Linear network coding - Revision history

en>Pixelpapst: dead link, point to published version of article. should become full cite

2014-12-07T18:38:49Z

dead link, point to published version of article. should become full cite

← Older revision		Revision as of 20:38, 7 December 2014
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	I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. ~~Alaska is exactly where~~ I've always ~~been~~ living. ~~She~~ is ~~truly fond of caving but~~ she ~~doesn't have the time recently~~. ~~Distributing manufacturing is exactly where my main income arrives from and it~~'s ~~some thing I truly enjoy~~.<br><br>~~Here is~~ my blog ~~post~~ - ~~clairvoyant~~ psychic ([http://~~1.234~~.36.~~240~~/~~fxac~~/~~m001_2~~/~~7330 1.234.36.240~~])		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. I've always loved living in Alaska. Doing ballet is something she would by no means give up. Since he was 18 he's been working as an information officer but he plans on altering it.<br><br>Also visit my blog - online psychic readings ([http://www.skullrocker.com/blogs/post/10991 additional hints])

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

2014-02-08T06:00:26Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9916)

← Older revision		Revision as of 08:00, 8 February 2014
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	In [[mathematics]], the '''Gibbs measure''', named after [[Josiah Willard Gibbs]], is a [[probability measure]] frequently seen in many problems of [[probability theory]] and [[statistical mechanics]]. It is the measure associated with the [[canonical ensemble]]. Gibbs measure implies the [[Markov property]] (a certain kind of statistical independence); and importantly, it implies the [[Hammersley–Clifford theorem]] that the energy function can be written as a multiplication of parts, thus leading to ~~its widespread appearance in many problems outside of [[physics]]~~, such as [[Hopfield network]]s, [[Markov network]]s, and [[Markov logic network]]s. In addition, the Gibbs measure is the unique measure that maximizes the [[entropy (general concept)\|entropy]] for a given expected energy; thus, the Gibbs measure underlies [[maximum entropy method]]s and ~~the algorithms derived therefrom.~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Alaska is exactly where I've always been living. She is truly fond of caving but she doesn't have the time recently. Distributing manufacturing is exactly where my main income arrives from and it's some thing I truly enjoy.<br><br>Here is my blog post - clairvoyant psychic ([http://1.234.36.240/fxac/m001_2/7330 1.234.36.240])

	~~The measure gives the probability of the system~~ '~~'X'' being in state ''x'' (equivalently, of the [[random variable]] ''X'' having value ''x'') as~~

	~~:<math>P(X=x) = \frac{1}{Z(\beta)} \exp \left( - \beta E(x) \right).</math>~~

	Here, <math>E(x)</math> is a function from the space of states to the real numbers; in physics applications, <math>E(x)</math> is interpreted as the energy of the configuration ''x''. The parameter <math>\beta</math> is a free parameter; in physics, it ~~is the [[inverse temperature]]. The [[normalizing constant]] <math>Z(\beta)</math> is the [[partition function (mathematics)\|partition function]].~~

	~~==Markov property==~~
	~~An example of the [[Markov property]] of the Gibbs measure can be seen in the [[Ising model]]~~. ~~Here, the probability of a given spin <math>\sigma_k</math> being in state ''s''~~ is~~, in principle, dependent on all other spins in the model; thus one writes~~

	~~:<math>P(\sigma_k = s\|\sigma_j,\, j\ne k)</math>~~

	~~for this probability. However, the interactions in the Ising model are nearest-neighbor interactions, and thus, one actually has~~

	~~:<math>P(\sigma_k = s\|\sigma_j,\, j\ne k) =~~
	~~P(\sigma_k = s\|\sigma_j,\, j\isin N_k)~~
	~~</math>~~

	where ~~<math>N_k</math> is the set of nearest neighbors of site <math>k</math>. That is, the probability at site <math>k</math> depends~~ '~~'only'' on the nearest neighbors~~. ~~This last equation~~ is ~~in the form~~ of ~~a Markov-type statistical independence. Measures with this property are sometimes called [[Markov random field]]s. More strongly, the converse is also true:~~ ''any'' positive probability distribution (non-zero everywhere) having the Markov property can be represented with the Gibbs measure, given an appropriate energy function;<ref>Ross Kindermann and J. Laurie Snell, [http://www.ams.org/online_bks/conm1/ Markov Random Fields and Their Applications] (1980) American Mathematical Society, ISBN 0-8218-5001-6</ref> this is the ~~[[Hammersley–Clifford theorem]]~~.

	~~==Gibbs measure on lattices==~~
	~~What follows~~ is ~~a formal definition for the special case of a random field on a group lattice. The idea of a Gibbs measure is, however, much more general than this.~~

	~~The definition of a '''Gibbs random field''' on a [[lattice (group)\|lattice]] requires some terminology:~~

	* The '''lattice''': A countable set <math>\mathbb{L}</math>.

	* The '''single-spin space''': A [[probability space]] <math>(S,\mathcal{S},\lambda)</math>.

	* The '''[[configuration space]]''': <math>(\Omega, \mathcal{F})</math>, where ~~<math>\Omega = S^{\mathbb{L}}</math>~~ and ~~<math>\mathcal{F} = \mathcal{S}^{\mathbb{L}}</math>.~~

	* Given a configuration <math>\omega \in \Omega</math> and a subset <math>\Lambda \subset \mathbb{L}</math>, the restriction of <math>\omega</math> to <math>\Lambda</math> is <math>\omega_\Lambda = (\omega(t))_{t\in\Lambda}</math>. If <math>\Lambda_1\cap\Lambda_2=\emptyset</math> and <math>\Lambda_1\cup\Lambda_2=\mathbb{L}</math>, then the configuration <math>\omega_{\Lambda_1}\omega_{\Lambda_2}</math> is the configuration whose restrictions to <math>\Lambda_1</math> and <math>\Lambda_2</math> are <math>\omega_{\Lambda_1}</math> and <math>\omega_{\Lambda_2}</math>, respectively. These will be used to define [[cylinder set]]s~~, below~~.

	* The set <~~math~~>~~\mathcal{L}~~<~~/math> of all finite subsets of <math>\mathbb{L}</math>.~~

	* For each subset <math>\Lambda\subset\mathbb{L}</math>, <math>\mathcal{F}_\Lambda</math> is ~~the [[sigma algebra\|<math>\sigma</math>~~-~~algebra]] generated by the family of functions <math>(\sigma(t))_{t\in\Lambda}</math>, where <math>\sigma~~(~~t)(\omega)=\omega(t)</math>. This sigma-algebra is just the algebra of~~ [~~[cylinder set]]s on the lattice.~~

	* The '''[[potential]]''': ~~A family <math>\Phi=(\Phi_A)_{A\in\mathcal{L}}<~~/~~math> of functions <math>\Phi_A:\Omega \to \mathbb{R}<~~/~~math> such that~~
	*# For each <math>A\in\mathcal{L}</math>, <math>\Phi_A</math> is <math>\mathcal{F}_A</math>-[[measurable]].
	*# For all <math>\Lambda\in\mathcal{L}</math> and <math>\omega\in\Omega</math>, the series <math>H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega)</math> exists.

	* The '''[[Hamiltonian mechanics#Mathematical formalism\|Hamiltonian]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math>, for the potential <math>\Phi</math>, is defined by

	~~<center><math>H_\Lambda^\Phi(\omega \| \bar\omega) = H_\Lambda^\Phi(\omega_\Lambda\bar\omega_{\Lambda^c})</math>,</center>~~

	~~:where <math>\Lambda^c = \mathbb{L}\setminus\Lambda</math>.~~

	* The '''[[partition function (mathematics)\|partition function]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math> and inverse temperature <math>\beta\in\mathbb{R}_+</math> (for the potential <math>\Phi</math> and <math>\lambda</math>) is defined by

	~~<center><math>Z_\Lambda^\Phi(\bar\omega) = \int \lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega \| \bar\omega))</math>~~.~~</center>~~

	~~:Here <math>\lambda^\Lambda(\mathrm{d}\omega)</math> is the product measure <math>\prod_{t\in\Lambda}\lambda(\mathrm{d}\omega(t))</math>~~.

	:A potential <math>\Phi</math> is <math>\lambda</math>-admissible if <math>Z_\Lambda^\Phi(\bar\omega)</math> is finite for all <math>\Lambda\in\mathcal{L}</math>, <math>\bar\omega\in\Omega</math> and <math>\beta>0</math>.

	~~A [[probability measure]] <math>\mu<~~/~~math> on <math>(\Omega,\mathcal{F})</math> is a '''Gibbs measure''' for a <math>\lambda<~~/~~math>-admissible potential <math>\Phi<~~/~~math> if it satisfies the '''[[Dobrushin-Lanford-Ruelle]] (DLR) equations'''~~

	~~<center><math>\int \mu(\mathrm{d}\bar\omega)Z_\Lambda^\Phi(\bar\omega)^{-~~1~~} \int\lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega \| \bar\omega)) 1_A(\omega_\Lambda\bar\omega_{\Lambda^c}) = \mu(A)</math>,</center>~~

	~~: for all <math>A\in\mathcal{F}</math> and <math>\Lambda\in\mathcal{L}</math>~~.

	~~===An example===~~
	To help understand the above definitions, here are the corresponding quantities in the important example of the [[Ising model]] with nearest-neighbour interactions (coupling constant <math>J</math>) and a magnetic field (<math>h</math>), on <math>\mathbb{Z}^d</math>:

	* The lattice is simply <math>\mathbb{L} = \mathbb{Z}^d</math>.
	* The single-spin space is <math>S=\{-1,1\}</math>.
	* The potential is given by
	~~<center><math>\Phi_A(\omega) = \begin{cases}~~
	~~-J\,\omega(t_1)\omega(t_2) & \mathrm{if\ } A=\{t_1,t_2\} \mathrm{\ with\ } \\|t_2-t_1\\|_1 = 1 \\~~
	~~-h\,\omega(t) & \mathrm{if\ } A=\{t\}\\~~
	~~0 & \mathrm{otherwise}~~
	~~\end{cases}</math></center>~~

	~~==See also==~~
	* [[Exponential family]]
	* [[Gibbs algorithm]]
	* [[Gibbs sampling]]
	* [[Interacting particle system]]
	* [[Stochastic cellular automata]]

	~~==References==~~
	~~<references/>~~
	* Georgii, H.-O. "Gibbs measures and phase transitions", de Gruyter, Berlin, 1988, 2nd edition 2011.

	~~{{Stochastic processes}}~~

	~~[[Category:Measures (measure theory~~)]]
	~~[[Category:Statistical mechanics]]~~

en>Ali fa1360: /* References */

2014-01-01T13:28:40Z

References

← Older revision		Revision as of 15:28, 1 January 2014
Line 1:		Line 1:
	The ~~writer~~ is called ~~Wilber Pegues~~. ~~I am~~ an ~~invoicing officer~~ and I'll be ~~promoted soon~~. The ~~favorite hobby for him and his children~~ is to ~~play lacross~~ and ~~he would never give it up~~. ~~North Carolina~~ is where we'~~ve been residing~~ for ~~years~~ and ~~will never move~~.<br><br>~~My webpage~~ ..~~. authentic psychic readings~~ ([~~http~~://~~netwk~~.~~hannam~~.ac.kr/xe/~~data_2~~/~~85669 More Signup bonuses~~])		In [[mathematics]], the '''Gibbs measure''', named after [[Josiah Willard Gibbs]], is a [[probability measure]] frequently seen in many problems of [[probability theory]] and [[statistical mechanics]]. It is the measure associated with the [[canonical ensemble]]. Gibbs measure implies the [[Markov property]] (a certain kind of statistical independence); and importantly, it implies the [[Hammersley–Clifford theorem]] that the energy function can be written as a multiplication of parts, thus leading to its widespread appearance in many problems outside of [[physics]], such as [[Hopfield network]]s, [[Markov network]]s, and [[Markov logic network]]s. In addition, the Gibbs measure is the unique measure that maximizes the [[entropy (general concept)\|entropy]] for a given expected energy; thus, the Gibbs measure underlies [[maximum entropy method]]s and the algorithms derived therefrom.

			The measure gives the probability of the system ''X'' being in state ''x'' (equivalently, of the [[random variable]] ''X'' having value ''x'') as

			:<math>P(X=x) = \frac{1}{Z(\beta)} \exp \left( - \beta E(x) \right).</math>

			Here, <math>E(x)</math> is a function from the space of states to the real numbers; in physics applications, <math>E(x)</math> is interpreted as the energy of the configuration ''x''. The parameter <math>\beta</math> is a free parameter; in physics, it is the [[inverse temperature]]. The [[normalizing constant]] <math>Z(\beta)</math> is the [[partition function (mathematics)\|partition function]].

			==Markov property==
			An example of the [[Markov property]] of the Gibbs measure can be seen in the [[Ising model]]. Here, the probability of a given spin <math>\sigma_k</math> being in state ''s'' is, in principle, dependent on all other spins in the model; thus one writes

			:<math>P(\sigma_k = s\|\sigma_j,\, j\ne k)</math>

			for this probability. However, the interactions in the Ising model are nearest-neighbor interactions, and thus, one actually has

			:<math>P(\sigma_k = s\|\sigma_j,\, j\ne k) =
			P(\sigma_k = s\|\sigma_j,\, j\isin N_k)
			</math>

			where <math>N_k</math> is the set of nearest neighbors of site <math>k</math>. That is, the probability at site <math>k</math> depends ''only'' on the nearest neighbors. This last equation is in the form of a Markov-type statistical independence. Measures with this property are sometimes called [[Markov random field]]s. More strongly, the converse is also true: ''any'' positive probability distribution (non-zero everywhere) having the Markov property can be represented with the Gibbs measure, given an appropriate energy function;<ref>Ross Kindermann and J. Laurie Snell, [http://www.ams.org/online_bks/conm1/ Markov Random Fields and Their Applications] (1980) American Mathematical Society, ISBN 0-8218-5001-6</ref> this is the [[Hammersley–Clifford theorem]].

			==Gibbs measure on lattices==
			What follows is a formal definition for the special case of a random field on a group lattice. The idea of a Gibbs measure is, however, much more general than this.

			The definition of a '''Gibbs random field''' on a [[lattice (group)\|lattice]] requires some terminology:

			* The '''lattice''': A countable set <math>\mathbb{L}</math>.

			* The '''single-spin space''': A [[probability space]] <math>(S,\mathcal{S},\lambda)</math>.

			* The '''[[configuration space]]''': <math>(\Omega, \mathcal{F})</math>, where <math>\Omega = S^{\mathbb{L}}</math> and <math>\mathcal{F} = \mathcal{S}^{\mathbb{L}}</math>.

			* Given a configuration <math>\omega \in \Omega</math> and a subset <math>\Lambda \subset \mathbb{L}</math>, the restriction of <math>\omega</math> to <math>\Lambda</math> is <math>\omega_\Lambda = (\omega(t))_{t\in\Lambda}</math>. If <math>\Lambda_1\cap\Lambda_2=\emptyset</math> and <math>\Lambda_1\cup\Lambda_2=\mathbb{L}</math>, then the configuration <math>\omega_{\Lambda_1}\omega_{\Lambda_2}</math> is the configuration whose restrictions to <math>\Lambda_1</math> and <math>\Lambda_2</math> are <math>\omega_{\Lambda_1}</math> and <math>\omega_{\Lambda_2}</math>, respectively. These will be used to define [[cylinder set]]s, below.

			* The set <math>\mathcal{L}</math> of all finite subsets of <math>\mathbb{L}</math>.

			* For each subset <math>\Lambda\subset\mathbb{L}</math>, <math>\mathcal{F}_\Lambda</math> is the [[sigma algebra\|<math>\sigma</math>-algebra]] generated by the family of functions <math>(\sigma(t))_{t\in\Lambda}</math>, where <math>\sigma(t)(\omega)=\omega(t)</math>. This sigma-algebra is just the algebra of [[cylinder set]]s on the lattice.

			* The '''[[potential]]''': A family <math>\Phi=(\Phi_A)_{A\in\mathcal{L}}</math> of functions <math>\Phi_A:\Omega \to \mathbb{R}</math> such that
			*# For each <math>A\in\mathcal{L}</math>, <math>\Phi_A</math> is <math>\mathcal{F}_A</math>-[[measurable]].
			*# For all <math>\Lambda\in\mathcal{L}</math> and <math>\omega\in\Omega</math>, the series <math>H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega)</math> exists.

			* The '''[[Hamiltonian mechanics#Mathematical formalism\|Hamiltonian]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math>, for the potential <math>\Phi</math>, is defined by

			<center><math>H_\Lambda^\Phi(\omega \| \bar\omega) = H_\Lambda^\Phi(\omega_\Lambda\bar\omega_{\Lambda^c})</math>,</center>

			:where <math>\Lambda^c = \mathbb{L}\setminus\Lambda</math>.

			* The '''[[partition function (mathematics)\|partition function]]''' in <math>\Lambda\in\mathcal{L}</math> with '''boundary conditions''' <math>\bar\omega</math> and inverse temperature <math>\beta\in\mathbb{R}_+</math> (for the potential <math>\Phi</math> and <math>\lambda</math>) is defined by

			<center><math>Z_\Lambda^\Phi(\bar\omega) = \int \lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega \| \bar\omega))</math>.</center>

			:Here <math>\lambda^\Lambda(\mathrm{d}\omega)</math> is the product measure <math>\prod_{t\in\Lambda}\lambda(\mathrm{d}\omega(t))</math>.

			:A potential <math>\Phi</math> is <math>\lambda</math>-admissible if <math>Z_\Lambda^\Phi(\bar\omega)</math> is finite for all <math>\Lambda\in\mathcal{L}</math>, <math>\bar\omega\in\Omega</math> and <math>\beta>0</math>.

			A [[probability measure]] <math>\mu</math> on <math>(\Omega,\mathcal{F})</math> is a '''Gibbs measure''' for a <math>\lambda</math>-admissible potential <math>\Phi</math> if it satisfies the '''[[Dobrushin-Lanford-Ruelle]] (DLR) equations'''

			<center><math>\int \mu(\mathrm{d}\bar\omega)Z_\Lambda^\Phi(\bar\omega)^{-1} \int\lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega \| \bar\omega)) 1_A(\omega_\Lambda\bar\omega_{\Lambda^c}) = \mu(A)</math>,</center>

			: for all <math>A\in\mathcal{F}</math> and <math>\Lambda\in\mathcal{L}</math>.

			===An example===
			To help understand the above definitions, here are the corresponding quantities in the important example of the [[Ising model]] with nearest-neighbour interactions (coupling constant <math>J</math>) and a magnetic field (<math>h</math>), on <math>\mathbb{Z}^d</math>:

			* The lattice is simply <math>\mathbb{L} = \mathbb{Z}^d</math>.
			* The single-spin space is <math>S=\{-1,1\}</math>.
			* The potential is given by
			<center><math>\Phi_A(\omega) = \begin{cases}
			-J\,\omega(t_1)\omega(t_2) & \mathrm{if\ } A=\{t_1,t_2\} \mathrm{\ with\ } \\|t_2-t_1\\|_1 = 1 \\
			-h\,\omega(t) & \mathrm{if\ } A=\{t\}\\
			0 & \mathrm{otherwise}
			\end{cases}</math></center>

			==See also==
			* [[Exponential family]]
			* [[Gibbs algorithm]]
			* [[Gibbs sampling]]
			* [[Interacting particle system]]
			* [[Stochastic cellular automata]]

			==References==
			<references/>
			* Georgii, H.-O. "Gibbs measures and phase transitions", de Gruyter, Berlin, 1988, 2nd edition 2011.

			{{Stochastic processes}}

			[[Category:Measures (measure theory)]]
			[[Category:Statistical mechanics]]

en>Lark ascending: /* Random network coding */

2012-08-02T01:31:12Z

Random network coding

New page

The writer is called Wilber Pegues. I am an invoicing officer and I'll be promoted soon. The favorite hobby for him and his children is to play lacross and he would never give it up. North Carolina is where we've been residing for years and will never move.<br><br>My webpage ... authentic psychic readings ([http://netwk.hannam.ac.kr/xe/data_2/85669 More Signup bonuses])