtual Photon Target and Factorization Scheme Dependence ∗Ken Sasaki a and Tsuneo Uematsu baDepartment of Physics,Faculty of Engineering,Yokohama National University Yokohama 240-8501,JapanbDepartment of Fundamental Sciences,FIHS,Kyoto University,Kyoto 606-8501,JapanWe investigate parton distributions in the virtual photon target,both polarized and unpolarized,up to the next-leading order (NLO)in QCD.Parton distributions can be predicted completely up to NLO,but they are factorization-scheme-dependent.We analyze parton distributions in several factorization schemes and discuss their scheme dependence.Particular attentions are paid to the axial anomaly effect on the first moments of the polarized quark parton distributions,and also to the large-x behaviors of polarized and unpolarized parton distributions.YNU-HEPTh-00-101KUCP-156May 20001.INTRODUCTIONIn e +e −collision experiments,we can measure the spin-independent and spin-dependent struc-ture functions,F γ2(x,Q 2,P 2)and g γ1(x,Q 2,P 2),of the virtual photon(Fig.1).e Figure 1.Deep inelastic scattering on a virtual photon in e +e −collision.The advantage in studying the virtual photon target is that,in the case Λ2≪P 2≪Q 2(1)2Let∆qγS (∆qγNS),∆Gγ,∆Γγbe theflavor singlet(non-singlet)-quark,gluon,and photon distribu-tion functions,respectively,in the longitudinally polarized virtual photon with mass−P2.In the leading order of the electromagnetic coupling con-stant,α=e2/4π,∆Γγdoes not evolve with Q2 and is set to be∆Γγ=δ(1−x).In terms of the Mellin moments of these pdf’s,the moment of the polarized virtual photon structure function gγ1(x,Q2,P2)is expressed in the QCD improved parton model asgγ1(n,Q2,P2)=∆Cγ(n,Q2)·∆qγ(n,Q2,P2)(2) where∆Cγ(n,Q2)=(∆CγS,∆CγG,∆CγNS,∆Cγγ)∆qγ(n,Q2,P2)=(∆qγS ,∆Gγ,∆qγNS,∆Γγ)and∆CγS(∆CγNS),∆CγG,and∆Cγγare the mo-ments of the coefficient functions corresponding to singlet(non-singlet)-quark,gluon,and photon, respectively,and they are independent of P2. The pdf’s∆qγsatisfy inhomogeneous evolu-tion equations.The explicit expressions of∆qγS ,∆Gγ,and∆qγNS up to the NLO are derived from Eq.(4.46)of Ref.[2].They are given in terms of one-(two-)loop hadronic anomalous dimen-sions∆γ(0),nij (∆γ(1),nij)(i,j=ψ,G)and∆γ(0),nNS(∆γ(1),nNS),one-(two-)loop anomalous dimensions∆K(0),ni (∆K(1),ni)(i=ψ,G,NS)which rep-resent the mixing between photon and three hadronic operators R n i(i=ψ,G,NS),andfi-nally∆A n i,the one-loop photon matrix elements of these hadronic operators renormalized atµ2= P2(=−p2),γ(p)|R n i(µ)|γ(p) |µ2=P2=αMS schemeto a new factorization scheme-a,is given by∆Cγ,nS,a=∆Cγ,nS,2π∆w a(n)∆Cγ,nG,a=∆Cγ,nG,2π∆z a(n)∆Cγ,nNS,a=∆Cγ,nNS,2π∆w a(n)(4)∆Cγ,nγ,a=∆Cγ,nγ,π3 e4 ∆ˆz a(n)where e2 = i e2i/N f, e4 = i e4i/N f,withN f being the number offlavors of active quarksand e i being the electric charge of i-flavor-quark.Once the relations(4)between the coefficientfunctions in the a-scheme andMS scheme to a-scheme for therelevant two-loop anomalous dimensions and alsofor the one-loop photon matrix elements,∆A nψand∆A n NS,of the quark operators.Note that,in one-loop order,the photon matrix elementsof gluonic operators R n G vanish in any scheme,∆A n G=0.We consider three different factorizationschemes both in the polarized and unpolarizedcases.3.1.The polarized case(i)[TheMS scheme,the QCD(QED)axial anomaly resides in the quark distri-butions and not in the gluon(photon)coefficientfunction[7,8].In fact we observe∆γ(1),n=1ψψ,MS=∆B n=1γ,3values,i.e.,∆A n=1ψ,e 4 − e 2 2∆A n=1NS,MS scheme to theCI scheme is achieved by ∆w CI (n )=0,∆z CI (n )=∆ˆz CI (n )=2N f1MS to the OS scheme is made by choosing ∆w OS (n )=C F S 1(n ) 2+3S 2(n )−S 1(n ) 1(n +1)−7n −3n 2+2n (n +1)S 1(n )+1n 2−4MS scheme;(ii)The off-shell (OS)scheme;and (iii)The DIS γscheme.The transformation fromn (n +1)(n +2)S 1(n )+1n 2+4(n +2)2.(10)The DIS γwas introduced [13]some time agofor the analysis of the unpolarized real photonstructure function F γ2(x,Q 2)in NLO.In thisscheme the direct-photon contribution to F γ2is absorbed into the photonic quark distributions,so that we takew DIS γ(n )=z DIS γ(n )=0ˆz DIS γ(n )=N fMS(11)=N f−n 2+n +2n +1n +1−64find ∆A n =1ψ,a =∆A n =1NS,a =0and this leads to ∆q γS (n =1)|a =∆q γNS (n =1)|a =0(13)up to NLO.In these schemes,the axial anomalyeffects are transfered to the gluon and photon co-efficient functions.On the other hand,in theMS= −αβ0αs (P 2)−αs (Q 2)π3 e 2N fis related to the QEDaxial anomaly and the term 2πN f is coming from the QCD axial anomaly [14].For gluon distribution,we obtain in NLO∆G γ(n =1)=12ααs (Q 2),(15)the same result for4π4π4−14π4π2−ln xMS scheme,wefind near x =1,∆q γS (x )|NLO ,4πN f e 2 6[−ln(1−x )],∆G γ(x )|NLO ,4πN f e 2 3[−ln x ].(17)It is remarkable that in theMS and ∆q γNS (x )|NLO ,MSfor large x .On the other hand,the OS scheme gives quite different behaviors near x =1for the quark pdf’s.We find that,in x space,∆q γS (x )|NLO ,OS does not diverge for x →1but approaches a constant value:∆q γS (x )|NLO ,OS →α8+3MS ,OS.In DIS γscheme,quark pdf’s becomes negative and divergent for x →1.In fact,we findq γS (x )|NLO ,DIS γ≈αMS,is absorbed into the quark pdf’s in the DIS γscheme.51–11LOOS0.5xQ 2 = 30 GeV 2P 2= 1 GeV 2N f = 3MS CI q (x , Q , P )/3N <e > − l n Q /P∆γs 2f 22απ22Figure 2.The polarized singlet quark distribution ∆q γS (x,Q 2,P 2)in LO (solid line)and beyond LO.The NLO results are from MS,CI and OS.The CI and OSlines cross the x -axis nearly at the same point,just below x =0.5,while theMS scheme.As x →1,we observethat theMS line is much differentfrom the LO one.We see that OS scheme predictsa better behavour for ∆q γS than other schemes in11Q 2 = 30 GeV2P 2 = 1 GeV 2N f = 30.5xLOMS DIS OSγq (x , Q , P )/3N <e > − l n Q /Pγs 2f 22απ22x Figure 3.The unpolarized singlet quark distribu-tion xq γS (x,Q 2,P 2)in LO (solid line)and beyond LO.The NLO results are fromMS,OS and DIS γ.As in the polar-ized case,the6to the LO line below x<0.7,but negatively di-verges as x→1.Again,in the unpolarized case, the OS scheme gives a better behavior for xqγS. Actually,the OS line is closer to the LO one and starts to drop to reach thefinite value for x→1.7.SUMMARYThe behaviors of the pdf’s inside the virtual photon target,polarized and unpolarized,can be predicted entirely up to NLO,but they are factorization-scheme-dependent.We have stud-ied the scheme dependence of the pdf’s in the virtual photon.In the case of polarized pdf’s, the scheme dependence is clearly seen in thefirst moments and the large-x behaviors of quark dis-tributions.In the umpolarized case,the scheme dependence is also observed in the large-x behav-iors of quark distributions.The NLO quark pdf’s predicted by the。
Twisted vertex representations via spin groups and the McKay correspondence
TWISTED VERTEX REPRESENTATIONS VIA SPIN GROUPS AND THE MCKAY CORRESPONDENCE
IGOR B. FRENKEL, NAIHUAN JING, AND WEIQIANG WANG Abstract. We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ and a virtual character of Γ we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products Γ ≀ Sn of Γ and a double cover of the symmetric group Sn for all n. When Γ is a subgroup of SL2 (C) with the McKay virtual character, our construction gives a group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras. When Γ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for Γ ≀ Sn .
Scattering and Inverse Scattering on ACH Manifolds
dt2 + h(t) t2 ∞ ¯ The function ρ := ψ∗ t is where h(t), t ∈ [0, ǫ) is a C 1-parameter family of C ∞ metrics on ∂ X. ¯ ¯ ¯ Note that a boundary defining function in X near ∂ X, which can be extended smoothly to X. n the boundary represents the geometric infinity of X , as does the sphere S for the hyperbolic space Hn+1 the Laplacian of (X, g ) was studied in [27]; it consists a finite pure point spectrum σpp (∆), which is the set of L2 (X ) eigenvalues, and an absolutely continuous spectrum σac (∆) satisfying The resolvent σac (∆) = n2 /4, ∞ and σpp (∆) ⊂ 0, n2 /4 .
R(λ) = (∆g − λ(n − λ))−1 ,
2000 Mathematics Subject Classification. Primary 58J50, Secondary 35P25. Key words and phrases. scattering, pseudoconvex domains, asymptotically complex hyperbolic manifolds. The first author was partially supported by French ANR grants no. [JC05-52556] and [JC05-46063], and an Australian National University postdoctoral fellowship. Both authors were supported in part by the National Science Foundation Grant DMS 0500788.
Dirichlet空间上的k阶斜Toeplitz算子
Dirichlet空间上的k阶斜Toeplitz算子【摘要】本文给出了dirichlet空间上的k阶斜toeplitz算子的定义，讨论了k阶斜toeplitz算子的交换性和谱等，证明了：若，则的充要条件是在中线性相关；若,则。
【关键词】dirichlet空间斜toeplitz算子交换性谱1引言及预备知识文中表示复平面上的单位开圆盘，表示的规范面积测度，即。
sobolev空间为上所有弱可导且满足的函数构成hilbert空间。
dirichlet空间是中满足的解析函数构成的闭子空间，其上的内积定义为。
dirichlet空间的再生核是。
sobolev空间定义为:。
是上的本性有界函数构成的空间。
表示上的解析函数空间，表示上的有界解析函数构成的空间，并记。
令为从到的正交投影，给定，定义d上符号为的toeplitz算子为，则是上的有界算子[1]。
对于，定义dirichlet空间上的算子为：基于的定义，我们有如下的：定义1.1:设，则称作符号为的斜toeplitz算子。
近年来，斜toeplitz算子备受人们的关注。
1996年，引进了hardy空间上记号为的斜topelitz算子的概念，介绍了这类算子的背景，讨论了该类算子的若干基本性质，如有界性、紧性、c*-代数、谱、本质谱等等[2]。
继而，他又展开了对斜topelitz算子更深入的讨论，介绍了具有连续符号的斜toeplitz算子的谱以及斜toeplitz算子的伴随算子[3-4]。
近年来，人们又研究bergman空间上的斜toeplitz算子，斜toeplitz算子的基本性质与符号的解析性质之间的内在联系，上斜toeplitz算子的紧性，并更进一步给出了上阶斜toeplitz算子的谱以及交换性等等[5-6]。
自然地，我们会想到在dirichlet空间上如何定义相应的概念以及dirichlet空间上的斜toeplitz算子有哪些基本性质，这些性质与符号的解析性质之间又有何内在联系。
Aether Compactification
a r X i v :0802.0521v 3 [h e p -p h ] 24 J u n 2008CALT-68-2670Aether CompactificationSean M.Carroll and Heywood Tam 11California Institute of Technology,Pasadena,CA 91125We propose a new way to hide large extra dimensions without invoking branes,based on Lorentz-violating tensor fields with expectation values along the extra directions.We investigate the case of a single vector “aether”field on a compact circle.In such a background,interactions of other fields with the aether can lead to modified dispersion relations,increasing the mass of the Kaluza-Klein excitations.The mass scale characterizing each Kaluza-Klein tower can be chosen independently for each species of scalar,fermion,or gauge boson.No small-scale deviations from the inverse square law for gravity are predicted,although light graviton modes may exist.I.INTRODUCTIONIf spacetime has extra dimensions in addition to the four we perceive,they are somehow hidden from us.For a long time,the only known way to achieve this goal was the classic Kaluza-Klein scenario:compactify the dimen-sions on a manifold of characteristic size ∼R .Momen-tum in the extra dimensions is then quantized in units of R −1,giving rise to a Kaluza-Klein tower of states;if R is sufficiently small,the extra dimensions only become evi-dent at very high energies.More recently,it has become popular to consider scenarios in which Standard Model fields are localized on a brane embedded in a larger bulk [1,2,3,4].In this picture,the extra dimensions are difficult to perceive because we can’t get there.In this paper we consider a new way to keep extra di-mensions hidden,or more generally to affect the propa-gation of fields along directions orthogonal to our macro-scopic dimensions:adding Lorentz-violating tensor fields (“aether”)with expectation values aligned along the ex-tra directions.Interactions with the aether modify the dispersion relations of other fields,leading (with appro-priate choice of parameters)to larger energies associated with extra-dimensional momentum.1This scenario has several novel features.Most impor-tantly,it allows for completely different spacings in the Kaluza-Klein towers of each species.If the couplings are chosen universally,the extra mass given to fermions will be twice that given to bosons.There will also be new degrees of freedom associated with fluctuations of the aether field itself;these are massless Goldstone bosons from the spontaneous breaking of Lorentz invariance,but can be very weakly coupled to ordinary matter.There is a sense in which the effect of the aether field is to distort the background metric,but in a way that is felt differ-−g−12Any configuration for which V ab =0everywhere will solve this equation.In particular,there is a background solution of the formu a =(0,0,0,0,v ),(5)so that the aether field points exclusively along the extra direction.We will consider this solution for most of this paper.Constraints on four-dimensional Lorentz violation via couplings to Standard Model fields have been extensively studied [7,8,9,10].The dynamicsofthe (typically time-like)aether fields themselves and their gravitational ef-fects have also been considered [6,11,12,13,14,15,16,19].More recently,attention has turned to the case of spacelike vector fields,especially in the early universe [20,21].The particular form of the Lagrangian (2)is chosen to ensure stability of the theory;for spacelike vector fields,a generic set of kinetic terms would generally give rise to negative-energy excitations.This specific choice prop-agates two positive-energy modes:one massless scalar,and one massless pseudoscalar [21].For purposes of this paper we will not investigate the fluctuations of u a in any detail.Although the modes are light,their couplings to Standard Model fields can be suppressed.Neverthe-less,we expect that traditional methods of constraining light scalars (such as limits from stellar cooling)will pro-vide interesting bounds on the parameter space of these models.III.ENERGY-MOMENTUM AND COMPACTIFICATIONA crucial property of aether fields is the dependence of their energy density on the spacetime geometry.The energy-momentum tensor takes the form T ab =V ac V b c −1b (x ).(8)It is straightforward to verify that this configuration sat-isfies the equation of motion (4),as well as the constraint (3),even though V ab does not vanish:V µ5=−V 5µ=v ∇µb.(9)We can then calculate the energy-momentum tensor associated with the aether:T (u )µν=v 22g µν∇σb ∇σbT (u )µ5=0T (u )55=v 2∇σ∇σb −12(∂φ)2−12µ2φu a u b∂a φ∂b φ,(11)with a corresponding equation of motion∂a ∂a φ−m 2φ=µ−2φ∂a (u a u b∂b φ).(12)Expanding the scalar in Fourier modes,φ∝e ik a x a=e ik µkµ+ik 5x 5,(13)yields a dispersion relation−k µk µ=m 2+1+α2φ k 25,(14)whereαφ=v/µφ.(15)3 Note that with our metric signature,−kµkµ=ω2− k2.This simple calculation illustrates the effect of the cou-pling to the spacelike vectorfipactifying thefifth dimension on a circle of radius R quantizes themomentum in that direction,k5=n/R.In standardKaluza-Klein theory,this gives rise to a tower of statesof masses m2KK=m2+(n/R)2.With the addition of the aetherfield,the mass spacing between different states in the KK tower is enhanced,m2AC=m2+(1+α2φ) n 4F ab F ab−1µ2ψu a u b¯ψγa∂bψ,(25)4 leading to an equation of motioniγa∂aψ−mψ−iµ2ψ(u a k a)2−1µu a¯ψ∂aψ.Followingthe same procedure as before,thefirst term leads to the dispersion relation−kµkµ=m2+v2+k25+2vk5=m2+(v+k5)2.(30) As usual,coupling to u a enhances the mass spacing of the KK tower,but now the spacing will depend on the direction of the5th-dimensional momentum as well as its magnitude.Meanwhile,the second term leads to the dispersion −kµkµ=m2−2mαk5+(1+α2)k25(31)=(m−αk5)2+k25,(32) whereα=v/µ.Interestingly,if(1+α2)/α<2mR, this coupling results in a reduction in m2for small n. However,it can be checked that these negative mass cor-rections are never sufficiently large to lead to tachyons. For n large,the mass spacing is enhanced,as usual.VII.GRA VITYThe aetherfield can couple nonminimally to gravity through an actionS=M∗ d5x√2R+αg u a u b R ab ,(33)where M P is the4-dimensional Planck scale andαg is di-mensionless.The gravitational equation of motion takes the formG ab=αg2Rg ab andW ab=R cd u c u d g ab+∇c∇a(u b u c)+∇c∇b(u a u c)−∇c∇d(u c u d)g ab−∇c∇c(u a u b).(35)Now we consider smallfluctuations of the metric,g ab=ηab+h ab.(36)The choice of backgroundfield u a=(0,0,0,0,v)spon-taneously breaks diffeomorphism invariance,so not allcoordinate transformations are open to us if we wantto preserve that form.Under an infinitesimal coor-dinate transformation parameterized by a vectorfield,x a→¯x a=x a+ξa,the metricfluctuation and aetherchange by h ab→h ab+∂aξb+∂bξa and u a→u a+∂5ξa.Therefore,we should limit our attention to gauge trans-formations satisfying∂5ξa=0.We can,for example,set hµ5=0.We then still have residual gauge freedomin the form ofξµ,as long as∂5ξµ=0.This amountsto the usual4-d gauge freedom for the massless four-dimensional graviton.Taking advantage of this gauge freedom,we can partlydecompose the metric perturbation ashµν=¯hµν+Ψηµν,h55=Φ,(37)whereηµν¯hµν=0.In this decomposition,¯hµνrepresentspropagating gravitational waves,Ψrepresents Newtoniangravitationalfields,andΦis the radionfield represent-ing the breathing mode of the extra dimension.The zeromode of thisfield is a massless scalar coupled to matterwith gravitational strength;in a phenomenologically vi-able model,it would have to be stabilized,presumbablyby bulk matterfields.The Einstein tensor becomesGµν=12 ∂5∂λ¯hλµ−3∂µ∂5Ψ ,(39)G55=1Φ=0=Ψand consider transverse waves,∂λ¯hλµ=0. The gravity equation(34)becomes∂25¯hµν.(44)−12M2PThis implies a dispersion relation−kµkµ= 1+αg v2[19]R.Bluhm,S.H.Fung and V. A.Kostelecky,arXiv:0712.4119[hep-th].[20]L.Ackerman,S.M.Carroll and M.B.Wise,Phys.Rev.D75,083502(2007)[arXiv:astro-ph/0701357].[21]T.R.Dulaney,M.I.Gresham and M. B.Wise,arXiv:0801.2950[astro-ph].。