Very Long Baseline Array - Revision history

en>Barrel Roll: Added a period to the end of the first section.

2014-12-15T16:21:55Z

Added a period to the end of the first section.

← Older revision		Revision as of 18:21, 15 December 2014
Line 1:		Line 1:
	~~In [[geometry]], two sets of points in~~ a ~~two-dimensional space are~~ '~~''linearly separable''' if they can be completely separated by a single line~~. ~~In general, two point sets are linearly separable in ''n''-dimensional space if they~~ can ~~be separated by a [[hyperplane]].~~		Eusebio Stanfill is what's displayed on my birth document although it is certainly not the name on particular birth certificate. Idaho is our birth city. I work as an pay for clerk. As a man what Post really like is representing but I'm thinking on starting something new. You can find my website here: http://Prometeu.net/<br><br>My website - [http://Prometeu.net/ clash of clans cheats deutsch]

	~~In more mathematical terms~~: ~~Let <math>X_{0}<~~/~~math> and <math>X_{1}<~~/~~math> be two sets of points in an n-dimensional space~~. ~~Then <math>X_{0}<~~/math> and <math>X_{1}</math> are linearly separable if there exists n+1 real numbers <math>w_{1}, w_{2},..,w_{n}, k</math>, such that every point <math>x \in X_{0}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i}\ge k</math> and every point <math>x \in X_{1}<~~/math~~> ~~satisfies~~ <~~math~~>~~\sum^{n}_{i=1} w_{i}x_{i} < k</math>, where <math>x_{i}</math> is the <math>i</math>-th component of <math>x</math>.~~

	~~Equivalently, two sets are linearly separable precisely when their respective [[convex hull]]s are [[disjoint sets\|disjoint]] (colloquially, do not overlap).~~

	~~== Example ==~~

	~~Three points in two classes ('+' and '~~-~~') are always linearly separable in two dimensions. This is illustrated by the three examples in the following figure:~~

	~~{\| align="center" border="0" cellpadding="4" cellspacing="0"~~
	~~\| align="center" \|~~ [~~[File~~:~~VC1~~.~~svg]]~~
	~~\| align="center" \| [[File:VC2.svg]]~~
	~~\| align="center" \| [[File:VC3.svg]]~~
	\|}

	~~However, not all sets of four points are linearly separable in two dimensions. The following example would need '''two''' straight lines and thus is not linearly separable:~~

	~~{\| align="center" border="0" cellpadding="4" cellspacing="0"~~
	~~\| [[File:VC4.svg]]~~
	\|}

	~~== Linear separability of hypercubes in n dimensions ==~~

	~~{\| class="wikitable"~~
	~~\|+<small>Number of linearly separable Boolean hypercubes in each dimension<~~/~~small><ref>~~
	~~{{cite paper~~
	~~\| last=Gruzling~~
	~~\| first=Nicolle~~
	~~\| title=Linear separability~~ of ~~the vertices of an n-dimensional hypercube. M.Sc Thesis~~
	~~\| publisher= University of Northern British Columbia~~
	~~\| year=2006~~
	~~}}</ref> {{OEIS\|id=A000609}}~~
	~~!Dimension~~
	~~!Linearly separable Boolean hypercubes~~
	\|-
	~~\| 2 \|\| 14~~
	\|-
	~~\| 3 \|\| 104~~
	\|-
	~~\| 4 \|\| 1882~~
	\|-
	~~\| 5 \|\| 94572~~
	\|-
	~~\| 6 \|\| 15028134~~
	\|-
	~~\| 7 \|\| 8378070864~~
	\|-
	~~\| 8 \|\| 17561539552946~~
	\|-
	~~\| 9 \|\| 144130531453121108~~
	\|}

	~~== Usage ==~~
	~~Linear separability allows simple [[Classification in machine learning~~]].

	~~== See also ==~~

	* [[Vapnik–Chervonenkis dimension]]

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Multi-dimensional geometry]]~~
	~~[[Category:Convex analysis]]~~


	~~{{Geometry-stub}}~~

89.98.76.207: /* Observations by the VLBA */ Group HSA information together

2014-02-03T21:21:11Z

Observations by the VLBA: Group HSA information together

← Older revision		Revision as of 23:21, 3 February 2014
Line 1:		Line 1:
	~~Introduction~~. ~~The author~~'~~s name is Dalton but it~~'~~s not the~~ a ~~lot of~~ [~~http~~://~~statigr~~.am/~~tag~~/~~masucline masucline] name out presently~~. ~~The ideal hobby for him then his kids is to finally drive~~ and ~~he's been doing it for several years. He actually works as a [http:~~//~~Photo.net~~/~~gallery~~/~~tag~~-~~search~~/~~search?query_string=cashier cashier~~]. ~~His wife~~ and ~~him chose to measure~~ in ~~South Carolina and as well as his family loves~~ the ~~program~~. ~~Go to his website acquire out more~~: ~~http~~:/~~/circuspartypanama.com~~<br><br>~~my website;~~ [~~http~~:~~//circuspartypanama.com clash of clans hack android download~~]		In [[geometry]], two sets of points in a two-dimensional space are '''linearly separable''' if they can be completely separated by a single line. In general, two point sets are linearly separable in ''n''-dimensional space if they can be separated by a [[hyperplane]].

			In more mathematical terms: Let <math>X_{0}</math> and <math>X_{1}</math> be two sets of points in an n-dimensional space. Then <math>X_{0}</math> and <math>X_{1}</math> are linearly separable if there exists n+1 real numbers <math>w_{1}, w_{2},..,w_{n}, k</math>, such that every point <math>x \in X_{0}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i}\ge k</math> and every point <math>x \in X_{1}</math> satisfies <math>\sum^{n}_{i=1} w_{i}x_{i} < k</math>, where <math>x_{i}</math> is the <math>i</math>-th component of <math>x</math>.

			Equivalently, two sets are linearly separable precisely when their respective [[convex hull]]s are [[disjoint sets\|disjoint]] (colloquially, do not overlap).

			== Example ==

			Three points in two classes ('+' and '-') are always linearly separable in two dimensions. This is illustrated by the three examples in the following figure:

			{\| align="center" border="0" cellpadding="4" cellspacing="0"
			\| align="center" \| [[File:VC1.svg]]
			\| align="center" \| [[File:VC2.svg]]
			\| align="center" \| [[File:VC3.svg]]
			\|}

			However, not all sets of four points are linearly separable in two dimensions. The following example would need '''two''' straight lines and thus is not linearly separable:

			{\| align="center" border="0" cellpadding="4" cellspacing="0"
			\| [[File:VC4.svg]]
			\|}

			== Linear separability of hypercubes in n dimensions ==

			{\| class="wikitable"
			\|+<small>Number of linearly separable Boolean hypercubes in each dimension</small><ref>
			{{cite paper
			\| last=Gruzling
			\| first=Nicolle
			\| title=Linear separability of the vertices of an n-dimensional hypercube. M.Sc Thesis
			\| publisher= University of Northern British Columbia
			\| year=2006
			}}</ref> {{OEIS\|id=A000609}}
			!Dimension
			!Linearly separable Boolean hypercubes
			\|-
			\| 2 \|\| 14
			\|-
			\| 3 \|\| 104
			\|-
			\| 4 \|\| 1882
			\|-
			\| 5 \|\| 94572
			\|-
			\| 6 \|\| 15028134
			\|-
			\| 7 \|\| 8378070864
			\|-
			\| 8 \|\| 17561539552946
			\|-
			\| 9 \|\| 144130531453121108
			\|}

			== Usage ==
			Linear separability allows simple [[Classification in machine learning]].

			== See also ==

			* [[Vapnik–Chervonenkis dimension]]

			== References ==
			{{reflist}}

			[[Category:Multi-dimensional geometry]]
			[[Category:Convex analysis]]


			{{Geometry-stub}}

en>Sleinen: /* Observations by the VLBA */ fixed typo (closing brace -> parenthesis)

2012-08-20T11:22:13Z

Observations by the VLBA: fixed typo (closing brace -> parenthesis)

New page

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