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| In [[mathematics]], the '''equilateral dimension''' of a [[metric space]] is the maximum number of points that are all at equal distances from each other.<ref name="DD09">{{harvtxt|Deza|Deza|2009}}</ref> Equilateral dimension has also been called "[[Metric dimension (disambiguation)|metric dimension]]", but the term "metric dimension" also has many other inequivalent usages.<ref name="DD09"/> The equilateral dimension of a ''d''-dimensional [[Euclidean space]] is {{nowrap|''d'' + 1}}, and the equilateral dimension of a ''d''-dimensional [[vector space]] with the [[Chebyshev distance]] (L<sup>∞</sup> norm) is 2<sup>''d''</sup>. However, the equilateral dimension of a space with the [[Taxicab geometry|Manhattan distance]] (L<sup>1</sup> norm) is not known; '''Kusner's conjecture''', named after [[Robert B. Kusner]], states that it is exactly 2''d''.<ref>{{harvtxt|Guy|1983}}; {{harvtxt|Koonen|Laurent|Schrijver|2000}}.</ref>
| | A cordless influence wrench is a variety of tool that is designed to supply higher torque energy with the power coming from a rechargeable battery pack. If Dale wants to just rotate tires with his tool, the Craftsman C3 impact will do that straightforward for $one hundred. A corded effect wrench could be an selection for the property user if he's concerned with longevity vs battery tools' lifespan and later costs Skil Oscillating Software Critiques of replacement batteries. Realistically, if he's a household user and just attempting to get past that his pancake compressor cannot run an influence wrench, I'd strongly advise utilizing a plug-in tool more than a [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=cordless+model&Submit=Go cordless model].<br><br><br><br>I would appreciate to see an update this summer on this article to include things like Snap On's new providing, the CT7850 - which supposedly ups the torque to 450ft-lbs. Also, for a reference, to measure a pneumatic influence wrench. The Household Depot presents additional than two,000 Eco Selections items on the internet and in store that can enable you save funds and are better selections for the environment. Please pay a visit to your nearest Home Depot place for particulars. If you want an effect to be reliable on maintain on searching.<br><br>If you want a wrench that you can use for your household, a 12V influence wrench could possibly be all you will need. But just to be clear, the 12V cordless version may be the lowest voltage but this does not imply that this device is weak. It 2013 Oscillating Software Testimonials can nonetheless do all the tasks that are anticipated from any common impact wrench. If you liked this posting and you would like to acquire additional facts with regards to [http://www.bestoscillatingtoolreviews.com/ BEST OSCILLATING TOOL REVIEWS – MULTI TOOL] kindly stop by the internet site. For this model of cordless influence wrench, you can anticipate to spend about $60 to as substantially as $800. Web pages like� are really valuable in a search for the ideal cordless influence wrench.<br><br>A 3/8" Ingersoll Rand air ratchet wrench is good tool to begin with in your collection. You can get this tool into tight places in your automobile and get those challenging nuts loosened. Employing an air ratchet wrench is like going from an old handheld screwdriver to an 18v cordless drill. Look into this page to understand additional: Influence Wrenches on Decide on exactly what you choose in the trigger, rocker and / or paddle lever.<br><br>Specialists looking for a versatile cordless screwdriver will be entirely happy with a cordless influence wrench from the Bosch Power Tools series. The tiny, potent tools for tough applications: with a torque of up to 650 Nm, the Bosch cordless impact drivers are actually effective tools when it comes to fitting and loosening screws. Cordless effect wrenches are also safer in the workplace as they leave no dangling cables to trip more than.<br><br>Effect ratchets use socket attachments so that they can perform on a range of bolt and nut sizes. The primary difference involving a regular effect wrench and an effect ratchet wrench is that the ratchet can slip into smaller spaces that the wrench can not and it consists of a toothed wheel that makes it possible for movement in one path only. Influence drivers are particularly made to do a single job, and that is to drill screws. Effect wrenches are extensively made use of in numerous industries. |
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| ==Lebesgue spaces== | |
| The equilateral dimension has been particularly studied for [[Lp space|Lebesgue spaces]], finite-dimensional [[normed vector space]]s with the L<sup>''p''</sup> norm
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| :<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\cdots+|x_d|^p\right)^{1/p}.</math>
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| The equilateral dimension of L<sup>''p''</sup> spaces of dimension ''d'' behaves differently depending on the value of ''p'':
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| *For ''p'' = 1, the L<sup>''p''</sup> norm gives rise to [[Taxicab geometry|Manhattan distance]]. In this case, it is possible to find 2''d'' equidistant points, the vertices of an axis-aligned [[cross polytope]]. The equilateral dimension is known to be exactly 2''d'' for {{nowrap|''d'' ≤ 4}},<ref>{{harvtxt|Bandelt|Chepoi|Laurent|1998}}; {{harvtxt|Koonen|Laurent|Schrijver|2000}}.</ref> and to be upper bounded by {{nowrap|O(''d'' log ''d'')}} for any ''d''.<ref name="AP03">{{harvtxt|Alon|Pudlák|2003}}.</ref> Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly 2''d'';<ref name="Guy83"/> this suggestion (together with a related suggestion for the equilateral dimension when ''p'' > 2) has come to be known as '''Kusner's conjecture'''.
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| *For 1 < ''p'' < 2, the equilateral dimension is at least {{nowrap|(1 + ε)''d''}} where ε is a constant that depends on ''p''.<ref name="S04">{{harvtxt|Swanepoel|2004}}.</ref>
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| *For ''p'' = 2, the L<sup>''p''</sup> norm is the familiar [[Euclidean distance]]. The equilateral dimension of ''d''-dimensional [[Euclidean space]] is {{nowrap|''d'' + 1}}: the {{nowrap|''d'' + 1}} vertices of an [[equilateral triangle]], [[regular tetrahedron]], or higher-dimensional regular [[simplex]] form an equilateral set, and every equilateral set must have this form.<ref name="Guy83">{{harvtxt|Guy|1983}}.</ref>
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| *For 2 < ''p'' < ∞, the equilateral dimension is at least {{nowrap|''d'' + 1}}: for instance the ''d'' [[basis vector]]s of the vector space together with another vector of the form {{nowrap|(−''x'', −''x'', ...)}} for a suitable choice of ''x'' form an equilateral set. Kusner's conjecture states that in these cases the equilateral dimension is exactly {{nowrap|''d'' + 1}}. Kusner's conjecture has been proven for the special case that {{nowrap|1=''p'' = 4}}.<ref name="S04"/> When ''p'' is an odd integer the equilateral dimension is upper bounded by {{nowrap|O(''d'' log ''d'')}}.<ref name="AP03"/>
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| *For ''p'' = ∞ (the limiting case of the ''L''<sup>''p''</sup> norm for finite values of ''p'', in the limit as ''p'' grows to infinity) the ''L''<sup>''p''</sup> norm becomes the [[Chebyshev distance]], the maximum absolute value of the differences of the coordinates. For a ''d''-dimensional vector space with the Chebyshev distance, the equilateral dimension is 2<sup>''d''</sup>: the 2<sup>''d''</sup> vertices of an axis-aligned [[hypercube]] are at equal distances from each other, and no larger equilateral set is possible.<ref name="Guy83"/>
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| ==Normed vector spaces==
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| Equilateral dimension has also been considered for [[normed vector space]]s with norms other than the [[Lp space|L<sup>''p''</sup>]] norms. The problem of determining the equilateral dimension for a given norm is closely related to the [[kissing number problem]]: the kissing number in a normed space is the maximum number of disjoint translates of a unit ball that can all touch a single central ball, whereas the equilateral dimension is the maximum number of disjoint translates that can all touch each other.
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| For a normed vector space of dimension ''d'', the equilateral dimension is at most 2<sup>''d''</sup>; that is, the L<sup>∞</sup> norm has the highest equilateral dimension among all normed spaces.<ref name="Petty71">{{harvtxt|Petty|1971}}.</ref> {{harvtxt|Petty|1971}} asked whether every normed vector space of dimension ''d'' has equilateral dimension at least {{nowrap|''d'' + 1}}, but this remains unknown. There exist normed spaces in any dimension for which certain sets of four equilateral points cannot be extended to any larger equilateral set<ref name="Petty71"/> but these spaces may have larger equilateral sets that do not include these four points. For norms that are sufficiently close in [[Banach–Mazur compactum|Banach–Mazur distance]] to an L<sup>''p''</sup> norm, Petty's question has a positive answer: the equilateral dimension is at least {{nowrap|''d'' + 1}}.<ref name="sv08">{{harvtxt|Swanepoel|Villa|2008}}.</ref>
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| It is not possible for high-dimensional spaces to have bounded equilateral dimension: for any integer ''k'', all normed vector spaces of sufficiently high dimension have equilateral dimension at least ''k''.<ref>{{harvtxt|Braß|1999}}; {{harvtxt|Swanepoel|Villa|2008}}.</ref> more specifically, according to a variation of [[Dvoretzky's theorem]] by {{harvtxt|Alon|Milman|1983}}, every ''d''-dimensional normed space has a ''k''-dimensional subspace that is close either to a Euclidean space or to a Chebyshev space, where
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| :<math>k\ge\exp(c\sqrt{\log d})</math> | |
| for some constant ''c''. Because it is close to a Lebesgue space, this subspace and therefore also the whole space contains an equilateral set of at least ''k'' + 1 points. Therefore, the same superlogarithmic dependence on ''d'' holds for the lower bound on the equilateral dimension of ''d''-dimensional space.<ref name="sv08"/>
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| ==Riemannian manifolds==
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| For any ''d''-dimensional [[Riemannian manifold]] the equilateral dimension is at least {{nowrap|''d'' + 1}}.<ref name="Guy83"/> For a ''d''-dimensional [[sphere]], the equilateral dimension is {{nowrap|''d'' + 2}}, the same as for a Euclidean space of one higher dimension into which the sphere can be embedded.<ref name="Guy83"/> At the same time as he posed Kusner's conjecture, Kusner asked whether there exist Riemannian metrics with bounded dimension as a manifold but arbitrarily high equilateral dimension.<ref name="Guy83"/>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation
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| | last1 = Alon | first1 = N. | author1-link = Noga Alon
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| | last2 = Milman | first2 = V. D. | author2-link = Vitali Milman
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| | doi = 10.1007/BF02804012
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| | mr = 720303
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| | issue = 4
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| | journal = Israel Journal of Mathematics
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| | pages = 265–280
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| | title = Embedding of <math>\scriptstyle l^{k}_{\infty}</math> in finite-dimensional Banach spaces
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| | volume = 45
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| | year = 1983}}.
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| *{{citation
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| | volume = 13
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| | year = 2003}}.
| |
| *{{citation
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| | last1 = Bandelt | first1 = Hans-Jürgen | |
| | last2 = Chepoi | first2 = Victor
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| | last3 = Laurent | first3 = Monique
| |
| | doi = 10.1007/PL00009370
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| | issue = 4
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| | journal = Discrete and Computational Geometry
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| | mr = 1620076
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| | pages = 595–604
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| | title = Embedding into rectilinear spaces
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| | url = http://www.lif-sud.univ-mrs.fr/~chepoi/rectilinear.pdf
| |
| | volume = 19
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| | year = 1998}}.
| |
| *{{citation
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| | last = Braß | first = Peter
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| | issue = 2
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| | journal = Contributions to Algebra and Geometry
| |
| | mr = 1720106
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| | pages = 303–307
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| | title = On equilateral simplices in normed spaces
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| | url = http://www.emis.ams.org/journals/BAG/vol.40/no.2/3.html
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| | volume = 40
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| | year = 1999}}.
| |
| *{{citation
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| | last1 = Deza | first1 = Michel Marie | author1-link = Michel Deza
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| | last2 = Deza | first2 = Elena
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| | page = 20
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| | publisher = Springer-Verlag
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| | title = Encyclopedia of Distances
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| | year = 2009}}.
| |
| *{{citation
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| | last = Guy | first = Richard K. | authorlink = Richard K. Guy
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| | issue = 3
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| | journal = American Mathematical Monthly
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| | mr = 1540158
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| | pages = 196–200
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| | title = An olla-podrida of open problems, often oddly posed
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| | jstor = 2975549
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| | volume = 90
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| | year = 1983}}.
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| *{{citation
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| | last3 = Schrijver | first3 = Alexander | author3-link = Alexander Schrijver
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| | doi = 10.1023/A:1008391712305
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| | journal = Designs, Codes and Cryptography
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| | pages = 149–164
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| | title = Equilateral dimension of the rectilinear space
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| | volume = 21
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| | year = 2000}}.
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| *{{citation
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| | last = Petty | first = Clinton M. | |
| | doi = 10.1090/S0002-9939-1971-0275294-8
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| | issue = 2
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| | journal = Proceedings of the American Mathematical Society
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| | pages = 369–374
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| | title = Equilateral sets in Minkowski spaces
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| | year = 1971}}.
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| *{{citation
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| | doi = 10.1007/s00013-003-4840-8
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| | arxiv = math/0309317
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| *{{citation
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| | year = 2008}}.
| |
| | |
| [[Category:Metric geometry]]
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| [[Category:Dimension theory]]
| |
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