formulasearchengine - User contributions [en]

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2014-12-11T19:20:29Z

128.111.17.54:

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2014-02-12T09:50:16Z

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2014-01-27T23:40:14Z

128.111.159.205: /* Multiplication of basis elements */

{{For|the film by Hollis Frampton|Zorns Lemma (film)}}

'''Zorn's lemma''', also known as the '''Kuratowski–Zorn lemma''', is a proposition of [[set theory]] that states:

<blockquote>
Suppose a [[partially ordered set]] ''P'' has the property that every [[Total order|chain]] (i.e. [[Total order|totally ordered]] [[subset]]) has an [[upper bound]] in ''P''. Then the set ''P'' contains at least one [[maximal element]].
</blockquote>

It is named after the [[mathematician]]s [[Max August Zorn|Max Zorn]] and [[Kazimierz Kuratowski]].

The terms are defined as follows. Suppose (''P'',≤) is a [[partially ordered set]]. A subset ''T'' is ''totally ordered'' if for any ''s'', ''t'' in ''T'' we have ''s'' ≤ ''t'' or ''t'' ≤ ''s''. Such a set ''T'' has an ''upper bound'' ''u'' in ''P'' if ''t'' ≤ ''u'' for all ''t'' in ''T''. Note that ''u'' is an element of ''P'' but need not be an element of ''T''. An element ''m'' of ''P'' is called a ''maximal element'' (or ''non-dominated'') if there is no element ''x'' in ''P'' for which ''m'' < ''x''.

Note that ''P'' is not required to be non-empty.
However, the empty set is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of ''P''.
An equivalent formulation of the lemma is therefore:

<blockquote>
Suppose a non-empty partially ordered set ''P'' has the property that every non-empty chain has an upper bound in ''P''. Then the set ''P'' contains at least one maximal element.
</blockquote>

The distinction may seem subtle, but proofs involving Zorn's lemma often involve taking a union of some sort to produce an upper bound.
The case of an empty chain, hence empty union is a boundary case that is easily overlooked.

Zorn's lemma is equivalent to the [[well-ordering theorem]] and the [[axiom of choice]], in the sense that any one of them, together with the [[Zermelo–Fraenkel axioms]] of [[set theory]], is sufficient to prove the others. It occurs in the proofs of several theorems of crucial importance, for instance the [[Hahn–Banach theorem]] in [[functional analysis]], the theorem that every [[vector space]] has a [[basis (linear algebra)|basis]], [[Tychonoff's theorem]] in [[topology]] stating that every product of [[compact space]]s is compact, and the theorems in [[abstract algebra]] that every nonzero [[ring (algebra)|ring]] has a [[maximal ideal]] and that every [[field (mathematics)|field]] has an [[algebraic closure]].

==An example application==
We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring ''R'' with [[Unital ring|unity]] contains a [[maximal ideal]]. The set ''P'' here consists of all (two-sided) [[ideal (ring theory)|ideal]]s in ''R'' except ''R'' itself, which is not empty since it contains at least the trivial ideal {0}. This set is partially ordered by [[subset|set inclusion]]. We are done if we can find a maximal element in ''P''. The ideal ''R'' was excluded because maximal ideals by definition are not equal to ''R''.

We want to apply Zorn's lemma, and so we take a non-empty totally ordered subset ''T'' of ''P'' and have to show that ''T'' has an upper bound, i.e. that there exists an ideal ''I'' ⊆ ''R'' which is bigger than all members of ''T'' but still smaller than ''R'' (otherwise it would not be in ''P''). We take ''I'' to be the [[union (set theory)|union]] of all the ideals in ''T''. Because ''T'' contains at least one element, and that element contains at least 0, the union ''I'' contains at least 0 and is not empty. Now to prove that ''I'' is an ideal: if ''a'' and ''b'' are elements of ''I'', then there exist two ideals ''J'', ''K'' ∈ ''T'' such that ''a'' is an element of ''J'' and ''b'' is an element of ''K''. Since ''T'' is totally ordered, we know that ''J'' ⊆ ''K'' or ''K'' ⊆ ''J''. In the first case, both ''a'' and ''b'' are members of the ideal ''K'', therefore their sum ''a'' + ''b'' is a member of ''K'', which shows that ''a'' + ''b'' is a member of ''I''. In the second case, both ''a'' and ''b'' are members of the ideal ''J'', and we conclude similarly that ''a'' + ''b'' ∈ ''I''. Furthermore, if ''r'' ∈ ''R'', then ''ar'' and ''ra'' are elements of ''J'' and hence elements of ''I''. We have shown that ''I'' is an ideal in ''R''.

Now comes the heart of the proof: why is ''I'' smaller than ''R''? The crucial observation is that an ideal is equal to ''R'' [[if and only if]] it contains 1. (It is clear that if it is equal to ''R'', then it must contain 1; on the other hand, if it contains 1 and ''r'' is an arbitrary element of ''R'', then ''r1'' = ''r'' is an element of the ideal, and so the ideal is equal to ''R''.) So, if ''I'' were equal to ''R'', then it would contain 1, and that means one of the members of ''T'' would contain 1 and would thus be equal to ''R'' – but we explicitly excluded ''R'' from ''P''.

The condition of Zorn's lemma has been checked, and we thus get a maximal element in ''P'', in other words a maximal ideal in ''R''.

Note that the proof depends on the fact that our ring ''R'' has a multiplicative unit 1. Without this, the proof wouldn't work and indeed the statement would be false. For example, the ring with <math>\Q</math> as additive group and trivial multiplication (i. e. <math>a b=0</math> for all <math>a,b</math>) has no maximal ideal (and of course no 1): Its ideals are precisely the additive subgroups. The [[factor group]] <math>\Q/A</math> by a proper subgroup <math>A</math> is a [[divisible group]], hence certainly not [[finitely generated abelian group|finitely generated]], hence has a proper non-trivial subgroup, which gives rise to a subgroup and ideal containing <math>A</math>.

== Sketch of the proof of Zorn's lemma (from the axiom of choice)==
A sketch of the proof of Zorn's lemma follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, ''P'' such that every totally ordered subset has an upper bound, and every element has a bigger one. For every totally ordered subset ''T'' we may then define a bigger element ''b''(''T''), because ''T'' has an upper bound, and that upper bound has a bigger element. To actually define the [[function (mathematics)|function]] ''b'', we need to employ the axiom of choice.

Using the function ''b'', we are going to define elements ''a''0 < ''a''1 < ''a''2 < ''a''3 < ... in ''P''. This sequence is '''really long''': the indices are not just the [[natural number]]s, but all [[ordinal number|ordinal]]s. In fact, the sequence is too long for the set ''P''; there are too many ordinals (a [[proper class]]), more than there are elements in any set, and the set ''P'' will be exhausted before long and then we will run into the desired contradiction.

The ''ai'' are defined by [[transfinite recursion]]: we pick ''a''0 in ''P'' arbitrary (this is possible, since ''P'' contains an upper bound for the empty set and is thus not empty) and for any other ordinal ''w'' we set ''a''''w'' = ''b''({''a''''v'': ''v'' < ''w''}). Because the ''a''''v'' are totally ordered, this is a well-founded definition.

This proof shows that actually a slightly stronger version of Zorn's lemma is true:
{{Quote|If ''P'' is a [[poset]] in which every [[well-order]]ed subset has an upper bound, and if ''x'' is any element of ''P'', then ''P'' has a maximal element that is greater than or equal to ''x''. That is, there is a maximal element which is comparable to ''x''.}}

==History==
The [[Hausdorff maximal principle]] is an early statement similar to Zorn's lemma.

[[Kazimierz Kuratowski|K. Kuratowski]] proved in 1922<ref>{{cite journal |first=Casimir |last=Kuratowski |title=Une méthode d'élimination des nombres transfinis des raisonnements mathématiques |trans-title=A method of disposing of transfinite numbers of mathematical reasoning |journal=[[Fundamenta Mathematicae]] |volume=3 |issue= |year=1922 |pages=76–108 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3114.pdf |format=pdf |accessdate=2013-04-24 |language=French}}</ref> a version of the lemma close to its modern formulation (it applied to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by [[Max August Zorn|Max Zorn]] in 1935,<ref>{{cite journal |first=Max |last=Zorn |title=A remark on method in transfinite algebra |journal=Bulletin of the American Mathematical Society |volume=41 |year=1935 |issue=10 |pages=667–670 |doi=10.1090/S0002-9904-1935-06166-X }}</ref> who proposed it as a new [[axiom]] of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.

The name "Zorn's lemma" appears to be due to [[John Tukey]], who used it in his book ''Convergence and Uniformity in Topology'' in 1940. [[Bourbaki]]'s ''Théorie des Ensembles'' of 1939 refers to a similar maximal principle as "le théorème de Zorn".<ref>{{harvnb|Campbell|1978|p=82}}.</ref> The name "[[:pl:lemat Kuratowskiego-Zorna|Kuratowski–Zorn lemma]]" prevails in Poland and Russia.

{{Portal|Mathematics}}

==Equivalent forms of Zorn's lemma==
Zorn's lemma is equivalent (in [[Zermelo–Fraenkel set theory|ZF]]) to three main results:
# [[Hausdorff maximal principle]]
# [[Axiom of choice]]
# [[Well-ordering theorem]].

Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
# Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the [[Hahn–Banach theorem]]
# Every vector space has a [[Hamel basis]], a result from linear algebra (to which it is equivalent<ref>{{cite journal
|last = Blass
|first = Andreas
|year = 1984
|title = Existence of bases implies the Axiom of Choice
|journal=Contemp. Math.
|volume = 31
|pages = 31–33
|ref=blass
}}</ref>)
# Every commutative unital ring has a [[maximal ideal]], a result from ring theory
# [[Tychonoff's theorem]] in topology (to which it is also equivalent<ref>{{cite journal
|last = Kelley
|first = John L.
|year = 1950
|title= The Tychonoff product theorem implies the axiom of choice
| journal= Fundamenta mathematica
| volume = 37
| pages = 75–76
| ref=kelley
}}</ref>)

In this sense, we see how Zorn's lemma can be seen as a powerful tool, especially in the sense of unified mathematics{{Clarify|date=June 2011}}.

== Notes ==
<references/>

==References==
* {{cite journal
| last = Campbell
| first = Paul J.
|date=February 1978
| title = The Origin of ‘Zorn's Lemma’
| journal = Historia Mathematica
| volume = 5
| issue = 1
| pages = 77–89
| doi = 10.1016/0315-0860(78)90136-2
| ref = harv
}}
*{{cite book |title=Set Theory for the Working Mathematician |last=Ciesielski |first=Krzysztof |location= |publisher=Cambridge University Press |year=1997 |isbn=0-521-59465-0 }}

==External links==
*[http://www.apronus.com/provenmath/choice.htm Zorn's Lemma at ProvenMath] contains a formal proof down to the finest detail of the equivalence of the axiom of choice and Zorn's Lemma.
*[http://us.metamath.org/mpegif/zorn.html Zorn's Lemma] at [[Metamath]] is another formal proof. ([http://us.metamath.org/mpeuni/zorn.html Unicode version] for recent browsers.)

{{DEFAULTSORT:Zorn's Lemma}}
[[Category:Axiom of choice]]
[[Category:Order theory]]
[[Category:Lemmas]]

Stochastic Eulerian Lagrangian method

2013-11-13T22:07:11Z

128.111.88.56:

{{lowercase title}}

The '''p-Cycle protection''' scheme is a technique to protect a [[mesh network]] from a failure of a link, with the benefits of ring like recovery speed and mesh-like capacity efficiency, similar to that of a shared backup path protection (SBPP). p-Cycle protection was discovered in late 1990s, with research and development done mostly by Wayne D. Grover, and D. Stamatelakis.<ref name="p-Cycle: An overview">Asthana, R.; Singh, Y.N.; Grover, W.D.; , "p-Cycles: An overview," Communications Surveys & Tutorials, IEEE , vol.12, no.1, pp.97-111, First Quarter 2010</ref><ref name="Announcing">{{cite web|last=Grover|first=Wayne|title=Announcing|url=http://www.ece.ualberta.ca/~grover/p-Cycles_book/Announcing_p_Cycles_book.htm|publisher=John Wiley & Sons|accessdate=3 December 2012}}</ref>

== Overview of the p-Cycle ==

In Transport [[communication networks]] two methods were developed and introduced for restoration and recovery, one was a ring-based protection and the other was mesh restoration.<ref name="networks with p-cycle">Claus G. Gruber and Dominic A. Schupke.; , "Capacity-efficient Planning of Resilient Networks with p-Cycles,". 2002.</ref> The ring based protection offered a quick recovery time at the expense of higher capacity redundancy, while the mesh restoration offered better capacity-efficiency at the expense of slower recovery times. In 1998 the ''p-Cycle'' became a promising technique for recovery in mesh networks because of the combined benefits of ring network recovery speed and mesh like capacity efficiency.<ref name="networks with p-cycle" /> In a mesh network, the spare capacity is used to create the ring like structures as shown in Figure 1. Due to the nature of the rings assuming bi-directional line switched ring (BLSR), only 2 end nodes are involved in a case of a link failure to switch traffic to a pre-planned cycle (path) and recover, as it is demonstrated in Figure 2.

[[File:P-cycle (B).png|center|thumb|Fig 1. p-Cycle in a mesh network creating ring like structure.]]
[[File:On cycle failure (B).png|right|thumb|Fig 2. Failure on a p-cycle showing the 2 nodes involved in recovery.]]
[[File:Straddling span failure (B).png|right|thumb|Fig 3. Failure of straddling span and recovery of that link by the p-cycle.]]
[[File:Hamiltonian.png|thumb|left|Fig 4. A Hamiltonian p-cycle, the protection path passes through all the nodes in the network only once]]
[[File:Nonsimple.png|thumb|left|Fig 5. A Non-Simple p-cycle, the protection path passes through the blue node more than once.]]

One of the key differences between a ring-based scheme and the ''p-cycle'' scheme is the ability of the ''p-cycle'' to protect links that are not on the ''p-cycle'' ring as shown in Figure 3. The ability to protect two channel for every spare channel that is assigned to the p-cycle allows to achieve mesh-like capacity efficiency. This feature gives the ''p-cycle'' the additional efficiency over the ring-based schemes.<ref name="hop limits">Kodian, A.; Sack, A.; Grover, W.D.; , "p-cycle network design with hop limits and circumference limits," Broadband Networks, 2004. BroadNets 2004. Proceedings. First International Conference on , vol., no., pp. 244- 253, 25-29 Oct. 2004</ref> ''"Another over looked feature of the ''p-Cycle'' is that working paths may be freely routed over the network graph and are not limited to follow the ring-constrained routings"''.<ref name="p-Cycle: An overview" />

=== P-Cycle Types ===

The p-cycles come in few variations depending on how they protect a given network and their underlying architecture. The types of p-cycles that are available are: ''Hamiltonian'', ''Simple'', ''Non-Simple'', ''Span'', ''Node encircling'', ''Path'', and ''Flow''. The [[Hamiltonian path|Hamiltonian]], Simple, and Non-Simple are named after their underlying architecture (In relationship to the Network). The Span, Node, Path, and Flow p-cycles are named after the type of protection offered to the network.

:*'''Hamiltonian''' - a p-cycle in which the protection path passes through all nodes in a network only once. This p-cycle is illustrated in Figure 4.

:*'''Simple''' - a p-cycle in which the protection path is not required to pass through all the nodes in the network. The p-cycle is allowed to pass through any one node only once shown in Figure 1.

:*'''Non-simple''' - a p-cycle in which the protection path is allowed to pass through any given node more than once. This is shown in Figure 5.

:*'''Span p-cycle''' - a p-cycle whose primary job is to protect spans or links not on the p-cycle itself. This type of p-cycle is shown in Figure 3.

:*'''Node encircling''' - a p-cycle that protects in case of a node failure. In this type, the traffic that used to pass through that node before a failure is rerouted to an adjacent node(s) encircling the failed node, but not through the failed node.

:*'''Path protecting p-cycle''' - a p-cycle that protects a complete path, from source to destination as long as all the nodes are on the p-cycle.

:*'''Flow p-cycle''' - a p-cycle that offers protection for links that are on the p-cycle, the opposite of the Span p-cycle protection scheme.

== Designs & Formation of p-cycles ==

To design p-cycle, a few methods may be used. The two main categories in which the p-cycles are formed are: ''Centralized'' or ''Distributed''. Further categorization is based on a number of factors including order of the p-cycle and working demands based on routing. The p-cycles can be created after the working demands are routed in the network or at the same time depending on the needs and requirements. There are a number of papers dealing with the p-cycle design, and the idea that p-cycle networks are based many times on the single Hamiltonian cycle seems to float around. While the idea may be good from management simplicity, it does not mean its the best possible solution.<ref name="citi5">Onguetou, D.P.; Grover, W.D.; , "p-cycle network design: From fewest in number to smallest in size," Design and Reliable Communication Networks, 2007. DRCN 2007. 6th International Workshop on , vol., no., pp.1-8, 7-10 Oct. 2007</ref>

=== Centralized ===

In the '''centralized''' method, the p-cycles can be determined and picked based on the possible candidate cycles from a large eligible set for the design in order to protect all the possible working channels and links. Another way in which the centralized method is used is based on network graphs. This way the p-cycles are chosen from a set of a network graph.<ref name="p-Cycle: An overview" /> For the centralized method, many techniques exist to accomplish the above computations. Some major ones are presented below:

==== Integer Linear Programming Models ====

In this model, there are a few techniques that are used for creating acceptable p-cycles in order to protect the network, some of those include:

:*'''''Spare Capacity Optimization''''' - The objective of this technique is to optimize the capacity used for the creation of the p-cycles (minimize) while insuring that all of the working channels are protected. This method creates p-cycles that protect off-cycle paths or spans.<ref name="p-Cycle: An overview" /> This model is able to provide an acceptable set of p-cycles that guarantees 100% protection in case of a single failure. It is possible to have more constrains to further specify and meet the required design specifications.

:*'''''Joint Capacity Optimization''''' - In this technique the optimization is extended not only to the spare capacity of the network but to the total capacity of the network. This includes the spare capacity and the working capacity of the network. Another difference is the routing on the working capacity is not done before the p-cycle formation. First a working route option is calculated for each source/destination pair, than from all the possible solutions found, a pair is selected along with the addition of spare capacity taken into consideration to optimize the total capacity of the network.<ref name="p-Cycle: An overview" /> The model for this technique can be found in [1].
:*'''''Protected Working Capacity Envelope Optimization''''' - This model different from the other 2 models because in this model the p-cycles are found first. There are some considerations when creating the p-cycles based on the idea of optimizing the general volume of the working channels which must be protected. After the p-cycles are found, the working demand is routed on the network within the p-cycle protection domain. This concept is known as protected working capacity envelope (PWCE).<ref name="p-Cycle: An overview" />

==== Heuristic Method ====

The first method of creating p-cycles is computationally intensive when the number of nodes is large.<ref name="citi7">Zhenrong Zhang; Wen-De Zhong; Mukherjee, B.; , "A heuristic method for design of survivable WDM networks with p-cycles," Communications Letters, IEEE , vol.8, no.7, pp. 467- 469, July 2004</ref> The ''[[Heuristic]]'' method presented called the ER-based unity-p-cycle, shows an attractive solution to solve the problem with creating p-cycles without the use of ILP. This method also has a solution that is close to that of an optimal solution, but without the extra computational time required. The general idea of the algorithm is to identify unity p-cycles that are able to protect as many working links as possible, this essentially reduces the number of spare units required for protection. A ''"unit p-cycle is able to protect one working link in opposite direction for every on cycle span and two working units for every straddling span. The number of spare units of a unity-p-cyle is equal to the number of the spans on the cycle."''<ref name="citi7" /> A ratio called ER is defined as the number of working links that are protected by the unity p-cycle to the number of spare units. The higher the ratio the better the efficiency of the protecting p-cycles and hence this is what the algorithm aims for.

The method can be explained as follows as given in [6] is shown here:
:#Based on algorithm in [7]<ref name="citi8">H. Hwang, S. Y. Ahn, Y. H. Yoo, and S. K. Chong, “Multiple shared backup cycles for survivable optical networks,” in Proc. ICCCN’01, Scottsdale, AZ, Oct. 2001, pp. 284–289.</ref> Find the possible cycles, and determine the working capacity for each based on one of the [[shortest path]] algorithms.
:#Calculate the ratio ER of the unit-cycles for the cycles computed in step 1.
:#Based on the ER calculation select the cycle with the highest ER.
:#Remove the working links that can be protected by the selected cycle from above, and update the working capacity.
:#Repeat the above steps until the working capacity on every span is 0.

==== Straddling Link Algorithm ====
The Integer Linear Programming (ILP) method for creating p-cycles requires that all possible sets of cycles are found first up to a certain size or circumference of the network. As a result this method is good for small or medium sized networks.<ref name="citi6">Doucette, J.; He, D.; Grover, W.D.; Yang, O.; , "Algorithmic approaches for efficient enumeration of candidate p-cycles and capacitated p-cycle network design," Design of Reliable Communication Networks, 2003. (DRCN 2003). Proceedings. Fourth International Workshop on , vol., no., pp. 212- 220, 19-22 Oct. 2003</ref> Because as the number of nodes increases, the network graph grows exponentially complicating the problem for ILP and substantially increasing the time required to compute the sets. Therefore this method is not suitable for large networks and a different method must be used. One solution is a '''Straddling Link Algorithm''' (SLA) method. This method is fast and simple to create a set of cycles, but suffers from inefficiency for the overall network design.<ref name="citi6" /> This is because the algorithm generates p-cycles that have only one straddling span.

The key feature of the SLA is the ability to find the p-cyles quickly. The Algorithm works by finding the [[shortest path]] between the nodes of a span, and than find another shortest path between the same set of the nodes that is disjoint from the first route. The p-cycle is than created by combining the previously found two routes into one.<ref name="citi6" /> The span is able to use the other route as a backup in case of a failure. This formation of p-cycle is called a primary p-cycle. The problem with this method is that most primary p-cycles contain only one straddling span, and therefore are inefficient as compared to other types of constructed p-cycles.

=== Distributed ===

The distributed method for creating p-cycles differs from the centralized approach in a number of ways. The major difference being in assumptions made in centralized methods. This assumption is based on the fact that p-cycles are always guaranteed to protect 100% of the working capacity. In other words it is assumed that its always possible to create the p-cycles that are able to protect the working capacity in full. The distributed method deals with logical configuration and assignment of already in-place physical capacities.<ref name="p-Cycle: An overview" /> this means that the distributed method is aimed towards real life operations where the physical links are fixed but logical distinction can be made of how the spare and working capacity can be used and or decided. This method does not always make it possible to be able to protect 100% of the working capacities as there may not be enough of the spare capacity to create the required p-cycles in order protect all of the working links on the network.
The distributed method can be done in one of the two ways:

==== Distributed cycle pre-configuration ====
This method is a based on rules and concepts adopted from the Selfhealing Network protocol.<ref name="citi9">Grover, W.D.; Stamatelakis, D.; , "Cycle-oriented distributed preconfiguration: ring-like speed with mesh-like capacity for self-planning network restoration," Communications, 1998. ICC 98. Conference Record. 1998 IEEE International Conference on , vol.1, no., pp.537-543 vol.1, 7-11 Jun 1998</ref> The idea behind the (DCPC) is as follows: each spare link has a state associated with it called a ''statelet'' with a number of states. The node sees each logical link with an incoming state and an outgoing state. The incoming state from the link to the node originates from an adjacent node that is connected by that link. Also each outbound state from a link has an incoming state which forms its precursor. Based on this idea, a number of statelets is sent throughout the network (broadcast) and forms a tree of states. ''"Each node in the tree, is rooted at the precursor port from which the outgoing statelets are propagated."''<ref name="citi9" /> This is called a state route. There are two node options in the algorithm namely ''Cycler'' and the ''Tandem'', each having it specific role. The ''Cycler'' is a sender/chooser role, in this mode the ''Cycler'' sends and receives parts of a state it initiated. All of the nodes adopt this behavior and this is accomplished in a [[round-robin scheduling|round-robin]] scheme. The other role is the ''Tandem'', which works by mediating the state broadcasts competition with new rules and criteria not found in Selfhealing networks.<ref name="citi9" /> Simply put, each node is allowed to explore the network and discover possible p-cycles. The ''Tandem'' role also dictates allowed discovery of p-cycles by the ''Cycler'' node type. Based on the DCPC, the p-cycles are self-organized in the spare capacity of the network and are found in a distributed way. The algorithm can be re-run each time a network change occurs to create an optimum use of spare capacity.<ref name="p-Cycle: An overview" /> For more information the reader is encouraged to read [9].

==== Swarm Intelligence System ====
This method is based on an intelligent system that is found in nature. Its a distributed method that relies on agents working independently, yet communicating with each other by a means of massages that are left or collected at each node which was visited by that agent. This behavior is similar to that of ants, and so called a p-cycle ant system. The aggregation of the messages left or generated by those ants is the basis of forming p-cycles in the system.<ref name="p-Cycle: An overview" /> This technique has high adaptability and redundancy in the network and as a result optimal solutions are possible.

== Efficiency of p-cycles ==
The efficiency of a p-cycle is based on type of p-cycle in use. The Hamiltonian p-cycle, where the p-cycle passes through all the nodes only once, can be very efficient when the working capacity that is unprotected is able to have all the relationships required by a full Hamiltonian implementation.<ref name="citi10">W.D. Grover, Mesh-based Survivable Networks: Options for Optical, MPLS, SONET and ATM Networking, Prentice-Hall, Aug. 2003.</ref> While the Hamiltonian seems to float around as the choice of p-cycle formation, it is not the only type allowed. In some network configurations a mix of the Hamiltonian p-cycle with other types is required to achieve optimal efficiency in the network design.<ref name="p-Cycle: An overview" /> A study preformed last years showed that an efficient way to create p-cycles can be achieved in mesh networks that are flat. This means that the number of links not on the p-cycle or spans is identical.

A type of network called a homogeneous network, where all of the spans have equal working capacity, showed an efficiency that was not quite optimal in terms of spare to working capacity ratio. This is due to the loss of an ability for a p-cycle to protect more than one straddling span.<ref name="p-Cycle: An overview" /> As an alternative a concept of semi-homogeneous mesh networks was developed. In this type of network the ability of the p-cycle to protect more than one straddling span allowed it to reach an efficiency of

<math>1 / ( \bar{d}-1 )</math>

which is a lower limit. Thus it was proven that with the use of Hamiltonian p-cycles in the semi-homogeneous networks, the theoretical efficiency could be reached, but with some exceptions as real network are different and a mix of different p-cycle is required to achieve optimal solutions for a given network topology and design.<ref name="p-Cycle: An overview" />

== Applications ==

The idea behind ''p-cycles protection'' was an ability to offer protection in mesh optical networks by combining the benefits of ring like recovery speed and the efficiency of a mesh network, however the concept is not only limited to transport optical networks and can be extended to higher levels and other network types:

*[[IP address|IP]]
*[[Wavelength-division multiplexing|WDM]]
*[[ASTN]]
*[[ASON]]
*[[Synchronous Digital Hierarchy|SDH]]
*[[MPLS]]
*[[SONET]]
*[[Segment protection]]
*[[Optical Mesh Networks]]
*Optical Multicast Media Traffic

== References ==
<references />

[[Category:Fiber-optic communications]]
[[Category:Network architecture]]
[[Category:Network protocols]]

Van Hove singularity

2013-05-02T19:12:36Z

128.111.193.4: /* Experimental observation */

In [[homological algebra]], a branch of [[mathematics]], a '''quasi-isomorphism''' is a morphism ''A'' → ''B'' of [[chain complex]]es (respectively, cochain complexes) such that the induced morphisms

:<math>H_n(A_\bullet) \to H_n(B_\bullet)\ (\text{respectively, } H^n(A^\bullet) \to H^n(B^\bullet))\ </math>

of [[homology (mathematics)|homology]] groups (respectively, of cohomology groups) are isomorphisms for all ''n''.

In the theory of [[model category|model categories]], quasi-isomorphisms are sometimes used as the class of [[weak equivalence (homotopy theory)|weak equivalence]]s when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of [[Bousfield localization]] in [[homotopy theory]].

==References==
*Gelfand, Manin. ''Methods of Homological Algebra'', 2nd ed. Springer, 2000.

[[Category:Algebraic topology]]
[[Category:Homological algebra]]

Mangler Transformation

2013-04-30T22:33:13Z

128.111.72.128:

'''Mismatch loss''' in [[transmission line]] theory is the amount of power expressed in [[decibels]] that will not be available on the output due to impedance mismatches and [[Reflections of signals on conducting lines|signal reflections]]. A transmission line that is properly terminated, that is, terminated with the same impedance as that of the [[characteristic impedance]] of the transmission line, will have no reflections and therefore no mismatch loss. Mismatch loss represents the amount of power wasted in the system. It can also be thought of as the amount of power gained if the system was perfectly matched. [[Impedance matching]] is an important part of RF system design; however, in practice there will likely be some degree of mismatch loss.<ref>Daniels, David J. (2004). Ground penetrating radar (2nd Edition). Institution of Engineering and Technology. ISBN 978-0-86341-360-5</ref> In real systems, relatively little loss is due to mismatch loss and is often on the order of 1dB.

== Calculation ==
Mismatch loss (ML) is the ratio of incident power to the difference between incident and reflected power:

[[File:T line img.jpg|thumb|right|Figure 1. Simple circuit showing characteristic impedance Zo and the load impedance ZL. In a perfectly matched system ZL=Zo, and there is no mismatch loss.]]

:<math>
ML_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_i}{P_i-P_r}\bigg) \,
</math>

:<math>
P_r=P_i - P_d\,
</math>

where

<math>P_i</math> = incident power 
<math>P_r</math> = reflected power 
<math>P_d</math> = delivered power (also called the ''accepted power'')

The fraction of incident power delivered to the load is

[[File:T line img2.jpg|thumb|right|Figure 2. Simple circuit showing incident power, Pi, on a load. The reflected power will be the difference between Pi and the power delivered, Pd.]]
:<math>
\frac{P_d}{P_i}=1-\rho^2
</math>
 
where
<math>\rho</math> is the magnitude of the [[reflection coefficient]]. Note that as the reflection coefficient approaches zero, power to the load is maximized.

If the reflection coefficient is known, mismatch can be calculated by 
 
:<math>
ML_\mathrm{dB} = - 10 \log_{10} \bigg(1-\rho^2\bigg) \,
</math>
In terms of the voltage standing wave ratio ([[VSWR]]):

:<math>
ML_\mathrm{dB} = - 10 \log_{10} \bigg(1-\bigg(\frac{VSWR-1}{VSWR+1}\bigg)^2\bigg) \,
</math>
<ref>Lo YT and Lee SW 1988 Antenna handbook: Theory, Applications, and Design</ref>

==Sources of mismatch loss==

Any component of the transmission line that has an input and output will contribute to the overall mismatch loss of the system. For example, in mixers mismatch loss occurs when there is an impedance mismatch between the RF port and IF port of the mixer.<ref>Carr, Joseph J. (2002). RF components and circuits. Oxford: Newnes. ISBN 0-7506-4844-9</ref> This is one of the principal reasons for losses in mixers. Likewise, a large amount of the loss in [[amplifiers]] comes from the mismatch between the input and output. Consequently, not all of the available power generated by the amplifier gets transferred to the load.<ref>Skolnik, Merrill I. (2001). Introduction to radar systems (3rd Edition). New York: McGraw-Hill. ISBN 0-07-288138-0</ref> This is most important in antenna systems where mismatch loss in the transmitting and receiving antenna directly contributes to the losses the system—including the system [[noise figure]]. Other common RF system components such as [[Filter (signal processing)|filters]], [[attenuator]]s, [[splitters]], and [[combiner]]s will generate some amount of mismatch loss. While completely eliminating mismatch loss in these components is near impossible, mismatch loss contributions by each component can be minimized by selecting quality components for use in a well designed system.

==Mismatch error==
<ref>White, Joseph F. (2004). High frequency techniques: An introduction to RF and microwave engineering. Hoboken: Wiley. ISBN 0-471-45591-1</ref> If there are two or more components in cascade as is often the case, the resultant mismatch loss is not only due to the mismatches from the individual components, but also from how the reflections from each component combine with each other. The overall mismatch loss cannot be calculated by just adding up the individual loss contributions from each component. The difference between the sum of the mismatch loss in each component and total mismatch loss due to the interactions of the reflections is known as mismatch error. Depending on how the multiple reflections combine, the overall system loss may be lower or higher than the sum of the mismatch loss from each component. Mismatch error occurs in pairs as the signal reflects off of each mismatched component. So for the example in Figure 3, there are mismatch errors generated by each pair of components.<ref>Briggs, John N. (2004). Target detection by marine radar. Institution of Engineering and Technology. ISBN 978-0-86341-359-9</ref> The mismatch uncertainty increases as the frequency increases, and in wide-band applications. The phasing of the reflections makes it particularly harder to model.

The general case for calculating mismatch error (ME) is:

[[File:T line img3.jpg|thumb|Figure 3. Simple circuit showing multiple reflections due to more than 1 mismatched device.]]

:<math>
ME_\mathrm{dB} = 20 \log_{10} \bigg(1-\rho_1\rho_2\,e^{-j2\theta}\bigg) \,
</math>

where <math>\theta</math> is the complex phase change due to the second reflection

==See also==
*[[VSWR]]
*[[Insertion loss]]
*[[Return loss]]
*[[Reflection coefficient]]

==References==
{{Reflist}}

[[Category:Telecommunications engineering]]

Conserved current

2012-01-06T22:47:11Z

128.111.14.177: in third line, I changed "large enough to have no currents through its surface" to "large enough to have no net currents through its surface"

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