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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.
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| == Formal definition ==
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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]])
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| :<math>d\colon M \times M \rightarrow \mathbb{R}</math>
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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:
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| # <math>d(x, y) \ge 0</math>
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| # <math>d(x, y) = 0</math> [[iff]] <math>x=y</math>
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| # <math>d(x, y) = d(y, x)</math> ('''symmetry''')
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| # <math>d(x, z) \le \max \left\{ d(x, y), d(y, z) \right\}</math> ('''strong triangle''' or '''ultrametric inequality''').
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| 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:
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| : <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>.
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| ==Properties==
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| [[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>:
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| * 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>.
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| 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>.
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| * 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>.
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| * 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>.
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| 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.
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| == Examples ==
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| # The [[discrete metric]] is an ultrametric.
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| # 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.
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| # The [[p-adic numbers]] form a complete ultrametric space.
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| # 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]]. — 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.)
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| # 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
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| | last = Leclerc | first = Bruno
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| | mr = 623034
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| | issue = 73
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| | journal = Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
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| | language = French
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| | pages = 5–37, 127
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| | title = Description combinatoire des ultramétriques
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| | year = 1981}}.</ref>
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| == Applications==
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| 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]].
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| 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>
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| Ultrametric distances are also utilized in [[Taxonomy (biology)|taxonomy]] and [[phylogenetic tree]] construction using the [[UPGMA]] and [[WPGMA]] methods.<ref name=physics_apps/>
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| == References ==
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| {{reflist}}
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| ==Further reading==
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| {{commons category|Non-Archimedean geometry}}
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| *{{citation
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| | last = Kaplansky | first = I. | author-link = Irving Kaplansky
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| | isbn = 0-8218-2694-8
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| | publisher = AMS Chelsea Publishing
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| | title = Set Theory and Metric Spaces
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| | year = 1977}}.
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| {{DEFAULTSORT:Ultrametric Space}}
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| [[Category:Metric geometry]]
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| [[Category:Metric spaces]]
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These are troubling economic times for many businesses. However, there are certain equipment upgrades that are necessary even when money is tight. If your company is looking to save money by shopping around for the best deal when replacing work surfaces like solid epoxy resin countertops, the following is an affordable solution.
Affordable Solutions for Solid Epoxy Resin Countertops
During a recession, one of the easiest ways to make cost-effective improvements to your facility is to invest in replacement countertops, as opposed to buying a whole new table or workbench, for example.
OnePointe Solutions supplies high-quality, epoxy resin replacement countertops and new custom epoxy resin worktables with exceptional work surfaces. And if customization is what you need, we can handle that for you too. By constructing solid epoxy resin tops and workbenches to your exact specifications, you save valuable time and money.
Why Solid Epoxy Resin Tops/Countertops are a Superior Investment
To understand fully why epoxy resin tabletops are such a sound investment, lets look at a number of features of solid epoxy resin work surfaces:
They are strong resistance too many chemicals, Water Resistant, Extremely hard material, Durable and easy to clean, Approved for Food Services (NSF), Non Porous, Never Requires Sealing, Made in the USA.
In addition, solid epoxy resin countertops are stain resistant, moisture resistant and resist chipping and scratching.
If you adored this article and you also would like to collect more info relating to resinas (click homepage) please visit our web site. For all of these reasons, solid epoxy resin countertops last for a long time, making them a sound investment in any economy.