formulasearchengine - User contributions [en]

Coefficient of determination

2014-12-19T19:09:21Z

18.111.19.65: /* Interpretation */ Correction. I am guessing that at some point the order of the two definitions of R was swapped.

== since it is behind you húnluàn Heaven's Soldiers ==

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== 'Wrath of Heaven's Soldiers' ==

Zhou instrument operation, then, that the ancient god of chaos vigor of any flaws, they can not escape the cold side of the eyes and ears and the concept of God. 'award seven type II style! souls triumph!' side cold face of the ancient god of chaos visions,プラダ財布値段, played second type again, Veuve Clicquot played, ruling the world,プラダハンドバッグ, the vast Light into the body of the ancient god of chaos, that the ancient god body immediately began to tremble, physically condense out of a layer chaos armor,prada 新作財布, resist the soul triumph impact. party does not stop cold hands mudra,財布プラダ, once again played the third type, the fourth type, strength superimposed waves,prada スタッズ財布, such as the Yangtze overlapping waves, higher and higher, Makino vast cast out seven style awards now and compared to him, is simply a child playing house in the game. between moment, he had reached the fifth type cast, 'Wrath of Heaven's Soldiers', Bang played out, I saw a fury, a vast expanse, no boundaries, directly put the ancient god of chaos
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<li>イェナン牙Qingwei日</li>

<li>body of a move</li>

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== 'だから、私のレイ強盗は、最も危険なの一つです ==

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<li>[http://www.mj-wg.com/forum.php?mod=viewthread&tid=33616&fromuid=10092 しかし、彼女の背後にある7は、古代、しかし神天皇は存在しない]</li>

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== 、ゆっくりと近くに来た ==

それ以外の場合は、シールされて相手が神の寒冷前線の時代に入ること、結果は悲惨である可能性があります。 主Dharmadhatuはもはや無力に感じる」、終了しなければなりません,[http://www.aseanacity.com/webalizer/prada-bags-31.html プラダ財布リボン]......低温側、終わりが来ないでしょうし、今、最後の栄光の人生は、素晴らしいです」、ゆっくりだけ強烈な怒りを残して姿を消した声は、彼はまた、今より専制低温側を知って、彼に自分自身を窒息している,[http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト]。驚天動地の戦いを 、ゆっくりと近くに来た,[http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト]。あなたが渡した場合、最終的にすべての天軍の年齢外のメイン広場に加えて、斬首 風邪は、この戦争は、私が10倍以上王朝の起源の破壊が大きな影響である可能が怖いです,[http://www.aseanacity.com/webalizer/prada-bags-24.html prada ベルト]。 ほとんどセージヨレを終了することができ、多くの古代天軍が同等の戦争を見事斬首。 アビス、長い時間は、センターとしてクリムゾン·プレーンに戦争を完全にショックだった飛行機の多くを十分に静めることはできません,[http://www.aseanacity.com/webalizer/prada-bags-31.html プラダ長財布]。 「最終
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== 」側清はリングドラゴンキングを見て ==

私は精錬を来るように、生きるための抑制が、良好な空は私が法律を破ることができ、彼は最高の不死を学んだ,[http://www.aseanacity.com/webalizer/prada-bags-29.html prada メンズ財布]！「彼の手の牙寒波は、側面から清は静かにレイ·シティーを持って、彼女の手を見つけ出す自分たちの生活に足を踏み入れた。 Baolei市全ての雷、密な形空洞の遮断と、この操作は、すべてがここに住んで投獄する,[http://www.aseanacity.com/webalizer/prada-bags-28.html prada トートバッグ]。 は「あなたは私を探していますか？」側清はリングドラゴンキングを見て： 'だから、私はあなたが望むだけのようだ。' 今回は、ゴースト皇帝ペンを 、静かに神の子から八尾の目の前に登場している,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダの財布]。 「トス？宇宙の法則プロトスを理解,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダ人気財布]？弱い人にああ。私はあなたがトスを行うどのくらいかわからないとき斬首 ,[http://www.aseanacity.com/webalizer/prada-bags-22.html プラダメンズベルト]。 もともと私は後輩をいじめたくない、誰が高齢者春陽ダンのあらゆる百万数字で、この子低温側を聞かせて行うために私に尋ねた？小さな蟻一般を見ているかのように「人々は、神の子から八尾黄ペンを見て
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== 第千四百三十IX ==

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<li>[http://art-modeling.net/cgi-bin/in.cgi それが彼の研究と昼夜の投影数、の理解にある静かでない限り]</li>

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== まだ生きて実際に、悪魔の神を見たことがありますか ==

その肉の化身を超え悪魔の神。それが終わっ驚いと冷たいエチケット側を失うことを拒否 た後、長い脂肪の約束：」若い視聴者Xingzhuレディ」を歌う 一目で、彼の妻は、サイドXingzhu冷えた体を見て、この時、マスタ側寒いが、魔法のファム、そして微笑んだ「ああファム超自然、神秘的な弟子,[http://www.aseanacity.com/webalizer/prada-bags-31.html プラダ財布新作 2014]？弟子出現のドアであることが判明？ '：'あなたのこれは、私はそれはそれの肉化身を超えているどのように見ている？ ' 「地下の後輩が神の悪魔の化身を見ている必要がありましたので、これは肉の化身を超えXingzhu夫人だと思います。」 低温側の信用を主張するため,[http://www.aseanacity.com/webalizer/prada-bags-25.html プラダの財布]。「ああ,[http://www.aseanacity.com/webalizer/prada-bags-35.html プラダスタッズ財布]？あなたはのように真の弟子として表示されたときに、本当に奇跡がドアをフェザリング、ありますか？まだ生きて実際に、悪魔の神を見たことがありますか？ ' 妻は低温側弟子をフェザリングされている理由Xingzhu知って、彼は祭服を着ていたので、スタイル,[http://www.aseanacity.com/webalizer/prada-bags-26.html プラダ財布迷彩]。 「若く清シニア姉妹弟子が一緒に行くことです,[http://www.aseanacity.com/webalizer/prada-bags-35.html prada 長財布]。 ' '四角い清
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<li>[http://reanimus.com/index.cgi 楽観主義は、影響力のあるネットワークのテキストの円のままで]</li>

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== 「裏切りの運命は、私たちは運命の罰を受けなければなりません ==

人生の大きな、より少ない消費。 「裏切りの運命は、私たちは運命の罰を受けなければなりません！ ' 側冷たい声は、より多くの、さらには彼の声の下の肉の「カオス元タイヤ」を中心に、すべては、口、血まみれの口の中の絵に凝縮され、彼は同じ声を送った,[http://www.aseanacity.com/webalizer/prada-bags-35.html prada 長財布]。だけではなく、彼の声、でも全世界、すべての強さをこの瞬間を ,[http://www.aseanacity.com/webalizer/prada-bags-31.html プラダ財布新作 2014]。混乱の元タイヤの肉、彼の同情の両方。 '運命,[http://www.aseanacity.com/webalizer/prada-bags-30.html プラダ財布新作]！運命,[http://www.aseanacity.com/webalizer/prada-bags-30.html プラダ財布新作]！罰、罰,[http://www.aseanacity.com/webalizer/prada-bags-31.html プラダ長財布]！' 元タイヤ肉は、小さなパーティーの力寒宿命手術で、叫んで、叫んで、テロXuepentaikou精神的な叫びの絵を登場混乱、天安功の息子このシーンを見て、彼の顔が変更されました片側だけの寒さを指摘し、お話したいと思います。 OOO、OOO、OOO、OOO パワー 指先が、彗星良いの雲のように、「天皇への直撃を打つ
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Seven-dimensional cross product

2014-12-13T06:40:03Z

18.111.41.143: correct "Caley" to "Cayley"

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Rate–distortion theory

2014-11-26T18:15:56Z

18.111.3.164: /* Rate–distortion functions */ Fixing both spacings.

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Connected space

2014-02-06T20:33:41Z

18.111.61.164: Undid revision 594213321 by 18.111.14.249 (talk)

It is the begin of another unique year, and folks have all made their New Year's Resolutions. If you are like most people at this time, then you have possibly solved to finally lose a excess fat and start living a healthy lifetime. Despite the fact which folks will constantly choose this resolution, few of them create it function. It is somewhat surprising to learn this. What we may conclude from this info is that persons are not using a way of fat reduction that may really function. One doesn't require to take in much of the caffeine that comes with coffee whenever taking in the [http://greencoffeeweightlossplan.blogspot.com green coffee weight loss] supplement. It only has 20mg of caffeine per serving that is way less compared to the normal cup that has 100mg of caffeine. No should worry regarding that nervous, jittery feeling which regular coffee gives. Just sit back, relax plus wait for the chlorogenic acid, the active component of this particular supplement, to work its method to achieve the ideal body weight. Aside from eating appropriate, you furthermore have to be more active when you want to lose weight. You need to burn the excess fats that we have and the number one way to do it happens to be to exercise. Try enrolling in the gym or engaging in a fresh sport. Doing those details frequently really can help a lot whenever burning a excess fat. You might equally opt to jog about the corner or bike a technique through. Folks whom tried this supplement have lost 17 pounds inside 12 weeks. A study was made with scientists and volunteers whom desired to get rid of fat rapidly and naturally and the results were astonishing! It showed that green coffee beans not merely reduced their weight however it also controlled their blood sugar degrees plus their bowel movements. Almost 10% of their weight was reduced plus 16% of their fat, all of these in simply 12 weeks. 2) As fat: stored under the skin plus about the organs. Fat is more of a lengthy expression stamina shop. When we are low on glucose, glycogen is selected inside preference to fat. This causes the beans to turn brown and it furthermore destroys the chlorogenic acid. A board certified nutritionist plus holistic doctor by the name of Dr. Lindsey Duncan was a guest speaker on Dr. Oz's show, which was aired on April 2012. 2) It has been shown to inhibit an significant enzyme found in the liver which is needed to break down glycogen: glucose-6-phosphatase (G6P). Now this has been shown in the lab[2], nevertheless not yet in a living animal, thus how do we learn it really does inhibit G6P when eaten? Well it has not been straight demonstrated, nevertheless scientists have shown that chlorogenic acid is absorbed into the blood, through the belly without getting degraded inside rats[3]. So it is very reasonable to assume that it arrives at the liver intact, where is could get to work inhibiting G6P. The fact is that if you are experiencing any side-effects from a inexpensive green coffee bean brand, it really is probably contaminated with ephedrine. Green coffee bean extract has no known side-effects. So anything less than the highest standard green coffee bean extract, with at least 50% Chlorogenic acid, within the supplier, whom is charging a premium for the product, is a total waste of money... as should you have income to waste, right?

Von Neumann–Morgenstern utility theorem

2013-12-14T13:45:19Z

18.111.2.184:

In [[mathematics]] and [[computer science]], the [[probabilistic method]] is used to prove the existence of mathematical objects with desired combinatorial properties. The proofs are probabilistic — they work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probability. Consequently, they are [[nonconstructive proof|nonconstructive]] — they don't explicitly describe an efficient method for computing the desired objects.

The '''method of conditional probabilities''' {{harv|Erdös|Selfridge|1973}}, {{harv|Spencer|1987}}, {{harv|Raghavan|1988}} converts such a proof, in a "very precise sense", into an efficient [[deterministic algorithm]], one that is guaranteed to compute an object with the desired properties. That is, the method [[derandomization|derandomizes]] the proof. The basic idea is to replace each random choice in a random experiment by a deterministic choice, so as to keep the conditional probability of failure, given the choices so far, below 1.

The method is particularly relevant in the context of [[randomized rounding]] (which uses the probabilistic method to design [[approximation algorithm]]s).

When applying the method of conditional probabilities, the technical term '''pessimistic estimator''' refers to a quantity used in place of the true conditional probability (or conditional expectation) underlying the proof.

== Overview ==
{{harv|Raghavan|1988}} gives this description:

: ''We first show the existence of a provably good approximate solution using the [[probabilistic method]]... [We then] show that the probabilistic existence proof can be converted, in a very precise sense, into a deterministic approximation algorithm.''

(Raghavan is discussing the method in the context of [[randomized rounding]], but it works with the probabilistic method in general.)

[[File:Method of conditional probabilities.png|thumb|450px|border|right|The method of conditional probabilities]]

To apply the method to a probabilistic proof, the randomly chosen object in the proof must be choosable by a random experiment that consists of a sequence of "small" random choices.

Here is a trivial example to illustrate the principle.

: '''Lemma:''' ''It is possible to flip three coins so that the number of tails is at least 2.''
: ''Probabilistic proof.'' If the three coins are flipped randomly, the expected number of tails is 1.5. Thus, there must be some outcome (way of flipping the coins) so that the number of tails is at least 1.5. Since the number of tails is an integer, in such an outcome there are at least 2 tails. ''QED''

In this example the random experiment consists of flipping three fair coins. The experiment is illustrated by the rooted tree in the diagram to the right. There are eight outcomes, each corresponding to a leaf in the tree. A trial of the random experiment corresponds to taking a random walk from the root (the top node in the tree, where no coins have been flipped) to a leaf. The successful outcomes are those in which at least two coins came up tails. The interior nodes in the tree correspond to partially determined outcomes, where only 0, 1, or 2 of the coins have been flipped so far.

To apply the method of conditional probabilities, one focuses on the ''conditional probability of failure, given the choices so far'' as the experiment proceeds step by step.
In the diagram, each node is labeled with this conditional probability. (For example, if only the first coin has been flipped, and it comes up tails, that corresponds to the second child of the root. Conditioned on that partial state, the probability of failure is 0.25.)

The method of conditional probabilities replaces the random root-to-leaf walk in the random experiment by a deterministic root-to-leaf walk, where each step is chosen to inductively maintain the following invariant:

:: ''the conditional probability of failure, given the current state, is less than 1.''

In this way, it is guaranteed to arrive at a leaf with label 0, that is, a successful outcome.

The invariant holds initially (at the root), because the original proof showed that the (unconditioned) probability of failure is less than 1. The conditional probability at any interior node is the average of the conditional probabilities of its children. The latter property is important because it implies that ''any interior node whose conditional probability is less than 1 has at least one child whose conditional probability is less than 1.'' Thus, from any interior node, one can always choose some child to walk to so as to maintain the invariant. Since the invariant holds at the end, when the walk arrives at a leaf and all choices have been determined, the outcome reached in this way must be a successful one.

== Efficiency ==

In a typical application of the method, the goal is to be able to implement the resulting deterministic process by a reasonably efficient algorithm (formally, one taking [[polynomial time|time polynomial]] in the input size), even though typically the number of possible outcomes is huge (exponentially large). (E.g., consider the example above, but extended to ''n'' flips for large ''n''.)

In the ideal case, given a partial state (a node in the tree), the conditional probability of failure (the label on the node) can be efficiently and exactly computed. (The example above is like this.) If this is so, then the algorithm can select the next node to go to by computing the conditional probabilities at each of the children of the current node, then moving to any child whose conditional probability is less than 1. As discussed above, there is guaranteed to be such a node.

Unfortunately, in most applications, the conditional probability of failure is not easy to compute efficiently. There are two standard and related techniques for dealing with this:

* '''Using a conditional expectation:''' Many probabilistic proofs work as follows: they implicitly define a random variable ''Q'', show that (i) the expectation of ''Q'' is at most (or at least) some threshold value, and (ii) in any outcome where ''Q'' is at most (at least) this threshold, the outcome is a success. Then (i) implies that there exists an outcome where ''Q'' is at most the threshold, and this and (ii) imply that there is an outcome that is a success. (In the example above, ''Q'' is the number of tails, which should be at least the threshold 1.5. In many applications, ''Q'' is the number of "bad" events (not necessarily disjoint) that occur in a given outcome, where each bad event corresponds to one way the experiment can fail, and the expected number of bad events that occur is less than 1.)

In this case, to keep the conditional probability of failure below 1, it suffices to keep the conditional expectation of ''Q'' below (or above) the threshold. To do this, instead of computing the conditional probability of failure, the algorithm computes the conditional expectation of ''Q'' and proceeds accordingly: at each interior node, there is some child whose conditional expectation is at most (at least) the node's conditional expectation; the algorithm moves from the current node to such a child, thus keeping the conditional expectation below (above) the threshold.

* '''Using a pessimistic estimator:''' In some cases, as a proxy for the exact conditional expectation of the quantity ''Q'', one uses an appropriately tight bound called a [[pessimistic estimator]]. The pessimistic estimator is a function of the current state. It should upper (or lower) bound the conditional expectation of ''Q'' given the current state, and it should be non-increasing (or non-decreasing) in expectation with each random step of the experiment. Typically, a good pessimistic estimator can be computed by precisely deconstructing the logic of the original proof.

== Example using conditional expectations ==

This example demonstrates the method of conditional probabilities using a conditional expectation.

=== Max-Cut Lemma ===

Given any undirected [[Graph (mathematics)|graph]] ''G'' = (''V'', ''E''), the [[Max cut]] problem is to color each vertex of the graph with one of two colors (say black or white) so as to maximize the number of edges whose endpoints have different colors. (Say such an edge is ''cut''.)

'''Max-Cut Lemma:''' In any graph ''G'' = (''V'', ''E''), at least |''E''|/2 edges can be cut.

<blockquote>'''Probabilistic proof.''' Color each vertex black or white by flipping a fair coin. By calculation, for any edge e in ''E'', the probability that it is cut is 1/2. Thus, by [[Expected value#Linearity|linearity of expectation]], the expected number of edges cut is |''E''|/2. Thus, there exists a coloring that cuts at least |''E''|/2 edges. ''QED''</blockquote>

=== The method of conditional probabilities with conditional expectations ===

To apply the method of conditional probabilities, first model the random experiment as a sequence of small random steps. In this case it is natural to consider each step to be the choice of color for a particular vertex (so there are |''V''| steps).

Next, replace the random choice at each step by a deterministic choice, so as to keep the conditional probability of failure, given the vertices colored so far, below 1. (Here ''failure'' means that finally fewer than |''E''|/2 edges are cut.)

In this case, the conditional probability of failure is not easy to calculate. Indeed the original proof did not calculate the probability of failure directly; instead, the proof worked by showing that the expected number of cut edges was at least |''E''|/2.

Let random variable ''Q'' be the number of edges cut. To keep the conditional probability of failure below 1, it suffices to keep the conditional expectation of ''Q'' at or above the threshold |''E''|/2. (This is because as long as the conditional expectation of ''Q'' is at least |''E''|/2, there must be some still-reachable outcome where ''Q'' is at least |''E''|/2, so the conditional probability of reaching such an outcome is positive.) To keep the conditional expectation of ''Q'' at |''E''|/2 or above, the algorithm will, at each step, color the vertex under consideration so as to ''maximize'' the resulting conditional expectation of ''Q''. This suffices, because there must be some child whose conditional expectation is at least the current state's conditional expectation (and thus at least |''E''|/2).

Given that some of the vertices are colored already, what is this conditional expectation? Following the logic of the original proof, the conditional expectation of the number of cut edges is

:: ''the number of edges whose endpoints are colored differently so far''
:: + (1/2)*(''the number of edges with at least one endpoint not yet colored'').

=== Algorithm ===

The algorithm colors each vertex to maximize the resulting value of the above conditional expectation. This is guaranteed to keep the conditional expectation at |''E''|/2 or above, and so is guaranteed to keep the conditional probability of failure below 1, which in turn guarantees a successful outcome. By calculation, the algorithm simplifies to the following:

1. For each vertex ''u'' in ''V'' (in any order):
2. Consider the already-colored neighboring vertices of ''u''.
3. Among these vertices, if more are black than white, then color ''u'' white.
4. Otherwise, color ''u'' black.

Because of its derivation, this deterministic algorithm is guaranteed to cut at least half the edges of the given graph. (This makes it a [[Maximum cut#Approximation algorithms|0.5-approximation algorithm for Max-cut]].)

== Example using pessimistic estimators ==

The next example demonstrates the use of pessimistic estimators.

=== Turán's theorem  ===

One way of stating [[Turán's theorem]] is the following:

: Any graph ''G'' = (''V'', ''E'') contains an [[Independent set (graph theory)|independent set]] of size at least |''V''|/(''D''+1), where ''D'' = 2|''E''|/|''V''| is the average degree of the graph.

=== Probabilistic proof of Turán's theorem ===

Consider the following random process for constructing an independent set ''S'':
1. Initialize ''S'' to be the empty set.
2. For each vertex ''u'' in ''V'' in random order:
3. If no neighbors of ''u'' are in ''S'', add ''u'' to ''S''
4. Return ''S''.
Clearly the process computes an independent set. Any vertex ''u'' that is considered before all of its neighbors will be added to ''S''. Thus, letting ''d''(''u'') denote the degree of ''u'', the probability that ''u'' is added to ''S'' is at least 1/(''d''(''u'')+1). By [[Expected value#Linearity|linearity of expectation]], the expected size of ''S'' is at least

: <math>\sum_{u\in V} \frac{1}{d(u)+1} ~\ge~\frac{|V|}{D+1}.</math>

(The inequality above follows because 1/(''x''+1) is [[Convex function|convex]] in ''x'', so the left-hand side is minimized, subject to the sum of the degrees being fixed at 2|''E''|, when each ''d''(''u'') = ''D'' = 2|''E''|/|''V''|.) ''QED''

=== The method of conditional probabilities using pessimistic estimators ===

In this case, the random process has |''V''| steps. Each step considers some not-yet considered vertex ''u'' and adds ''u'' to ''S'' if none of its neighbors have yet been added. Let random variable ''Q'' be the number of vertices added to ''S''. The proof shows that ''E''[''Q''] ≥ |''V''|/(''D''+1).

We will replace each random step by a deterministic step that keeps the conditional expectation of ''Q'' at or above |''V''|/(''D''+1). This will ensure a successful outcome, that is, one in which the independent set ''S'' has size at least |''V''|/(''D''+1), realizing the bound in Turán's theorem.

Given that the first t steps have been taken, let ''S''(''t'') denote the vertices added so far. Let ''R''(''t'') denote those vertices that have not yet been considered, and that have no neighbors in ''S''(''t''). Given the first t steps, following the reasoning in the original proof, any given vertex ''w'' in ''R''(''t'') has conditional probability at least 1/(''d''(''w'')+1) of being added to ''S'', so the conditional expectation of ''Q'' is ''at least''

: <math>|S^{(t)}| ~+~ \sum_{w\in R^{(t)}} \frac{1}{d(w)+1}. </math>

Let ''Q''(''t'') denote the above quantity, which is called a '''pessimistic estimator''' for the conditional expectation.

The proof showed that the pessimistic estimator is initially at least |''V''|/(''D''+1). (That is, ''Q''(0)> ≥ |''V''|/(''D''+1).) The algorithm will make each choice to keep the pessimistic estimator from decreasing, that is, so that ''Q''(''t''+1) ≥ ''Q''(''t'') for each ''t''. Since the pessimistic estimator is a lower bound on the conditional expectation, this will ensure that the conditional expectation stays above |''V''|/(''D''+1), which in turn will ensure that the conditional probability of failure stays below 1.

Let ''u'' be the vertex considered by the algorithm in the next ((''t''+1)-st) step.

If ''u'' already has a neighbor in ''S'', then ''u'' is not added to ''S'' and (by inspection of ''Q''(''t'')), the pessimistic estimator is unchanged. If ''u'' does ''not'' have a neighbor in ''S'', then ''u'' is added to ''S''.

By calculation, if ''u'' is chosen randomly from the remaining vertices, the expected increase in the pessimistic estimator is non-negative. [The calculation: Conditioned on choosing a vertex in ''R''(''t''), the probability that a given term 1/(''d''(''w'')+1) is dropped from the sum in the pessimistic estimator is at most (''d''(''w'')+1)/|''R''(''t'')|, so the expected decrease in each term in the sum is at most 1/|''Q''(''t'')|. There are ''R''(''t'') terms in the sum. Thus, the expected decrease in the sum is at most 1. Meanwhile, the size of ''S'' increases by 1.]

Thus, there must exist some choice of ''u'' that keeps the pessimistic estimator from decreasing.

=== Algorithm maximizing the pessimistic estimator ===

The algorithm below chooses each vertex ''u'' to maximize the resulting pessimistic estimator. By the previous considerations, this keeps the pessimistic estimator from decreasing and guarantees a successful outcome.

Below, ''N''(''t'')(''u'') denotes the neighbors of ''u'' in ''R''(''t'')(that is, neighbors of ''u'' that are neither in ''S'' nor have a neighbor in ''S'').
1. Initialize ''S'' to be the empty set.
2. While there exists a not-yet-considered vertex ''u'' with no neighbor in ''S'':
3. Add such a vertex ''u'' to ''S'' where ''u'' minimizes <math>\sum_{w\in N^{(t)}(u)\cup\{u\}} \frac{1}{d(w)+1}</math>.
4. Return ''S''.

=== Algorithms that don't maximize the pessimistic estimator ===

For the method of conditional probabilities to work, it suffices if the algorithm keeps the pessimistic estimator from decreasing (or increasing, as appropriate). The algorithm does not necessarily have to maximize (or minimize) the pessimistic estimator. This gives some flexibility in deriving the algorithm. The next two algorithms illustrate this.

1. Initialize ''S'' to be the empty set.
2. While there exists a vertex ''u'' in the graph with no neighbor in ''S'':
3. Add such a vertex ''u'' to ''S'', where ''u'' minimizes ''d''(''u'') (the initial degree of ''u'').
4. Return ''S''.

1. Initialize ''S'' to be the empty set.
2. While the remaining graph is not empty:
3. Add a vertex ''u'' to ''S'', where ''u'' has minimum degree in the ''remaining'' graph.
4. Delete ''u'' and all of its neighbors from the graph.
5. Return ''S''.

Each algorithm is analyzed with the same pessimistic estimator as before. With each step of either algorithm, the net increase in the pessimistic estimator is

: <math>1 - \sum_{w\in N^{(t)}(u)\cup\{u\}} \frac{1}{d(w)+1},</math>

where ''N''(''t'')(''u'') denotes the neighbors of ''u'' in the remaining graph (that is, in ''R''(''t'')).

For the first algorithm, the net increase is non-negative because, by the choice of ''u'',

: <math>\sum_{w\in N^{(t)}(u)\cup\{u\}} \frac{1}{d(w)+1} \le (d(u)+1) \frac{1}{d(u)+1} = 1 </math>,

where ''d''(''u'') is the degree of ''u'' in the original graph.

For the second algorithm, the net increase is non-negative because, by the choice of ''u'',

: <math>\sum_{w\in N^{(t)}(u)\cup\{u\}} \frac{1}{d(w)+1} \le (d'(u)+1) \frac{1}{d'(u)+1} = 1 </math>,

where ''d′''(''u'') is the degree of ''u'' in the remaining graph.

== See also ==

* [[Probabilistic method]]
* [[Derandomization]]

{{no footnotes|date=June 2012}}

== References ==

* {{Citation

| title = On a combinatorial game
| first1 = Paul | last1 = Erdös | authorlink1= Paul Erdös
| first2 = J. L. | last2 = Selfridge
| journal = [[Journal of Combinatorial Theory, Series A]]
| volume = 14
| issue = 3
| pages = 298–301
| year = 1973
| doi = 10.1016/0097-3165(73)90005-8}}.

* {{Citation

| title=Ten lectures on the probabilistic method
| last=Spencer|first=Joel H.|authorlink=Joel Spencer
| url=http://books.google.com/books?id=Kz0B0KkwfVAC
| year=1987
| publisher=SIAM
| isbn=978-0-89871-325-1}}

* {{Citation

| title= Probabilistic construction of deterministic algorithms: approximating packing integer programs
| first = Prabhakar | last = Raghavan | authorlink=Prabhakar Raghavan
| journal=[[Journal of Computer and System Sciences]]
| volume=37
| issue=2
| pages=130–143
| year = 1988
| doi = 10.1016/0022-0000(88)90003-7}}.

== Further reading ==

* {{Cite book |first1=Noga |last1= Alon |authorlink1=Noga Alon

| first2=Joel |last2=Spencer |authorlink2=Joel Spencer
| series=Wiley-Interscience Series in Discrete Mathematics and Optimization
| title=The probabilistic method
| url=http://books.google.com/books?id=q3lUjheWiMoC&q=%22method+of+conditional+probabilities%22#v=snippet&q=%22method%20of%20conditional%20probabilities%22&f=false
| year=2008
| edition=third
| publisher=John Wiley and Sons
| location=Hoboken, NJ
| isbn=978-0-470-17020-5 , (Second 9780471370468)
| pages=250 et seq. (Second edition)
| mr=2437651 }}

* {{Cite book

| first1=Rajeev |last1=Motwani |authorlink1=Rajeev Motwani
| first2=Prabhakar |last2=Raghavan |authorlink2=Prabhakar Raghavan
| title=Randomized algorithms
| url=http://books.google.com/books?id=QKVY4mDivBEC&q=%22method+of+conditional+probabilities%22#v=snippet&q=%22method%20of%20conditional%20probabilities%22&f=false
| publisher=[[Cambridge University Press]]
| pages=120-
| isbn=978-0-521-47465-8}}

* {{Citation

| first=Vijay |last=Vazirani
| authorlink=Vijay Vazirani
| title=Approximation algorithms
| url=http://books.google.com/books?id=EILqAmzKgYIC&q=%22method+of+conditional%22#v=snippet&q=%22method%20of%20conditional%22&f=false
| publisher=[[Springer Verlag]]
| pages=130-
| isbn=978-3-540-65367-7}}



== External links ==

* [http://greedyalgs.info/blog/method-of-conditional-probabilities/ The probabilistic method — method of conditional probabilities], blog entry by Neal E. Young, accessed 19/04/2012.

{{DEFAULTSORT:Method Of Conditional Probabilities}}

[[Category:Approximation algorithms]]
[[Category:Probabilistic arguments]]

Peter Minkowski

2013-11-25T06:45:05Z

18.111.7.212:

{{redirect3|Square knot|For the mathematical version of this knot, see [[Square knot (mathematics)]]|for another knot known as "square knot"|Friendship knot}}
{{Infobox knot
| image=Square_knot.svg
| name=Reef knot
| names=Square knot, Hercules knot
| type=binding
| type2=bend
| origin=Ancient
| related=[[Thief knot]], [[Granny knot]], [[Grief knot]], [[Surgeon's knot]], [[Shoelace knot]]
| releasing=Jamming
| strength=
| caveat=Not secure as a bend. [[Knot#Capsizing|Spills]] easily if one of the free ends is pulled outward. Does not hold well if the two lines are not the same thickness.
| uses=Joining two ends of a single line to bind around an object.
| abok_number= #74, #75, #460, '''#1204''', '''#1402''', #2096, #2573, #2574, #2577, #2580
| instructions= [http://www.animatedknots.com/reef/]
}}
The '''reef knot''' or '''square knot''' is an ancient and simple [[binding knot]] used to secure a rope or line around an object. Although the reef knot is often seen used for [[List_of_bend_knots|tying two ropes together]], it is not recommended for this purpose due to potential instability of the knot.

A reef [[knot]] is formed by tying a left-handed [[overhand knot]] and then a right-handed overhand knot, or vice versa. A common [[mnemonic]] for this procedure is "right over left; left over right", which is often appended with the rhyming suffix "... makes a knot both tidy and tight". Two consecutive overhands of the same [[handedness]] will make a [[granny knot]]. The [[working end]]s of the reef knot must emerge both at the top or both at the bottom, otherwise a [[thief knot]] results.

==Naming==
The reef knot is at least between 4,000 and 9,000 years old. The name "reef knot" dates from at least 1794<ref name="steel">{{citation|title=The Elements and Practice of Rigging and Seamanship|year=1794|place=London|publisher=David Steel|author=David Steel|url=http://www.hnsa.org/doc/steel/index.htm|page=183}}</ref> and originates from its common use to [[Reefing|reef]] sails,<ref name="lever">{{Citation|last=Lever|first=Darcy|authorlink=Darcy Lever (author)|title=The Young Sea Officer's Sheet Anchor|year=1998|origyear=1819|edition=2nd|publisher=Dover Publications|location=Mineola, NY|page=83|ISBN=978-0-486-40220-8}}</ref><ref name="aksday">{{citation|author=Cyrus Lawrence Day|title=The Art of Knotting and Splicing|edition=4th|location=Annapolis|publisher=Naval Institute Press|year=1986|page=42}}</ref> that is to tie part of the [[sail]] down to decrease its effective surface area in strong [[wind]]s. To release the knot a [[sailor]] could collapse it with a pull of one hand; the sail's weight would make the collapsed knot come apart. It is specifically this behavior which makes the knot unsafe for connecting two ropes together.<ref name="ashley258">{{citation |last=Ashley |first=Clifford W. |title=The Ashley Book of Knots |url= |accessdate= |origyear= |year= 1944 |publisher=Doubleday |location=New York |isbn=978-0-385-04025-9 |page=258}}</ref>

The name "square knot" is found in [[Richard Henry Dana, Jr.|Dana's]] 1841 maritime compendium ''A Seaman's Friend'', which also gives "reef knot" as an alternative name.<ref name="ashley220">Ashley, p. 220.</ref><ref name="seamans">{{citation|author=Richard Henry Dana, Jr.|url=http://books.google.com/?id=8VwBAAAAQAAJ&pg=PA30#v=onepage&q=Square%20knot|title=The Seaman's Friend: A Treatise on Practical Seamanship|edition=14th revised and corrected|place=Mineola, NY|publisher=Dover|origyear=1879|year=1997|page=49|isbn=0-486-29918-X}}</ref>

== Uses ==
The reef knot is used to tie the two ends of a single line together such that they will secure something, for example a bundle of objects, that is unlikely to move much. In addition to being used by sailors for [[reefing]] and [[Furl (sailing)|furling]] sails, it is also one of the key knots of [[macrame]] textiles.<ref name="ashley399">Ashley, pp. 399-400.</ref>

The knot lies flat when made with cloth and has been used for tying [[bandage]]s for millennia. As a binding knot it was known to the [[Ancient Greece|ancient Greeks]] as the [[Hercules]] knot (''Herakleotikon hamma'') and is still used extensively in medicine.<ref name="hage">{{citation|doi=10.1007/s00268-007-9359-x|last=Hage|first=J. Joris|periodical=World Journal of Surgery|date=April 2008|volume=32|issue=4|pages=648–655|title=Heraklas on Knots: Sixteen Surgical Nooses and Knots from the First Century A.D.|url=http://www.springerlink.com/content/ft83233774k358v1/|pmid=18224483}}</ref> In his ''[[Natural History]]'', [[Pliny the Elder|Pliny]] relates the belief that [[wound healing|wounds heal]] more quickly when bound with a "Hercules knot".<ref name="plinynh">{{citation|title=The Natural History|author=[[Pliny the Elder]]|url=http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.02.0137%3Abook%3D28%3Achapter%3D17|page=28.17|editor1-first=John|editor1-last=Bostock|editor1-link=John Bostock (physician)|editor2-last=Riley|editor2-first=H. T.|editor2-link=Henry Thomas Riley|accessdate=2009-08-23}}</ref>

It has also been used since ancient times to tie [[belt (clothing)|belt]]s and [[sash]]es. A modern use in this manner includes tying the [[Obi (sash)|obi]] (or belt) of a [[martial arts]] [[keikogi]].

With both ends tucked (slipped) it becomes a good way to tie [[shoelaces]], whilst the non-slipped version is useful for shoelaces that are excessively short. It is appropriate for tying plastic garbage or trash bags, as the knot forms a handle when tied in two twisted edges of the bag.

The reef knot figures prominently in [[Scouting]] worldwide. It is included in the international membership badge<ref name="scoutemblem">See [[:File:World Scout Emblem 1955.svg]] for an image of the emblem.</ref> and many [[Square knot insignia|scouting awards]].<ref name="squareawards">{{citation|title=Square Knots - Meaning and Placement|url=http://www.boyscouttrail.com/square-knots.asp|accessdate=2009-08-17}}</ref> In the Boy Scouts of America demonstrating the proper tying of the square knot is a requirement for all boys joining the program.<ref name="bsareq">{{citation|url=http://www.scouting.org/scoutsource/BoyScouts/AdvancementandAwards/joining.aspx|title=Boy Scout Joining Requirements|author=[[Boy Scouts of America]]|accessdate=2009-08-23}}</ref> In [[Pioneering (Scouting)]], it is commonly used as a [[binding knot]] to finish off specialized [[lashing (ropework)]] and [[whipping knot]]s.<ref name="Square Knot">{{cite web |url=http://scoutpioneering.com/2013/06/15/foolproof-way-to-always-tie-a-square-knot-right/ | title=Foolproof Way to ALWAYS Tie a Square Knot Right | work=www.scoutpioneering.com | accessdate=2013-06-17}}</ref>

<gallery>
File:Egypte louvre 279 couple detail reef knot.jpg|Detail of [[Ancient Egypt|Egyptian]] statue dating from 2350 BC depicting a reef knot securing a belt
File:Ancient Greek jewelry Pontika (Ukraina) 300 bC.jpg|Ancient Greek jewelry from [[Chersonesos Taurica|Pontika]] (now in Ukraine), 300 BC, in the form of a reef knot
File:Kreuzknoten-slip.jpg|Singly slipped reef knot
File:Shoelace_knot.svg|Diagram of common shoelace bow knot, a doubly slipped reef knot
</gallery>

==Misuse as a bend==
[[Image:Capsizereefknot111.jpg|thumb|150px|upright|right|The reef knot can capsize (spill) when one of the free ends is pulled outward.]]
The reef knot's familiarity, ease of tying, and visually appealing symmetry conceal its weakness. The [[International Guild of Knot Tyers]] warns that this knot should never be used to [[bend knot|bend]] two ropes together.<ref name="igktwarn">{{citation|author=[[International Guild of Knot Tyers]]|title=Sea Cadet Knots|url=http://www.igkt.net/beginners/sea-cadet-knots.php|accessdate=2009-08-23}}</ref> A proper bend knot, for instance a [[sheet bend]] or [[double fisherman's knot]], should be used instead. Knotting authority [[Clifford Ashley]] claimed that misused reef knots have caused more deaths and injuries than all other knots combined.<ref name="ashley18">Ashley, p. 18.</ref> Further, it is easily confused with the [[granny knot]], which is a very poor knot.

==Physical analysis==
An approximate physical analysis<ref>Maddocks, J.H. and Keller, J. B., "Ropes in Equilibrium," SIAM J Appl. Math., 47 (1987), pp. 1185-1200</ref> predicts that a reef knot will hold if <math>2\mu e^{\mu\pi} \ge 1</math>, where μ is the relevant [[coefficient of friction]]. This inequality holds if <math>\mu \gtrsim 0.24</math>. Experiments show that the critical value of μ is actually somewhat lower.<ref>Crowell, "The physics of knots," http://www.lightandmatter.com/article/knots.html</ref>

==Related knots==
{{Reef knot family}}

== See also ==
{{commons category|Square knot}}
{{Wiktionary|square knot}}
* [[Shoelace knot]]
* [[Granny knot]]
* [[Thief knot]]
* [[Surgeon's knot]]
* [[List of bend knots]]
* [[List of binding knots]]
* [[List of knots]]
* [[Square knot (emblem or insignia)]]
* [[Reefing]]

==Notes and references==
{{reflist}}

==External links==
* [http://symboldictionary.net/?p=3030 Knot of Hercules]
* [http://www.solnyc.com/webcat_retail/wc_greek/info/asymbols.htm Ancient symbolism of Hercules knot]

{{Knots}}

{{DEFAULTSORT:Reef Knot}}
[[Category:Scoutcraft]]

Noether normalization lemma

2013-10-23T16:35:08Z

18.111.122.84:

{{Infobox person
|name = Vijay Vazirani
|image = Vijay Vazirani.jpg
|image_size = 200px
|caption = Vijay Vazirani in 2010 visiting the [[University of California, Berkeley]].
|birth_date = 1957
|birth_place =
|death_date =
|death_place =
|nationality = [[Indian American]]
|field = [[algorithms]], [[computational complexity theory]], [[algorithmic game theory]].

|work_institution =
|alma_mater = [[MIT]] (Bachelor's degree) [[University of California, Berkeley]] (PhD)
|doctoral_advisor = [[Manuel Blum]]
|doctoral_students =
|occupation = Professor of Computer Science at [[Georgia Institute of Technology|Georgia Tech]].
|prizes = [[Guggenheim Fellowship]]
|religion =
|footnotes = He is the brother of [[Umesh Vazirani]]
}}

'''Vijay Virkumar [[Vazirani]]''' ({{lang-hi|विजय वीरकुमार वज़ीरानी}}; b. 1957<ref>[http://d-nb.info/gnd/122932196/about/html'' Deutsche Nationalbibliothek'']</ref>) is an [[Indian American]] Professor of Computer Science at [[Georgia Institute of Technology|Georgia Tech]].<ref>[http://www.cc.gatech.edu/fac/Vijay.Vazirani/ Faculty page at Georgia Tech]</ref>

He received his [[Bachelor's degree]] from [[MIT]] in 1979 and his [[Ph.D.]] from the [[University of California, Berkeley]] in 1983. During the early to mid nineties, he was a Professor of Computer Science at the [[Indian Institute of Technology, Delhi]]. Vijay Vazirani was also a McKay Visiting Professor at the [[University of California, Berkeley]], and a Distinguished SISL Visitor at the Social and Information Sciences Laboratory at the [[California Institute of Technology]].

==Career==
His research career has been centered around the design of [[algorithms]], together with work on [[computational complexity theory]], [[cryptography]], and [[algorithmic game theory]].

During the 1980s, he made seminal contributions to the classical [[maximum matching]] problem,<ref>Three of his papers on the subject from that time period have over 100 citations each, according to Google scholar: {{citation | last1 = Micali | first1 = S. | author1-link = Silvio Micali | last2 = Vazirani | first2 = V. V. | contribution = An <math>\scriptstyle O(\sqrt{|V|}\cdot|E|)</math> algorithm for finding maximum matching in general graphs | doi = 10.1109/SFCS.1980.12 | pages = 17–27 | title = [[Symposium on Foundations of Computer Science|Proc. 21st IEEE Symp. Foundations of Computer Science]] | year = 1980}}; {{citation|title=Matching is as easy as matrix inversion|first1=Ketan|last1=Mulmuley|authorlink=Ketan Mulmuley|first2=Umesh V.|last2=Vazirani|authorlink2=Umesh Vazirani|first3=Vijay V.|last3=Vazirani|journal=Combinatorica|volume=7|issue=1|year=1987|pages=105–113|doi=10.1007/BF02579206}}; {{citation|first1=Richard M.|last1=Karp|author1-link=Richard M. Karp|first2=Umesh V.|last2=Vazirani|first3=Vijay V.|last3=Vazirani|contribution=An optimal algorithm for on-line bipartite matching|title=[[Symposium on Theory of Computing|Proc 22nd ACM Symp. Theory of Computing]]|year=1990|pages=352–358|doi=10.1145/100216.100262|isbn=0-89791-361-2}}.</ref> and some key contributions to [[computational complexity theory]], e.g., the [[Valiant-Vazirani theorem]]. During the 1990s he worked mostly on [[approximation algorithms]], championing the primal-dual schema, which he applied to problems arising in network design, facility location and web caching, and clustering. In July 2001 he published what is widely regarded as the definitive book on [[approximation algorithms]] (Springer-Verlag, Berlin). Since 2002, he has been at the forefront
of the effort to understand the computability of market equilibria, with an extensive body of work on the topic.

Two of his most significant research results were proving, along with [[Leslie Valiant]], that if [[Boolean satisfiability problem#Extensions of SAT|UNIQUE-SAT]] is in [[P (complexity)|P]], then [[NP (complexity)|NP]] = [[RP (complexity)|RP]] ([[Valiant–Vazirani theorem]]), and obtaining in 1980, along with [[Silvio Micali]], an algorithm for finding maximum matchings in general graphs; the latter is still the most efficient known algorithm for the problem.

He is the brother of [[UC Berkeley]] computer science professor [[Umesh Vazirani]]. In 2005 they both were inducted as Fellows of the [[Association for Computing Machinery]].<ref>[http://fellows.acm.org/fellow_citation.cfm?id=UK41587&srt=all ACM Fellows Award: Umesh Vazirani].</ref><ref>[http://fellows.acm.org/fellow_citation.cfm?id=UK17331&srt=all ACM Fellows Award: Vijay Vazirani].</ref>
In 2011, he was awarded a [[Guggenheim Fellowship]].

== References ==
<references/>

==External links==
*[http://www.scs.gatech.edu/people/vijay-vazirani "Vijay Vazirani"], ''Georgia Tech School of Computer Science''
*[http://genealogy.math.ndsu.nodak.edu/id.php?id=78255 "Vijay V. Vazirani"], ''Mathematics Genealogy Project''

{{Persondata 
| NAME = Vazirani, Vijay
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = American theoretical computer scientist
| DATE OF BIRTH = 1951
| PLACE OF BIRTH =
| DATE OF DEATH =
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Vazirani, Vijay}}
[[Category:1951 births]]
[[Category:Living people]]
[[Category:Georgia Institute of Technology faculty]]
[[Category:Fellows of the Association for Computing Machinery]]
[[Category:Massachusetts Institute of Technology alumni]]
[[Category:University of California, Berkeley alumni]]
[[Category:University of California, Berkeley faculty]]
[[Category:Theoretical computer scientists]]
[[Category:American Hindus]]
[[Category:Sindhi people]]
[[Category:Guggenheim Fellows]]

Talk:Locally convex topological vector space

2013-02-24T02:30:41Z

18.111.56.150:

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Endoreversible thermodynamics

2013-02-18T13:28:21Z

18.111.105.211: more punctuation

{{Multiple issues|cleanup=May 2009|notability=May 2009|unreferenced =May 2009|
{{Expert-subject|Computer science|date=May 2009}}
}}

The '''Suzuki-Kasami algorithm'''<ref>Ichiro Suzuki, Tadao Kasami, ''A distributed mutual exclusion algorithm'', ACM Transactions on Computer Systems, Volume 3 Issue 4, Nov. 1985 (pages 344 - 349)</ref> is a [[access token|token]]-based [[algorithm]] for achieving mutual exclusion in [[distributed systems]]. The process holding the token is the only process able to enter its critical section.

If a process wants to enter its critical section and it does not have the token, it broadcasts a request message to all other processes in the system. The process that has the token, if it is not currently in a critical section, will then send the token to the requesting process. The algorithm makes use of increasing Request Numbers to allow messages to arrive out-of-order.

== Algorithm description ==

Let <math>n</math> be the number of processes. Each process is identified by an integer in <math>1, ..., n</math>.

=== Data structures ===

Each process <math>i</math> maintains one data structure:

* an array <math>RN_i[n]</math> (for Request Number), where <math>RN_i[j]</math> stores the last Request Number received from <math>j</math>

The token contains two data structures:

* an array <math>LN[n]</math> (for Last request Number), where <math>LN[j]</math> stores the most recent Request Number of process <math>j</math> for which the token was successfully granted
* a queue Q, storing the ID of processes waiting for the token

=== Algorithm ===

==== Requesting the critical section (CS) ====

When process <math>i</math> wants to enter the CS, if it does not have the token, it:

* increments its sequence number <math>RN_i[i]</math>
* sends a request message containing new sequence number to all processes in the system

==== Releasing the CS ====

When process <math>i</math> leaves the CS, it:

* sets <math>LN[i]</math> of the token equal to <math>RN_i[i]</math>. This indicates that its request <math>RN_i[i]</math> has been executed
* for every process <math>k</math> not in the token queue <math>Q</math>, it appends <math>k</math> to <math>Q</math> if <math>RN_i[k] = LN[k] + 1</math>. This indicates that process <math>k</math> has an outstanding request
* if the token queue <math>Q</math> is nonempty after this update, it pops a process ID <math>j</math> from <math>Q</math> and sends the token to <math>j</math>
* otherwise, it keeps the token

==== Receiving a request ====

When process <math>i</math> receives a request from <math>j</math> with sequence number <math>s</math>, it:

* sets <math>RN_i[j]</math> to <math>max(RN_i[j], s)</math> (if <math>s < RN_i[j]</math>, the message is outdated)
* if process <math>i</math> has the token and is not in CS, and if <math>RN_i[j] == LN[j] + 1</math> (indicating an outstanding request), it sends the token to process <math>j</math>

==== Executing the CS ====

A process enters the CS when it has acquired the token.

== Notes on the algorithm ==

* Only the site currently holding the token can access the CS
:* All processes involved in the assignment of the CS
* [[Hypertext Transfer Protocol|Request]] messages sent to all [[Node (networking)|nodes]]
:* Not based on [[Lamport timestamps|Lamport’s logical clock]]
:* The algorithm uses sequence numbers instead
* Used to keep track of outdated requests
* They advance independently on each site

The main design issues of the algorithm:
* Telling outdated requests from current ones
* Determining which site is going to get the token next

Data structures used to deal with these two aspects:
* Each site Si has an array RNi[1..N] to store the sequence
* Number of the latest requests received from other sites

The token contains two data structures:
* The token array LN[1..N] keeps track of the request executed most recently on each site
* The token queue Q is a queue of requesting sites

=== Requesting the CS ===
* If the site does not have the token, then it increases its sequence number RNi[i] and sends a request(i, sn) message to all other sites (sn= RNi[i])
* When a site Sj receives this message, it sets RNj[i] to max(RNj[i], sn). If Sj has the idle token, them it sends the token to Si if RNj[i] = LN[i]+1

=== Executing the CS ===
* Site Si executes the CS when it has received the token

=== Releasing the CS ===
* When done with the CS, site Si sets LN[i] = RNi[i]
* For every site Sj whose ID is not in the token queue, it appends its ID to the token queue if RNi[j] =LN[j]+1
* If the queue is not empty, it extracts the ID at the head of the queue and sends the token to that site

=== Performance ===
* either 0 or n messages for CS invocation (no messages if process holds the token; otherwise <math>N - 1</math> requests and <math>1</math> reply)
* Synchronization delay is 0 or N

==References==
{{Reflist}}

[[Category:Distributed algorithms]]

Acoustic wave equation

2012-07-12T01:21:05Z

18.111.113.59: /* Spherical coordinates */ The wording was bad, so I added the word "solution".

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Representation theory of the symmetric group

2012-06-15T19:19:04Z

18.111.112.239:

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Scalar field solution

2012-05-21T00:02:03Z

18.111.68.236: /* Einstein tensor */ The Einstein tensor given disagreed with the eigenvalue equation below, and gave a stress energy tensor with negative pressures.

In [[semi-Riemannian geometry]], the '''Ricci decomposition''' is a way of breaking up the [[Riemann tensor|Riemann curvature tensor]] of a [[pseudo-Riemannian manifold]] into pieces with useful individual algebraic properties. This decomposition is of fundamental importance in Riemannian- and pseudo-Riemannian geometry.

==The pieces appearing in the decomposition==
The decomposition is
:<math>R_{abcd}= \, S_{abcd}+E_{abcd}+C_{abcd}.</math>
The three pieces are:
# the ''scalar part'', the tensor <math>S_{abcd}</math>
# the ''semi-[[traceless]] part'', the tensor <math>E_{abcd}</math>
# the ''fully traceless part'', the [[Weyl tensor]] <math>C_{abcd}</math>
Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties.

The decomposition can have different signs, depending on the Ricci curvature convention, and only makes sense if the dimension satisfies <math>n>2</math>.

The scalar part
:<math> S_{abcd} = \frac{R}{n \, (n-1)} \, H_{abcd}</math>
is built using the [[scalar curvature]] <math>R = {R^m}_m</math>, where <math>R_{ab}={R^c}_{acb}</math> is the [[Ricci curvature]], and a tensor constructed algebraically from the [[metric tensor]] <math>g_{ab}</math>,
:<math>H_{abcd} = g_{ac} \, g_{db} - g_{ad} \, g_{cb} = 2g_{a[c} \, g_{d]b}.</math>

The semi-traceless part
:<math>E_{abcd} = \frac{1}{n-2} \, \left( g_{ac} \, S_{bd} - g_{ad} \, S_{bc} + g_{bd} \, S_{ac} - g_{bc} \, S_{ad} \right) =
\frac{2}{n-2} \, \left( g_{a[c} \, S_{d]b} - g_{b[c} \, S_{d]a} \right) </math>
is constructed algebraically using the metric tensor and the ''traceless part'' of the Ricci tensor
:<math> S_{ab} = R_{ab} - \frac{1}{n} \, g_{ab} \, R</math>
where <math>g_{ab}</math> is the [[metric tensor]].

The [[Weyl tensor]] <math>C_{abcd}</math> or ''conformal curvature tensor'' is completely traceless, in the sense that taking the trace, or [[tensor contraction|contraction]], over any pair of indices gives zero. [[Hermann Weyl]] showed that this tensor measures the deviation of a semi-Riemannian manifold from ''conformal flatness''; if it vanishes, the manifold is (locally) [[conformal equivalence|conformally equivalent]] to a flat manifold.

No additional differentiation is needed anywhere in this construction.

In the case of a [[Lorentzian manifold]], <math>n=4</math>, the [[Einstein tensor]] <math>G_{ab} = R_{ab} - 1/2 \, g_{ab} R</math> has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.

:<math> S_{ab} = R_{ab} - \frac{1}{4} \, g_{ab} \, R = G_{ab} - \frac{1}{4} \, g_{ab} \, G</math>

''Terminological note:'' the notation <math>R_{abcd}, \, C_{abcd}</math> is standard in the modern literature, the notations <math>S_{ab}, \, E_{abcd}</math> are commonly used but not standardized, and there is no standard notation for the scalar part.

==Mathematical definition==
Mathematically, the Ricci decomposition is the decomposition of the space of all [[tensor]]s having the symmetries of the Riemann tensor into its [[irreducible representation]]s for the action of the [[orthogonal group]] {{harv|Besse|1987|loc=Chapter 1, §G}}. Let ''V'' be an ''n''-dimensional [[vector space]], equipped with a [[metric tensor]] (of possibly mixed signature). Here ''V'' is modeled on the [[cotangent space]] at a point, so that a curvature tensor ''R'' (with all indices lowered) is an element of the [[tensor product]] ''V''&otimes;''V''&otimes;''V''&otimes;''V''. The curvature tensor is skew symmetric in its first and last two entries:
:<math>R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\,</math>
and obeys the interchange symmetry
:<math>R(x,y,z,w) = R(z,w,x,y),\,</math>
for all ''x'',''y'',''z'',''w'' ∈ ''V''&lowast;. As a result ''R'' is an element of the subspace ''S''2Λ2''V'', the second [[symmetric power]] of the second [[exterior power]] of ''V''. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the [[kernel (algebra)|kernel]] of the linear map
:<math>b(R)(x,y,z,w) = R(x,y,z,w) + R(y,z,x,w) + R(z,x,y,w).\,</math>

The space {{nowrap|1='''R'''''V'' = ker ''b''}} in ''S''2Λ2''V'' is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping
:<math>c : S^2\Lambda^2 V \to S^2V</math>
is given by
:<math>c(R)(x,y) = \operatorname{tr}R(x,\cdot,y,\cdot).</math>
This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms ''h'' and ''k'', the [[Kulkarni–Nomizu product]] of ''h'' and ''k''
:<math>(h {~\wedge\!\!\!\!\!\!\bigcirc~} k)(x,y,z,w) = h(x,z)k(y,w)+h(y,w)k(x,z) -h(x,w)k(y,z)-h(y,z)k(x,w)</math>
produces an algebraic curvature tensor.

If ''n'' > 4, then there is an orthogonal decomposition into (unique) irreducible subspaces
:{{nowrap|1='''R'''''V'' = '''S'''''V'' &oplus; '''E'''''V'' &oplus; '''C'''''V''}}
where
:<math>\mathbf{S}V = \mathbb{R} g {~\wedge\!\!\!\!\!\!\bigcirc~} g</math>, where <math>\mathbb{R}</math> is the space of [[real number|real]] scalars
:<math>\mathbf{E}V = g {~\wedge\!\!\!\!\!\!\bigcirc~} S^2_0V</math>, where ''S''{{su|p=2|b=0}}''V'' is the space of trace-free symmetric 2-forms
:<math>\mathbf{C}V = \ker c \cap \ker b.</math>

The parts ''S'', ''E'', and ''C'' of the Ricci decomposition of a given Riemann tensor ''R'' are the orthogonal projections of ''R'' onto these invariant factors. In particular,
:<math>R = S + E + C</math>
is an orthogonal decomposition in the sense that
:<math>|R|^2 = |S|^2 + |E|^2 + |C|^2.</math>
This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an [[irreducible representation]] for the [[orthogonal group]] {{harv|Singer|Thorpe|1968}}, and thus the Ricci decomposition is a special case of the splitting of a module for a [[semisimple Lie group]] into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the [[special orthogonal group]]: the [[self-dual]] and [[antiself-dual]] parts ''W''+ and ''W''−.

==Physical interpretation==
The Ricci decomposition can be interpreted physically in Einstein's theory of [[general relativity]], where it is sometimes called the ''Géhéniau-Debever decomposition''. In this theory, the [[Einstein field equation]]
:<math> G_{ab} = 8 \pi \, T_{ab}</math>
where <math>T_{ab}</math> is the [[stress–energy tensor]] describing the amount and motion of all matter and all nongravitational field energy and momentum, states that the Ricci tensor—or equivalently, the Einstein tensor—represents that part of the gravitational field which is due to the ''immediate presence'' of nongravitational energy and momentum. The Weyl tensor represents the part of the gravitational field which can propagate as a [[gravitational wave]] through a region containing no matter or nongravitational fields. Regions of spacetime in which the Weyl tensor vanishes contain no [[gravitational radiation]] and are also conformally flat.

==See also==

*[[Bel decomposition]] of the [[Riemann tensor]]
*[[Conformal geometry]]
*[[Petrov classification]]
*[[Plebanski tensor]]
*[[Ricci calculus]]
*[[Schouten tensor]]
*[[Trace-free Ricci tensor]]

==References==
*{{Citation | last1=Besse | first1=Arthur L. | title=Einstein manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10 | isbn=978-3-540-15279-8 | year=1987 | pages=xii+510}}.

*{{cite book | author=Hawking, S. W.; and Ellis, G. F. R.| title = The Large Scale Structure of Space-Time | publisher=Cambridge: Cambridge University Press | year=1973| isbn=0-521-09906-4}} See ''section 2.6'' for the decomposition. This book uses opposite signature but the same ''Landau-Lifshitz spacelike'' sign convention used in the Wikipedia.

*{{cite book | author=Weinberg, Steven | title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity | publisher=New York: John Wiley & Sons | year=1972 | isbn=0-471-92567-5}} See ''section 6.7'' for a discussion of the decomposition (but note different sign conventions).

*{{cite book | author=Wald, Robert M. | title=General Relativity | publisher=The University of Chicago Press | year=1984 | isbn=0-226-87033-2}} See ''section 3.2'' for a discussion of the decomposition.

* {{citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | id = ISBN 0-387-94732-9}}. Section 6.1 discusses the decomposition. Versions of the decomposition also enter into the discussion of conformal and projective geometries, in chapters 7 and 8.

* {{citation | first1=I.M.|last1=Singer|authorlink1=Isadore Singer|first2=J.A.|last2=Thorpe|title=Global Analysis (Papers in Honor of K. Kodaira)|contribution=The curvature of 4-dimensional Einstein spaces|publisher=Univ. Tokyo Press|year=1969|pages=355–365}}.

{{DEFAULTSORT:Ricci Decomposition}}
[[Category:Differential geometry]]
[[Category:Riemannian geometry]]
[[Category:Tensors in general relativity]]

Talk:Product topology

2012-05-10T21:51:26Z

18.111.59.229: /* Product space page */ new section

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Circuits over sets of natural numbers

2012-02-05T10:19:12Z

18.111.123.90: /* Examples */

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