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Gbcast-History-Gbcast is somewhat unusual in that most solutions to the state machine replication problem are closely integrated with the application being replicated. Gbcast, in contrast, is designed as a multicast API and implemented by a library that delivers messages to group members. Lamport, Malkhi and Zhou note ... | milkshake721/2.1M-wiki-STEM |
Gbcast-History-Examples of systems that used Isis in production settings include the New York Stock Exchange, where it was employed for approximately a decade to manage a configurable, fault-tolerant and self-healing reporting infrastructure for the trading floor, to relay quotes and trade reports from the "back office... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-The fundamental problem solved by Gbcast is this: we are given an initial set of group members and wish to support a multicast abstraction, permitting members of the group to send messages that encode various commands or requests. The protocol must agree on the messages to deliver, and on their... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-The set of group members changes each time a member fails or joins, and Gbcast is also used to maintain group membership by means of special multicasts that are delivered to the application as "new view" events, but that also adjust the group membership list maintained by the Gbcast protocol li... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Network partitions can split a group into two or more disjoint subgroups, creating the risk of split brain behavior, in which some group members take a decision (perhaps, to launch the rocket) without knowing that some other partition of the group has taken a different, conflicting decision. Gb... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Should a failed member recover (or if a partitioning failure caused some member to be incorrectly sensed as faulty and hence dropped from the view), after communication is restored, that member can rejoin. An incarnation number is used to avoid ambiguity: a counter that will be incremented each... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Any dynamically reconfigurable system, including both Gbcast and Paxos, can enter states from which no further progress is possible. For example, this could happen if operational processes are wrongly removed from the configuration, and then too many real crashes occur within the remaining memb... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Processes Processes run on processors that operate at arbitrary speed.
Processes may experience crash (halting) failures.
A process is uniquely identified by a three-tuple: (processor-id, process-id, incarnation-number).
Processes with stable storage may re-join the protocol after failures (fol... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Messages are delivered without corruption.These are weak assumptions: a network that never delivers any messages would satisfy them (we would say that such a network is experiencing a complete and permanent partitioning failure). The network conditions required for Gbcast to guarantee progress ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-To simplify the presentation, we assume that a TCP-like acknowledgement / retransmission scheme is employed, creating the illusion of a reliable, sequenced, non-repeating message channel between each pair of processes. A timeout occurs if this channel abstraction retries repeatedly and is unabl... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Process Groups and Views Gbcast is defined with respect to a "process group:" a set of processes. In a deployed system such a group might have a name (like a file name), a way to initially contact the group, and other attributes such as flow-control parameters. However, those kinds of details a... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Multicast Messages Members of a view can request that multicast messages be sent to a process group without knowledge of the membership that will apply at the time of delivery.The Gbcast protocol carries out these operations with a series of guarantees, discussed below.Delivery is by upcall to ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Application An application corresponds to a program which can be launched on one or more processors. Each application process then joins one or more process groups.An application process belonging to a group initiates new multicasts by invoking Gbcast. The protocol is considered to have termina... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Leader The leader of a group is defined with respect to some view of the group, and is the member with lowest rank in the view. As noted, the rank is age-ordered (with older members having lower rank), and ties are broken using a lexicographic sort. | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Failure detection All components of the system are permitted to participate in the role of "detecting" failures. Detection is distinct from the reporting of the failure (which occurs through a new view and is ordered with respect to message deliveries).The channel abstraction supported by the n... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Failed process Any member of the current view that has been detected as failed is considered to be a failed process.
An operational process that learns that it is considered to have failed (by attempting to communicate with some other process that rejects the message, thereby "shunning" it) mig... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Quorums Quorums are used to guarantee the safety properties of Gbcast by ensuring that there is a single globally agreed-upon sequence of group views and multicast messages and by preventing progress in more than one partition if a group becomes fragmented into two or more partitions (disjoint ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Problem statement-Given view i with n members {A,B,C….}, a quorum of the view is any majority subset of the members of that view. Notice that this is in contrast to the way the term is defined in systems that have a static underlying membership: for Gbcast, the quorum size will change over time as the membership... | milkshake721/2.1M-wiki-STEM |
Gbcast-Safety and liveness properties-In order to guarantee safety, Gbcast defines three safety properties and ensures they hold, regardless of the pattern of failures: Non-triviality Only multicasts actually sent by some group member are delivered. If a process receives a message from a group member that it considers ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Safety and liveness properties-Conditional liveness If multicast M is sent in some view and the sender remains operational, then eventually all members of that view (with the exception of any that crash) will deliver M. Liveness cannot be guaranteed under all conditions, hence we impose a further condition: we r... | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-This protocol is the one used under normal conditions. | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-Recall that in Gbcast, each operational process has a current view, and each view defines a leader. Only a process that believes itself to be the leader in the current view can initiate a new multicast; other members must relay multicasts by sending them to the leader, over 1-to-1 connections, and t... | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-Should the leader fail while some member that is not the leader is attempting to relay a multicast, the sender must determine the status of its pending request. This is accomplished as follows: Notice that members observe the delivery of their own multicasts. Accordingly, if a new view becomes defin... | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-Prepare step The leader proposes some sequence of one or more multicast messages by using the 1-to-all reliable network layer to send the message(s) to the members of the most current view, identifying each by means of an integer sequence number. The sequence numbers reset to 1 as each new view is d... | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-Promise step Each recipient retains a copy of the message(s) and responds with a promise to deliver them (such a promise will be fulfilled so long as the recipient itself remains a member of the group view, but if the recipient fails, the promise might not be carried out). During recovery, a recipie... | milkshake721/2.1M-wiki-STEM |
Gbcast-Basic Gbcast-Commit step The leader collects promise messages until, for each member of the group, it either has a promise message or a timeout has occurred causing the leader to suspect the corresponding member as faulty (recall that in this latter case, the leader will shun the suspected member, and because th... | milkshake721/2.1M-wiki-STEM |
Gbcast-Delivery step-If a member receives a Commit message, it delivers the associated message(s) or new view(s) to the application, in the order that they were proposed by the leader. The leader learns that this step has succeeded when the acknowledgements used by the reliable 1-to-1 channel are received. | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, simplest case-(Quorum size = 2, view1={A,B,C}) Member Leader Members Application Layer A A B C A B C | | | | | | | | X-------->| | | | | | | Request that the leader send a multicast M | X--------->|->|->| | | | Propose(1.1: M) (View 1, sequence 1, message M) | |<---------X--X--X | | |... | milkshake721/2.1M-wiki-STEM |
Gbcast-Error cases in basic Gbcast-The simplest error cases are those in which one or more members fail, but a quorum remains active. In the example below, the group consists of {A,B,C} with A playing the leader role. C fails during the promise phase and a timeout occurs within the reliable channel from the leader to p... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, failure of member other than the Leader-(Quorum size = 2, view1={A,B,C}) Member Leader Members Application Layer A A B C A B C | | | | | | | | X-------->| | | | | | | Request(M) | X--------->|->|->| | | | Propose(1.1: M) | | | | * | | * !! C FAILS !! | |<---------X--X | | Promise(1.1... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, add members {D,E,F}, failure of member other than the Leader-In the example shown below, a group that initially contains members {A,B,C} is asked to add {D,E,F}, but member C fails during the protocol. Membership change requests are treated as a special kind of multicast and the seque... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, add members {D,E,F}, failure of member other than the Leader-(Quorum size = 2, view1={A,B,C}) Member Leader Members Application Layer A A B C D E F A B C D E F | | | | | | | | | | | X-------->| | | | | | | | | | Request("add D,E,F") | X--------->|->|->| | | | | | | Propose(1.1: "add ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-The next failure case is when a leader fails, resulting in a new leader. To take over as the leader, the new leader first runs a takeover protocol, and then the new leader can run basic Gbcast as above. The takeover protocol is as follows: Inquiry Step The new leader... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Promise-List Step Each recipient sends the current list of promised messages to the leader. If a recipient lacks its initial view, it sends a request for an initial view to the leader.The new leader waits until it has either received a promise-list from each of the m... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Repeat If Necessary The new leader examines the promise-list, looking for membership-change messages that add new members. If any are present, it iterates the inquiry phase and promise-list collection phase, sending inquiries to the new members. This in turn could le... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Check for Quorums At the end of the inquiry phase, the leader has received promise-list responses from some of the processes it contacted; any unresponsive members will now be suspected. The new leader constructs a list of proposed views. To advance to the next step ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Start as New Leader Having successfully checked for quorums, the new leader becomes the leader. It can now run the basic protocol. It re-proposes any promised messages or view-changes, in the order it learned them from the promise-lists, following them with a new vie... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Dueling Leaders It is possible that the promise-lists include two or more distinct proposals for the same slot. This happens (only) if a first leader A became partitioned from the system, but nonetheless made a proposal X that was seen only by a small (non quorum) se... | milkshake721/2.1M-wiki-STEM |
Gbcast-Takeover protocol, used when the leader fails-Failure Suspicions Piggyback on Outgoing Messages Notice that the new leader believes the old leader to have failed, and may also believe that other members have failed. Thus, the inquiry phase, and or the new propose phase, may also carry piggybacked failure message... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, failure of Leader, TakeOver, Basic Gbcast by the new leader-(Quorum size = 2, view 1={A,B,C}) Member Leader Members Application Layer A B A B C A B C | | | | | | | | X----->| | | | | | | Request(M) | X------------>|->| | | | | Propose(1.1: M) !! Leader fails during send, Propose does... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, Add members {D,E,F}, failure of the Leader-As an example of a more complex case, here the leader fails in the middle of a commit that increases the size of the view (Quorum size = 2, view 1={A,B,C}) Member Leader Members Application Layer A B A B C D E F A B C D E F | | | | | | | | |... | milkshake721/2.1M-wiki-STEM |
Gbcast-Message flow: Basic Gbcast, Add members {D,E,F}, failure of the Leader-In this example, the final commit actually causes two views to be delivered in succession at members B and C. Even though the two proposals were sent concurrently, the commit for view2 requires a promise from a quorum of view1, whereas the co... | milkshake721/2.1M-wiki-STEM |
Gbcast-Correctness-To show that Gbcast satisfies non-triviality we start by tracing backwards from an arbitrary delivery action to the point at which a client requested the corresponding event; clearly, only messages that were legitimately sent will be delivered. However, nontriviality for this protocol goes further: w... | milkshake721/2.1M-wiki-STEM |
Gbcast-Correctness-To establish consistency we begin by analysis of the case in which there is just a single leader that never fails or loses connectivity with a quorum. Since the leader sequences the events and includes each member starting with the first view that contains that member, all members deliver the identic... | milkshake721/2.1M-wiki-STEM |
Gbcast-Correctness-When a new leader takes over, the inquiry is required to reach a quorum of members for the most recent committed view. This quorum necessarily will include at least one process that received any proposal that the old leader could have committed. Thus the new leader will learn of any potentially commi... | milkshake721/2.1M-wiki-STEM |
Gbcast-Correctness-We can show that a joining member will receive its initial view by analysis of the two relevant cases. If the leader doesn't fail, it sends the initial view on an eventually reliable channel. If the leader does fail and some member lacks its initial view, the new leader sends that view after receipt ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Correctness-A logical partitioning of the group is impossible because of the shunning rule. In order to commit any new view, the old leader must obtain promises from a quorum of the current view. A new leader, taking over, will learn of any view that could have become committed. To commit its own proposed next v... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness-The Gbcast protocol will make progress provided that at all times in the execution, if view v holds at time t, then less than a quorum of members of v fail (or are suspected as failing) within some subset of the members of the view. To maximize progress, it is important that excluded but still live memb... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness-This property is similar to but "stronger" than <>W, the "weakest failure detector" for achieving consensus, as defined by Chandra and Toueg. To see this, consider a run in which a mutually suspecting quorum arises "too quickly" for processes that have been wrongly excluded from the view to rejoin it. G... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness-Arguably, such runs would be unlikely in the kinds of data centers where Gbcast is typically used, but clearly they can be constructed in an adversarial manner. | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Failure sensing-The Gbcast protocol presumes that the probability of incorrect failure suspicions will be low; the scheme breaks down if failure suspicions occur frequently and operational processes are often suspected as faulty. By analogy, consider the TCP protocol, in which the failure to receive ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Failure sensing-In contrast, there are other failure detection schemes, such as the one explored by Chandra and Toueg, that can yield high rates of incorrect failure suspicions. Some protocols, including Paxos, are able to tolerate incorrect failure suspicions without any costly consequence. Whether ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Failure sensing-One extreme scenario is worthy of further mention: network partitioning events. Modern data centers and networks often experience events in which a single machine and all the processes on it becomes transiently partitioned from a larger pool of machines that remain connected to one an... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Failure sensing-A more extreme kind of partitioning is sometimes seen in data centers: in this situation, a network switch might fail, causing a collection of machines (perhaps, a whole rack or even an entire container) to become disconnected from the Internet and from the remainder of the data cente... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Failure sensing-The Transis system explored extensions to the Gbcast protocol that permit multiple partitions to form, to make independent progress, and then to remerge. This topic, however, is beyond the scope of the present discussion. | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Dueling leaders-In the Paxos protocol, a situation can arise in which two or more leaders "duel" by proposing different commands for the same slot. This can also occur in Gbcast.
In the normal sequence of events, one leader takes over because the prior leader has failed, learns of any proposals the p... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Dueling leaders-The closest situation to a duel is seen if the old leader has become partitioned from the majority and the new leader, taking over, is unable to contact some set of members (but does obtain the required quorum during the INQUIRE phase). Here the new leader may be unaware of some propo... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Dueling leaders-The shunning mechanism resolves such duels. When the new leader obtained a quorum during the INQUIRE phase, it also blocked the old leader from ever again achieving a quorum for any new PROPOSE it might initiate: a majority of members are now shunning the old leader. Thus if any propo... | milkshake721/2.1M-wiki-STEM |
Gbcast-Discussion: Dueling leaders-Shunning of leaders in Gbcast occurs in the pre-determined order of leader ranks: a higher-ranking leader only shuns a lower-ranking leader when it tries to take-over its place. The Paxos ballots mechanism serves precisely the same purpose, but differs in allowing participants to atte... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-Although superficially quite different, upon close study Gbcast is seen to be surprisingly similar to Paxos. Indeed, Paxos can be "transformed" into Gbcast with the following (reversible) sequence of steps. For brevity we describe these steps informally and omit a detailed proo... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-Note that this transformation does not address durability. Gbcast treats durable state as a property of the application, not the protocol, whereas Paxos logs events to a set of durable command logs, and hence can still recover its state even after the whole service is shut down... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-Start with the basic Paxos protocol. Add a process incarnation number to distinguish a rejoining process from one that has been continuously a member of the view. Impose an age-based ordering on the members of the group, designate the oldest member (breaking ties lexicographic)... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-Define a category of view-modifying commands that operate by adding or removing processes from the group membership. Introduce a failure detection mechanism as used in Gbcast, asking the leader to remove any timed-out members. A member removed from the group that reestablishes ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-Basic Paxos can propose a multicast to just a quorum of group members, hence a typical member may have gaps in its command list. This is why, in Paxos, a learner must read a quorum of members and merge their command lists. In our modified protocol, any multicast is proposed to ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-It follows that Gbcast and Paxos can be transformed, each to the other, without changing assumptions and with the identical correctness properties. Obviously, the protocols don't look very similar, but they have a deep connection. Indeed, one can make a stronger claim: any sequ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Bi-simulation equivalence to Paxos-The type of proof outlined above is formally called a bi-simulation: one shows that any (input-sequence, output-behavior) pair that one protocol can exhibit is also possible with the other protocol. Notice that in carrying out a bisimulation, features that one protocol supports... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-Gbcast has no durable state: the protocol does not maintain a log of events on disk, and durability is treated as an application-specific property. In contrast, Paxos guarantees durability: after recovering from a complete shutdown of the system, a Paxos applicatio... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-In the propose phase, Gbcast must wait for responses from all participants (or for the maximal timeout and then suspect the remaining ones), instead of making progress with the fastest quorum. In Gbcast, the cost of a failure suspicion is high and the protocol may ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-With Gbcast, if an error does occur (e.g. an operational process is suspected and shunned), that process must drop out (it can rejoin under a different name). With Paxos, if f>0, should a process be unable to participate in a protocol instance, it can continue to p... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-With Paxos, to propose multiple commands we use alpha>1, but in this case commands can be committed in a different order from the order in which they were initiated (one case in which this problematic scenario is seen involves dueling leaders; leader A proposes com... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-With Gbcast, a command is delivered in the view in which it was initiated. Reconfigurable Paxos can commit a command in a slot associated with a membership view prior to the active membership view at the time when the commit occurs. Thus, in Paxos, if an applicatio... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-Gbcast does not require that the protocol be halted when changing configurations: the rate of new proposals can be constant even across membership changes. For many implementations of reconfigurable Paxos, this would not be the case. | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-With both Gbcast and Paxos, reconfiguration is only possible if a quorum of the prior view is accessible and can acknowledge the new view. However, in Paxos, the requirement also extends to learning the outcomes of commands proposed for slots associated with the ol... | milkshake721/2.1M-wiki-STEM |
Gbcast-Summary of differences between Paxos and Gbcast-Gbcast does not require a garbage collection protocol because, as each message or view is committed and reported to the application it can be discarded. Paxos maintains state using a quorum scheme in the command logs at its acceptors, and requires a garbage collect... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness comparison-Both Paxos and Gbcast are subject to the FLP impossibility result. Thus neither protocol can be guaranteed live under all possible conditions. At best we can talk about the conditions under which liveness is guaranteed, expressed as predicates on the failure detection mechanism: if the condit... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness comparison-In Gbcast, progress will never resume if a circle of mutual suspicions arises, as noted above: once a quorum of mutually shunning processes arises, the shunning mechanism makes it impossible for any leader to obtain a quorum of promises. | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness comparison-With an (unmodified) Paxos protocol, this problem will not arise: once the excessive level of mutual suspicions ends, progress resumes. Thus Paxos makes progress with any failure-detection mechanism satisfying the <>W condition, even if periods arise during which more than a quorum of mutual ... | milkshake721/2.1M-wiki-STEM |
Gbcast-Liveness comparison-In Paxos deployments, it is common to require human operator intervention to reconfigure Paxos. In such settings, Gbcast may be able to make progress during period when Paxos cannot. Suppose that a group has membership that slowly drops to less than a quorum of the original group size. Gbcast... | milkshake721/2.1M-wiki-STEM |
Gbcast-Need for state transfer-Systems such as Isis that implement Gbcast typically provide a state transfer mechanism: at the instant the new view showing some joining member is delivered, some existing member makes a checkpoint of its copy of the group state. This is then copied to the new member, which loads the che... | milkshake721/2.1M-wiki-STEM |
Gbcast-Need for state transfer-Notice that this is in contrast to Paxos. In that protocol, gaps can arise as a consequence of the basic quorum update scheme, which doesn't guarantee that every member will see every update and can run over unreliable message passing layers that might never deliver some messages. The Pax... | milkshake721/2.1M-wiki-STEM |
Gbcast-Which dynamically reconfigurable state machine replication protocol came first?-The Gbcast protocol was published early in a period when several state machine protocols capable of managing their own membership were introduced: Gbcast, View-Stamped Replication (Oki and Liskov ), Basic Paxos (Lamport ), the partia... | milkshake721/2.1M-wiki-STEM |
Gbcast-Which dynamically reconfigurable state machine replication protocol came first?-If equivalence is formalized using bisimulation, in which any run that one protocol can exhibit is also exhibited by the other, and in which the assumptions made and the conditions for progress are identical, the comparison becomes m... | milkshake721/2.1M-wiki-STEM |
S-Ethylisothiouronium diethylphosphate-S-Ethylisothiouronium diethylphosphate-S-Ethylisothiouronium diethylphosphate (brand name Difetur) is an S-alkylisothiouronium derivative used as an antihypotensive drug. The S-alkylisothiouronium compounds are used in processes of treating acute hypotension, which may result, for... | milkshake721/2.1M-wiki-STEM |
S-Ethylisothiouronium diethylphosphate-Indications-It is used for increasing arterial blood pressure in cases of acute arterial hypotension due to surgical interference, trauma, poisoning, shock condition, hemorrhages; in conjunction with epidural anesthesia; in overdose of ganglion blockers, alpha-adrenergic blockers,... | milkshake721/2.1M-wiki-STEM |
S-Ethylisothiouronium diethylphosphate-Mechanism of action-S-Ethylisothiouronium diethylphosphate is a specific inhibitor of inducible NO synthase on hepatic NO production level. S-Ethylisothiouronium diethylphosphate affects systemic hemodynamic indices in the following way: the peripheral vascular resistance increase... | milkshake721/2.1M-wiki-STEM |
S-Ethylisothiouronium diethylphosphate-Forms-Injectable solution 10%-1 ml in ampules N 10. | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Maximal compact subgroup-In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.
Maximal compact subgroups play an important role in the classification of Lie groups and especial... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Example-An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as any inner p... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Definition-A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a maximal (compact subgroup) – rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a compact (maximal subgroup), but in any case is no... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another. For a semisimple Lie group uniqueness is a consequence of the Car... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-Maximal compact subgroups of connected Lie groups are usually not unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g ∈ G such that gKg−1 = L. Hence a maximal compact subgroup is essentially uniq... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition. The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space G/K ... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-Proof of uniqueness for semisimple groups Mostow (1955) also related the general problem for semisimple groups to the case of GL(n, R). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary prop... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-If H is another compact subgroup of G, then averaging the inner product over H with respect to the Haar measure gives an inner product invariant under H. The operators Ad p with p in P are positive symmetric operators. This new inner produst can be written as (S⋅X,Y)σ, ... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-So for h in H, S∘Ad(σ(h))=Ad(h)∘S.
For X in p define f(eX)=TrAd(eX)S. | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-If ei is an orthonormal basis of eigenvectors for S with Sei = λi ei, then min λi)⋅TreadX, so that f is strictly positive and tends to ∞ as |X| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad X and each non-zero eigenvalue ... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-If fi is an orthonormal basis of eigenvectors of ad X with corresponding real eigenvalues μi, then g(t)=f(eY/2etXeY/2)=∑eμit‖Ad(eY/2)fi‖σ2. | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Existence and uniqueness-Since the right hand side is a positive combination of exponentials, the real-valued function g is strictly convex if X ≠ 0, so has a unique minimum. On the other hand, it has local minima at t = 0 and t = 1, hence X = 0 and p = exp Y is the unique global minimum. By co... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Applications-Representation theory Maximal compact subgroups play a basic role in the representation theory when G is not compact. In that case a maximal compact subgroup K is a compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier.
The operat... | milkshake721/2.1M-wiki-STEM |
Maximal compact subgroup-Applications-Topology The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact K and a Euclidean space – G = K × Rd – thus in partic... | milkshake721/2.1M-wiki-STEM |
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