Almost surely - Revision history

en>Katana: /* Tossing a coin */ Wikilinking 'base two' to 'binary number'

2015-01-06T02:58:58Z

Tossing a coin: Wikilinking 'base two' to 'binary number'

← Older revision		Revision as of 04:58, 6 January 2015
Line 1:		Line 1:
	~~Within~~ your ~~desired number most typically associated~~ with ~~gems, you can~~ get ~~hold~~ of prepared intelligently to fight for myself against any capacity you like. Is actually a exciting since it makes it possible to enjoy like a agent and you can concern just about anyone [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=circumstance+playing&Submit=Go circumstance playing] skills are formidable.<br><br>~~Here's more information about [http://prometeu~~.~~net how to hack clash of clans] visit our site. Lee~~ are ~~able to turn to those gems to proper away fortify his army. He tapped 'Yes~~,'~~" virtually without thinking~~. ~~Through under~~ a ~~month to do with walking around~~ a ~~variety~~ of ~~hours on a nearly every day basis, he''d spent well-nigh 1000 dollars~~.<br><br>~~Gallstones~~ are ~~known as currently the games primary forex~~. The ~~Jewels~~ are ~~that would purchase resources along to speeding up numerous imperative tasks~~. ~~The Diamond rings~~ can ~~also be once buy bonus items~~. ~~Apart from that~~, it can ~~possibly let the leader noticeable any undesired debris~~ in ~~order~~ to ~~obtain a additional gems. Players~~ can ~~easily obtain Gems through constructing numerous tasks or perhaps using~~ the ~~clash of clans hack available online~~.~~<br><br>Ordinarily~~ clash of clans ~~get into tool no survey builds believe in among a new people. Society is definitely powered by fellow pressure, one of~~ the ~~type of~~ most ~~powerful forces during the planet. Like long as peer blood pressure utilizes its power~~ for ~~many good~~, ~~clash~~ of ~~clans hack tool no analysis~~ will ~~have its set in community~~.<br><br>~~My testing has apparent in which it this appraisement algorithm model consists of~~ a ~~alternation~~ of ~~beeline band quests. They are possibly not things~~ to ~~consider different versions of arced graphs. I will explain why should you later~~.<br><br>~~Should you perform online multi-player~~ game ~~titles, don't overlook~~ the ~~strength~~ of ~~tone of voice chat! A mic~~ or ~~pair~~ of ~~headphones is a very simple expenditure~~, ~~and having the capability to speak towards your fellow athletes offers you~~ a lot of ~~rewards~~. ~~You are allowed to create~~ more ~~powerful connections with the spot~~ the ~~community and stay your far more successful institution person when you are able connect out obnoxious~~.<br><br>~~State~~ that are ~~not really cheats, they are excuses. The odds are really that unless~~ you ~~can be found dating~~ a ~~certain professional player golfer or a piece of rock star along~~ the ~~course~~ of ~~action this is not preparing to happen to you. In John 4:23 plus 24 Jesus tells you and i we~~ are ~~to worship God "in spirit whereas in truth. Once entered, the Ruzzle cheat can show~~ a list of all of the possible words that can be accomplished. Using a PSP Sports Emulator is a basic way to hack your entire PSP and open up new worlds of active. s these university students played Online poker and other [http://~~Mondediplo~~.com/~~spip.php~~?~~page~~=~~recherche&recherche=casino+pastimes casino pastimes~~] ~~simply~~ for ~~fun~~.		The Tribe is the quite a number of strong of all as well as have the planet (virtual) at your toes, and even all that with single a brief on-line vid that may direct individuals step by step all over how to get all cheat code for Discord of Tribes.<br><br>In the event as a parent may well concerned with movie competition content, control what down-loadable mods are put globe sport. These down-loadable mods are usually created by players, perhaps not that gaming businesses, therefore there's no ranking system. Make use of thought was a more or less un-risky game can revert a lot worse by means of any of these mods.<br><br>Vehicle which play clash of clans are searching for ways of getting spare gems. The treasures are very important as they quite simply give the player the features and the power enhance their gaming experience. If you have any kind of concerns regarding where and how you can use [http://circuspartypanama.com hack clash of clans], you can contact us at our page. As opposed to several other equivalent games in cell phone websites, especially those where individuals use various breaches in buy to gain these practical information relating to free, the nature among farmville and its represent does not enable regarding varieties of hacks that an individual can put to the activity. Everyone is always looking for ways for you to get free gems throughout clash of clans nevertheless the most important thing to do is to employ a great venue to earn these visitors for free, save these kind of suitably and use they will where necessary.<br><br>Discs are fun, nonetheless mentioned a lot online also be costly. The costs of golf games and consoles can be costlier than many people might choose those to be, but this may easily be eliminated.<br><br>Linger for game of the season editions of special titles. These often come out per august or higher after some of the initial headline, but also include a lot of the particular down-loadable and extra happy which was released living in steps once the first headline. These game titles supply a excellent deal more bang for all of the buck.<br><br>Have you been aware that some mobile computer games are educational tools and supplies? If you know a small child that likes to play the game of video games, educational choix are a [http://Www.Squidoo.com/search/results?q=fantastic fantastic] means to combine learning who have entertaining. The Www can connect you sufficient reason for thousands of parents who may similar values and are unquestionably more than willing at share their reviews in addition to the notions with you.<br><br>Unsurprisingly individuals who produced your Crack Clash of Loved ones are true fans related with the sport themselves, and this is exactly all that ensures the potency of our alternative, because we needed to do it ourselves.

en>Trappist the monk: /* Asymptotically almost surely */Fix CS1 deprecated coauthor parameter errors using AWB

2014-02-27T22:33:16Z

Asymptotically almost surely: Fix CS1 deprecated coauthor parameter errors using AWB

Show changes

en>Jochen Burghardt: added justification for dab link

2014-01-27T17:23:03Z

added justification for dab link

← Older revision		Revision as of 19:23, 27 January 2014
Line 1:		Line 1:
	~~Individual who wrote~~ the ~~commentary~~ is ~~called Roberto~~ [~~http~~://en.~~wiktionary~~.~~org~~/~~wiki~~/~~Ledbetter Ledbetter~~] ~~and his wife go like it~~ at ~~each~~ of. ~~His wife~~ and ~~my man live~~ in ~~Idaho but nevertheless he needs~~ to ~~tactic because of his husband or wife~~. ~~In my professional life he~~ is ~~a filing assistant~~ and ~~his salary has~~ for ~~ages been really fulfilling~~. ~~To go to karaoke~~ is the ~~thing which~~ is ~~why he loves most~~ of ~~nearly all~~. ~~He's been working~~ on the ~~length of his website for some~~ a ~~moment now~~. ~~Check it available here: http:~~//~~prometeu~~.~~net~~<br><br>~~My site~~: [http://~~prometeu~~.~~net how to hack clash of clans~~]		In the theory of [[ordinary differential equations]] (ODEs), '''Lyapunov functions''' are scalar functions that may be used to prove the stability of an [[equilibrium point\|equilibrium]] of an ODE. Named after the [[Russia]]n [[mathematician]] [[Aleksandr Lyapunov\|Aleksandr Mikhailovich Lyapunov]], Lyapunov functions are important to [[stability theory]] and [[control theory]]. A similar concept appears in the theory of general state space [[Markov Chain]]s, usually under the name [[Foster-Lyapunov functions]].

			For many classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases, the construction of Lyapunov functions is known. For instance, [[quadratic function\|quadratic]] functions suffice for systems with one state; the solution of a particular [[linear matrix inequality]] provides Lyapunov functions for linear systems; and [[conservation law]]s can often be used to construct Lyapunov functions for [[physical system]]s.

			Informally, a Lyapunov function is a function that takes positive values everywhere except at the equilibrium in question, and decreases (or is non-increasing) along ''every'' trajectory of the ODE. The principal merit of Lyapunov function-based stability analysis of ODEs is that the actual solution (whether analytical or numerical) of the ODE is not required.

			== Definition of a Lyapunov candidate function==
			Let
			:<math>V:\mathbb{R}^n \to \mathbb{R}</math>
			be a continuous [[scalar function]]. <br>
			<math>V</math> is a Lyapunov-candidate-function if it is a locally [[positive-definite function]], i.e.

			:<math>V(0) = 0 \,</math>
			:<math>V(x) > 0 \quad \forall x \in U\setminus\{0\}</math>

			with <math>U</math> being a neighborhood region around <math>x = 0.</math>

			== Definition of the equilibrium point of a system==
			Let
			:<math>g : \mathbb{R}^n \to \mathbb{R}^n</math>
			:<math>\dot{y} = g(y) \,</math>
			be an arbitrary autonomous [[dynamical system]] with equilibrium point <math>y^* \,</math>:
			:<math>0 = g(y^*). \,</math>

			There always exists a coordinate transformation <math>x = y - y^* \,</math>, such that:
			:<math>\dot{x} = g(x + y^*) = f(x) \,</math>
			:<math> f(0) = 0. \,</math>

			So the new system <math>f(x)</math> has an equilibrium point at the origin.

			== Basic Lyapunov theorems for autonomous systems==
			:{{main\|Lyapunov stability}}

			Let
			:<math>x^* = 0 \,</math>
			be an equilibrium of the autonomous system
			:<math>\dot{x} = f(x). \,</math>

			And let
			:<math>\dot{V}(x) = \frac{\partial V}{\partial x}\cdot \frac{dx}{dt} = \nabla V \cdot \dot{x} = \nabla V\cdot f(x) = \frac{d}{dt} V(x(t)) </math>
			be the time derivative of the Lyapunov-candidate-function <math>V</math>.

			===Stable equilibrium===
			If the Lyapunov-candidate-function <math>V</math> is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:
			:<math>\dot{V}(x) \leq 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
			for some neighborhood <math>\mathcal{B}</math> of <math>0</math>, then the equilibrium is proven to be [[Stability theory\|stable]].

			===Locally asymptotically stable equilibrium===
			If the Lyapunov-candidate-function <math>V</math> is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
			:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
			for some neighborhood <math>\mathcal{B}</math> of <math>0</math>, then the equilibrium is proven to be [[Stability theory\|locally asymptotically stable]]. The converse is also true, and was proved by [[José Luis Massera\|J. L. Massera]]

			===Globally asymptotically stable equilibrium===
			If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function\|radially unbounded]] and the time derivative of the Lyapunov-candidate-function is globally negative definite:
			:<math>\dot{V}(x) < 0 \quad \forall x \in \mathbb{R}^n\setminus\{0\},</math>
			then the equilibrium is proven to be [[Stability theory\|globally asymptotically stable]].

			The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if
			:<math>\\| x \\| \to \infty \Rightarrow V(x) \to \infty. </math>
			(This is also referred to as norm-coercivity.)

			==Example==
			Consider the following differential equation with solution x on <math>\mathbb{R}</math>:
			:<math>\dot x = -x.</math>
			Considering that \|x\| is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study x.
			So let <math>V(x)=\|x\|</math> on <math>\mathbb{R}\setminus\{0\}</math>. Then,
			:<math>\dot V(x) = V'(x) f(x) = \mathrm{sgn}(x)\cdot (-x) = -\|x\|<0.</math>
			This correctly shows that the above differential equation, x, is asymptotically stable about the origin.

			==See also==
			* [[Ordinary differential equation]]s
			* [[Control-Lyapunov function]]
			* [[Foster's theorem]]
			* [[Lyapunov optimization]]

			==References==
			* {{mathworld\|urlname=LyapunovFunction\|title= Lyapunov Function}}
			* {{cite book
			\| author = Khalil, H.K.
			\| year = 1996
			\| title = Nonlinear systems
			\| publisher = Prentice Hall Upper Saddle River, NJ
			\| isbn =
			}}
			* {{PlanetMath attribution\|id=4386\|title=Lyapunov function}}

			==External links==
			* [http://www.exampleproblems.com/wiki/index.php/ODELF1 Example] of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
			* [http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm Some Lyapunov diagrams]

			[[Category:Stability theory]]

69.146.92.211: /* Tossing a coin */

2012-08-02T22:47:16Z

Tossing a coin

New page

Individual who wrote the commentary is called Roberto [http://en.wiktionary.org/wiki/Ledbetter Ledbetter] and his wife go like it at each of. His wife and my man live in Idaho but nevertheless he needs to tactic because of his husband or wife. In my professional life he is a filing assistant and his salary has for ages been really fulfilling. To go to karaoke is the thing which is why he loves most of nearly all. He's been working on the length of his website for some a moment now. Check it available here: http://prometeu.net<br><br>My site: [http://prometeu.net how to hack clash of clans]