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In [[mathematics]], an '''ultrametric space''' is a special kind of [[metric space]] in which the [[triangle inequality]] is replaced with <math>d(x,y)\leq\max\left\{d(x,z),d(z,y)\right\}</math>. Sometimes the associated metric is also called a '''non-Archimedean metric''' or '''super-metric'''. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications.
== リビングルームで ==


== Formal definition ==
「蘭シュウは、。ない '秦Yuは深呼吸を取った。<br><br>「愚かな少年、あなたは結婚に子供と一緒に立って、どのような場合に、私は人々が土地を破壊させません。これらのサードの場所 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 靴 メンズ]。汝<br><br>は、被写体さて、ストップ気性でなければならない。」蘭シュウ甘く子供を叱るかのように。<br>「いいえ。」<br><br><br>秦ゆう私は彼の気性を失っていないんだけど、、あなたは古代のアイアンウッドツリーを言うのを聞くために」、彼の頭を横に振った、私はすでに持っているものと考える<br><br>は把握第三の場所を獲得した。 '<br><br>'ああ？'蘭シュウ少し驚い、 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 価格] '何？'<br><br>「Forber。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 偽物] '秦Yuは突然、大声で叫んだ秦ゆうがあっForber所有者、フーボー秦ゆう突然感知された心であり、<br>リビングルームで<br> [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタンジャパン]。<br><br>江蘭心まで推測、秦YuはForberが何をしましょう​​ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ]？<br><br>「蘭シュウは、私と一緒に来る。 '秦ゆう蘭シュウは、その後の動きを気にし、右折言う。<br><br>江蘭シフトが空間的に感じる
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Formally, an ultrametric space is a [[Set (mathematics)|set]] of points <math>M</math> with an associated distance function (also called a [[metric (mathematics)|metric]])
== 秦Yuは地味なこの時間でした ==


:<math>d\colon M \times M \rightarrow \mathbb{R}</math>
図ボリュームは、驚くべきことではないので、多くのアーティファクトを持っています。軽く [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン ブーツ] '華ヤン。<br><br>秦ゆう、現時点ではこれらの言葉を見て、だけでなく、無謀、物語に耳を傾け、誰の心には多くのことを知っている。<br><br>「中央皇帝西安に対しては、最大のアーティファクト皇帝西安を誇っています。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 銀座] '秦ゆう心があっても妖精の魔法の鬼の世界の巨人がうらやましく作るために、その神TUファン」、書き留め、貴重度は想像することができます。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京] '<br>動かなかった<br>そして今回は、無謀な、 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン アウトレット]。彼らは、ペースなしで移動することはしなかった青風水を、この記念碑にも手書き文字を持っていますが、風が蘭ペースを移動しない、無謀推測していた [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_7.php クリスチャンルブタン 東京]。<br><br>「蘭風水熊、中央皇帝西安。それを書く左ネガティブだけでなく、反対」青鳳声を出して「リマインダー」の道。<br><br>秦Yuは地味なこの時間でした。<br><br>瞬間に風が語られたながら,,秦ゆう蘭心含み笑いを「もともとこれらの人々は、ああ私を待って。 '：'かというと、もちろん、私はこれを知っているが、私は非常に私を怖がって見えたあなたのことを考える」<br><br>蘭風が見に群衆に話を聞いた。<br><br>誰もが笑顔
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(where <math>\mathbb{R}</math> is the set of [[real number]]s), such that for all <math>x,y,z\in M</math>, one has:
== and walked toward the northwest Sheng Huang Temple. ==
# <math>d(x, y) \ge 0</math>
# <math>d(x, y) = 0</math> [[iff]] <math>x=y</math>
# <math>d(x, y) = d(y, x)</math> ('''symmetry''')
# <math>d(x, z) \le \max \left\{ d(x, y), d(y, z) \right\}</math> ('''strong triangle''' or '''ultrametric inequality''').


In the case of [[normed vector spaces]], the last property can be made stronger using the [[Wolfgang Krull|Krull]] sharpening<ref>Planet Math: [http://planetmath.org/encyclopedia/NonArchimedeanTriangleInequality.html Non Archimedean Triangle Inequality]</ref> to:
That ...... '<br><br>'I understand.'<br><br>week was laugh loudly. [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン バッグ] Eyes with cold. 'Estimate That Qin Yu cast FIGHTING space. Then been hiding FIGHTING space bar. This 'FIGHTING space' supernatural powers. Places really quite tough, but that Qin Yu, was also willing to faint bird turtle as general I do not know, he chickened bird this turtle to hide what day! '<br><br>week was suddenly remembered what it lightly against Liu Shan said:. 'you go back to it.'<br><br>'yes. Highness.' Liu Shan immediately back away from the living room. The week was thought for a moment in the living room, 'Fu Huang these days, has been looking at other space Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 価格] Yu, now seems to think this fear [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン ブーツ] is Fu Huang wrong.'<br><br>weeks and then I was out of the living room, and walked toward the northwest Sheng Huang Temple.<br>eastern<br>spirit world, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 偽物] Yuchi town.<br><br>into a restaurant.<br><br>punishment far are accompanied his wife Ziyun eating [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン メンズ] places. They also heard the couple these days Qin Yu to do. Qin Yu heart also worried.
: <math>\|x+y\|\le \max \left\{ \|x\|, \|y\| \right\}</math> with equality if <math>\|x\| \ne \|y\|</math>.
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We want to prove that if <math>\|x+y\| \le \max \left\{ \|x\|, \|y\|\right\}</math>, then the equality occurs if <math>\|x\| \ne \|y\|</math>. [[Without loss of generality]], let us assume that <math>\|x\| > \|y\|</math>. This implies that <math>\|x + y\| \le \|x\|</math>. But we can also compute <math>\|x\|=\|(x+y)-y\| \le \max \left\{ \|x+y\|, \|y\|\right\}</math>. Now, the value of <math>\max \left\{ \|x+y\|, \|y\|\right\}</math> cannot be <math>\|y\|</math>, for if that is the case, we have <math>\|x\| \le \|y\|</math> contrary to the initial assumption. Thus, <math>\max \left\{ \|x+y\|, \|y\|\right\}=\|x+y\|</math>, and <math>\|x\| \le \|x+y\|</math>. Using the initial inequality, we have <math>\|x\| \le \|x + y\| \le \|x\|</math> and therefore <math>\|x+y\| = \|x\|</math>.
== already guessed everything ==


==Properties==
Dillon touches his face calm, asked: '? Qin Yu, you said it seems that you had witnessed in general, how do you know.'<br><br>'Master told.' Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ] Yu replied smiling.<br><br>Dillon continue asked: 'Master And how do you know if you Master spiritual knowledge so amazing, you can even find things through, why not kill the mysterious man to win an eighth [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ] of the Yu Jian yet.? '<br><br>'Master brave terrible, but how will care Nabing eighth to Yu Jian? these nine legendary sword, Master has long been known that when he found black Yuhe spiritual knowledge, already guessed everything, Master Training Heaven , extremely aloof and mysterious is the mysterious person who claimed to fate. 'Qin Yu suddenly sigh.<br><br>'Alas, the war when I do not [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] know if I knew in advance, I asked Master for help, an estimated eight Highness will be able to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン ブーツ] save their lives up. Master [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 直営店] though eccentric, for me it only a disciple but also with great interest.'<br><br>Dillon nodded.<br><br>suddenly -<br><br>'youngest, you
[[File:Strong triangle ineq.svg|thumb|right|208px|Even some isosceles triangles cannot exist in an ultrametric space]]
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From the above definition, one can conclude several typical properties of ultrametrics. For example, in an ultrametric space, for all <math>x,y,z\in M</math> and <math>r,s\in\mathbb{R}</math>:
<ul>
* Every triangle is an acute [[isosceles]] or [[equilateral]], i.e. <math>d(x,y)=d(y,z)</math> or <math>d(x,z)=d(y,z)</math> or <math>d(x,y)=d(z,x)</math>.
 
In the following, the concept and notation of an [[open ball|(open) ball]] is the same as in the article about [[metric space]]s, i.e.
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: <math>B(x;r)=\{y\in M|d(x,y)<r\}</math>.
 
* Every point inside a ball is its center, i.e. if <math>d(x,y)<r</math> then <math>B(x;r)=B(y;r)</math>.
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* Intersecting balls are contained in each other, i.e. if <math>B(x;r)\cap B(y;s)</math> is non-empty then either <math>B(x;r)\subseteq B(y;s)</math> or <math>B(y;s)\subseteq B(x;r)</math>.
 
* All balls are both open and closed sets in the induced [[topology]]. That is, open balls are also closed, and closed balls (replace <math><</math> with <math>\leq</math>) are also open.
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* The set of all open balls with radius ''r'' and center in a closed ball of radius <math>r>0</math> forms a [[partition of a set|partition]] of the latter, and the mutual distance of two distinct open balls is again equal to <math>r</math>.
 
 
  </ul>
Proving these statements is an instructive exercise. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.
 
== Examples ==
 
# The [[discrete metric]] is an ultrametric.
# Consider the [[formal language|set of words]] of arbitrary length (finite or infinite) over some alphabet Σ. Define the distance between two different words to be 2<sup>-''n''</sup>, where ''n'' is the first place at which the words differ. The resulting metric is an ultrametric.
# The [[p-adic numbers]] form a complete ultrametric space.
# If ''r''=(''r<sub>n</sub>'') is a sequence of [[real numbers]] [[decreasing]] to zero, then |''x''|<sub>''r''</sub> := [[lim sup]]<sub>''n''→∞</sub> |''x<sub>n</sub>''|<sup>''r<sub>n</sub>''</sup> induces an ultrametric on the space of all complex sequences for which it is finite. (Note that this is not a [[seminorm]] since it lacks [[homogeneous function|homogeneity]]. &mdash; If the ''r<sub>n</sub>'' are allowed to be zero, one should use here the rather unusual convention that 0<sup>0</sup>=0.)
# If ''G'' is an edge-weighted [[undirected graph]], all edge weights are positive, and ''d''(''u'',''v'') is the weight of the [[widest path problem|minimax path]] between ''u'' and ''v'' (that is, the largest weight of an edge, on a path chosen to minimize this largest weight), then the vertices of the graph, with distance measured by ''d'', form an ultrametric space, and all finite ultrametric spaces may be represented in this way.<ref>{{citation
| last = Leclerc | first = Bruno
| mr = 623034
| issue = 73
| journal = Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
| language = French
| pages = 5–37, 127
| title = Description combinatoire des ultramétriques
| year = 1981}}.</ref>
 
== Applications==
A [[contraction mapping]] may then be thought of as a way of approximating the final result of a computation (which can be guaranteed to exist by the [[Banach fixed point theorem]]). Similar ideas can be found in [[domain theory]]. [[P-adic analysis]] makes heavy use of the ultrametric nature of the [[p-adic metric]].
 
Applications are also known in solid-state physics, namely in the treatment of [[spin glasses]] by the replica-theory of [[Giorgio Parisi]] and coworkers,<ref>Mezard, M; Parisi, G; and Virasoro, M: ''SPIN GLASS THEORY AND BEYOND'', World Scientific, 1986. ISBN 978-9971-5-0116-7</ref> and also in the theory of aperiodic solids.<ref name=physics_apps>{{cite journal|last=Rammal|first=R.|coauthors=Toulouse, G., Virasoro, M.|title=Ultrametricity for physicists|journal=Reviews of Modern Physics|year=1986|volume=58|issue=3|pages=765–788|doi=10.1103/RevModPhys.58.765|url=http://rmp.aps.org/abstract/RMP/v58/i3/p765_1|accessdate=20 June 2011}}</ref>  
 
Ultrametric distances are also utilized in [[Taxonomy (biology)|taxonomy]] and [[phylogenetic tree]] construction using the [[UPGMA]] and [[WPGMA]] methods.<ref name=physics_apps/>
 
== References ==
{{reflist}}
 
==Further reading==
{{commons category|Non-Archimedean geometry}}
*{{citation
| last = Kaplansky | first = I. | author-link = Irving Kaplansky
| isbn = 0-8218-2694-8
  | publisher = AMS Chelsea Publishing
| title = Set Theory and Metric Spaces
| year = 1977}}.
 
{{DEFAULTSORT:Ultrametric Space}}
[[Category:Metric geometry]]
[[Category:Metric spaces]]

Revision as of 13:50, 16 February 2014

リビングルームで

「蘭シュウは、。ない '秦Yuは深呼吸を取った。

「愚かな少年、あなたは結婚に子供と一緒に立って、どのような場合に、私は人々が土地を破壊させません。これらのサードの場所 クリスチャンルブタン 靴 メンズ。汝

は、被写体さて、ストップ気性でなければならない。」蘭シュウ甘く子供を叱るかのように。
「いいえ。」


秦ゆう私は彼の気性を失っていないんだけど、、あなたは古代のアイアンウッドツリーを言うのを聞くために」、彼の頭を横に振った、私はすでに持っているものと考える

は把握第三の場所を獲得した。 '

'ああ？'蘭シュウ少し驚い、 クリスチャンルブタン 価格 '何？'

「Forber。 クリスチャンルブタン 偽物 '秦Yuは突然、大声で叫んだ秦ゆうがあっForber所有者、フーボー秦ゆう突然感知された心であり、
リビングルームで
クリスチャンルブタンジャパン。

江蘭心まで推測、秦YuはForberが何をしましょう​​ クリスチャンルブタン ブーツ？

「蘭シュウは、私と一緒に来る。 '秦ゆう蘭シュウは、その後の動きを気にし、右折言う。

江蘭シフトが空間的に感じる 相关的主题文章：

秦Yuは地味なこの時間でした

図ボリュームは、驚くべきことではないので、多くのアーティファクトを持っています。軽く クリスチャンルブタン ブーツ '華ヤン。

秦ゆう、現時点ではこれらの言葉を見て、だけでなく、無謀、物語に耳を傾け、誰の心には多くのことを知っている。

「中央皇帝西安に対しては、最大のアーティファクト皇帝西安を誇っています。 クリスチャンルブタン 銀座 '秦ゆう心があっても妖精の魔法の鬼の世界の巨人がうらやましく作るために、その神TUファン」、書き留め、貴重度は想像することができます。 クリスチャンルブタン 店舗 東京 '
動かなかった
そして今回は、無謀な、 クリスチャンルブタン アウトレット。彼らは、ペースなしで移動することはしなかった青風水を、この記念碑にも手書き文字を持っていますが、風が蘭ペースを移動しない、無謀推測していた クリスチャンルブタン 東京。

「蘭風水熊、中央皇帝西安。それを書く左ネガティブだけでなく、反対」青鳳声を出して「リマインダー」の道。

秦Yuは地味なこの時間でした。

瞬間に風が語られたながら,,秦ゆう蘭心含み笑いを「もともとこれらの人々は、ああ私を待って。 '：'かというと、もちろん、私はこれを知っているが、私は非常に私を怖がって見えたあなたのことを考える」

蘭風が見に群衆に話を聞いた。

誰もが笑顔 相关的主题文章：

and walked toward the northwest Sheng Huang Temple.

That ...... '

'I understand.'

week was laugh loudly. クリスチャンルブタン バッグ Eyes with cold. 'Estimate That Qin Yu cast FIGHTING space. Then been hiding FIGHTING space bar. This 'FIGHTING space' supernatural powers. Places really quite tough, but that Qin Yu, was also willing to faint bird turtle as general I do not know, he chickened bird this turtle to hide what day! '

week was suddenly remembered what it lightly against Liu Shan said:. 'you go back to it.'

'yes. Highness.' Liu Shan immediately back away from the living room. The week was thought for a moment in the living room, 'Fu Huang these days, has been looking at other space Qin クリスチャンルブタン 価格 Yu, now seems to think this fear クリスチャンルブタン ブーツ is Fu Huang wrong.'

weeks and then I was out of the living room, and walked toward the northwest Sheng Huang Temple.
eastern
spirit world, クリスチャンルブタン 偽物 Yuchi town.

into a restaurant.

punishment far are accompanied his wife Ziyun eating クリスチャンルブタン メンズ places. They also heard the couple these days Qin Yu to do. Qin Yu heart also worried. 相关的主题文章：

already guessed everything

Dillon touches his face calm, asked: '? Qin Yu, you said it seems that you had witnessed in general, how do you know.'

'Master told.' Qin クリスチャンルブタン ブーツ Yu replied smiling.

Dillon continue asked: 'Master And how do you know if you Master spiritual knowledge so amazing, you can even find things through, why not kill the mysterious man to win an eighth クリスチャンルブタン ブーツ of the Yu Jian yet.? '

'Master brave terrible, but how will care Nabing eighth to Yu Jian? these nine legendary sword, Master has long been known that when he found black Yuhe spiritual knowledge, already guessed everything, Master Training Heaven , extremely aloof and mysterious is the mysterious person who claimed to fate. 'Qin Yu suddenly sigh.

'Alas, the war when I do not クリスチャンルブタン セール know if I knew in advance, I asked Master for help, an estimated eight Highness will be able to クリスチャンルブタン ブーツ save their lives up. Master クリスチャンルブタン 直営店 though eccentric, for me it only a disciple but also with great interest.'

Dillon nodded.

suddenly -

'youngest, you 相关的主题文章：

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