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{{for|the linearization in concurrent computing|Linearizability}}
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In [[mathematics]] '''linearization''' refers to finding the [[linear approximation]] to a [[function (mathematics)|function]] at a given point. In the study of [[dynamical system]]s, linearization is a method for assessing the local [[stability theory|stability]] of an [[equilibrium point]] of a [[system]] of [[nonlinear]] [[differential equation]]s or discrete [[dynamical system]]s.<ref>[http://www.scholarpedia.org/article/Siegel_disks/Linearization The linearization problem in complex dimension one dynamical systems at Scholarpedia]</ref>  This method is used in fields such as [[engineering]], [[physics]], [[economics]], and [[ecology]].


==Linearization of a function==
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Linearizations of a [[function (mathematics)|function]] are [[linear function|lines]] — ones  that are usually used for purposes of calculation. Linearization is an effective method for approximating the output of a function <math>y = f(x)</math> at any <math>x = a</math> based on the value and [[slope]] of the function at <math>x = b</math>, given that <math>f(x)</math> is continuous on <math>[a, b]</math> (or <math>[b, a]</math>) and that <math>a</math> is close to <math>b</math>. In short, linearization approximates the output of a function near <math>x = a</math>.
 
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For example, <math>\sqrt{4} = 2</math>. However, what would be a good approximation of <math>\sqrt{4.001} = \sqrt{4 + .001}</math>?
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For any given function <math>y = f(x)</math>, <math>f(x)</math> can be approximated if it is near a known continuous point. The most basic requisite is that, where <math>L_a(x)</math> is the linearization of <math>f(x)</math> at <math>x = a</math>, <math>L_a(a) = f(a)</math>. The [[Linear equation#Point–slope form|point-slope form]] of an equation forms an equation of a line, given a point <math>(H, K)</math> and slope <math>M</math>. The general form of this equation is: <math>y - K = M(x - H)</math>.
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Using the point <math>(a, f(a))</math>, <math>L_a(x)</math> becomes <math>y = f(a) + M(x - a)</math>. Because continuous functions are [[Local linearity|locally linear]], the best slope to substitute in would be the slope of the line [[tangent]] to <math>f(x)</math> at <math>x = a</math>.
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While the concept of local linearity applies the most to points [[Limit of a function#Limit of a function at a point|arbitrarily close]] to <math>x = a</math>, those relatively close work relatively well for linear approximations. The slope <math>M</math> should be, most accurately, the slope of the tangent line at <math>x = a</math>.
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[[Image:Tangent-calculus.svg|thumb|300px|An approximation of f(x)=x^2 at (''x'', ''f''(''x''))]]
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Visually, the accompanying diagram shows the tangent line of <math>f(x)</math> at <math>x</math>. At <math>f(x+h)</math>, where <math>h</math> is any small positive or negative value, <math>f(x+h)</math> is very nearly the value of the tangent line at the point <math>(x+h, L(x+h))</math>.


The final equation for the linearization of a function at <math>x = a</math> is:
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<math>y = f(a) + f'(a)(x - a)\,</math>
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For <math>x = a</math>, <math>f(a) = f(x)</math>. The [[derivative]] of <math>f(x)</math> is <math>f'(x)</math>, and the slope of <math>f(x)</math> at <math>a</math> is <math>f'(a)</math>.
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  <li>[http://coalition.movementcamp.org/circle/content/tim-rayners-shout-out-invite-movementcamp#comment-17656942 http://coalition.movementcamp.org/circle/content/tim-rayners-shout-out-invite-movementcamp#comment-17656942]</li>
 
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==Example==
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To find <math>\sqrt{4.001}</math>, we can use the fact that <math>\sqrt{4} = 2</math>. The linearization of  <math>f(x) = \sqrt{x}</math> at <math>x = a</math> is <math>y = \sqrt{a} + \frac{1}{2 \sqrt{a}}(x - a)</math>, because the function <math>f'(x) = \frac{1}{2 \sqrt{x}}</math> defines the slope of the function <math>f(x) = \sqrt{x}</math> at <math>x</math>. Substituting in <math>a = 4</math>, the linearization at 4 is <math>y = 2 + \frac{x-4}{4}</math>. In this case <math>x = 4.001</math>, so <math>\sqrt{4.001}</math> is approximately <math>2 + \frac{4.001-4}{4} = 2.00025</math>. The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent.


==Linearization of a multivariable function==
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The equation for the linearization of a function <math>f(x,y)</math> at a point <math>p(a,b)</math> is:
 
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<math> f(x,y) \approx f(a,b) + \left. {\frac{{\partial f(x,y)}}{{\partial x}}} \right|_{a,b} (x - a) + \left. {\frac{{\partial f(x,y)}}{{\partial y}}} \right|_{a,b} (y - b)</math>
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The general equation for the linearization of a multivariable function <math>f(\mathbf{x})</math> at a point <math>\mathbf{p}</math> is:
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<math>f({\mathbf{x}}) \approx f({\mathbf{p}}) + \left. {\nabla f} \right|_{\mathbf{p}}  \cdot ({\mathbf{x}} - {\mathbf{p}})</math>
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where <math>\mathbf{x}</math> is the vector of variables, and <math>\mathbf{p}</math> is the linearization point of interest
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.<ref>[http://www.ece.jhu.edu/~pi/Courses/454/linear.pdf Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering]</ref>
 
 
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==Uses of linearization==
 
 
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Linearization makes it possible to use tools for studying [[linear system]]s to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its [[Taylor expansion]] around the point of interest. For a system defined by the equation
 
 
</ul>
:<math>\frac{d\bold{x}}{dt} = \bold{F}(\bold{x},t)</math>,
 
the linearized system can be written as
 
:<math>\frac{d\bold{x}}{dt} = \bold{F}(\bold{x_0},t) + D\bold{F}(\bold{x_0},t)  \cdot (\bold{x} - \bold{x_0})</math>
 
where <math>\bold{x_0}</math> is the point of interest and <math>D\bold{F}(\bold{x_0})</math> is the [[Jacobian]] of <math>\bold{F}(\bold{x})</math> evaluated at <math>\bold{x_0}</math>.
 
===Stability analysis===
In [[stability theory|stability]] analysis of [[Autonomous Systems|autonomous systems]], one can use the [[eigenvalue]]s of the [[Jacobian matrix]] evaluated at a [[hyperbolic equilibrium point]] to determine the nature of that equilibrium. This is the content of [[linearization theorem]]. For time-varying systems, the linearization requires additional justification.<ref>G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107</ref>
 
===Microeconomics===
In [[microeconomics]], [[decision rules]] may be approximated under the state-space approach to linearization.<ref name="statespace">Moffatt, Mike. (2008) [[About.com]] ''[http://economics.about.com/od/economicsglossary/g/statespace.htm State-Space Approach]'' Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.</ref> Under this approach, the [[Euler_equations#Linearized_form|Euler equations]] of the [[utility maximization problem]] are linearized around the stationary steady state.<ref name="statespace"/> A unique solution to the resulting system of dynamic equations then is found.<ref name="statespace"/>
 
==See also==
* [[Linear stability]]
* [[Tangent stiffness matrix]]
* [[Stability derivatives]]
* [[Linearization theorem]]
* [[Taylor approximation]]
* [[Functional equation (L-function)]]
 
==References==
{{reflist}}
 
==External links==
 
===Linearization tutorials===
* [http://www.mathworks.com/discovery/linearization.html Linearization for Model Analysis and Control Design]
 
[[Category:Differential calculus]]
[[Category:Dynamical systems]]

Latest revision as of 15:18, 14 December 2014

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It took about 50 % a day to get used to being without a menu button. This sure would be nice if app makers followed Yahoo guidelines to accommodate the removal of that will soft key. As far as multitask, double tapping the home essential took about half a day to develop accustomed to.

And yes, I got my open up token from HTC, nevertheless I'm staying stock for now. I have yet to notice almost any lag. The software feels tremendous smooth and fluid. Most likely more so than my fiancee's i5.

Mine shipped today via FedEx Standard Overnight from Colorado front range, wish they had Saturday transport with that, but I'll understand it Monday which I guess will be quick enough. Brian is familiar with his stuff, and when they and Anand both expressed their particular pleasant experiences with it, I knew I had to get it.

I'm a big Windows Phone person, but the aluminum body as well as good camera performance taught me to be switch from Cheap Nike Shoes Verizon to help ATT and from WP so that you can Android. I originally bought excited a few months ago when I discovered and saw images of your Samsung Ativ S/Odyssey. and then I found out your brushed aluminum look was really plastic :( Though after converting to ATT I heard (in regards to week ago) that the Lumia 928 is due Verizon, and now I virtually wish I was getting that will. Actually, I wish I had their HTC One and Lumia 920, funny !.

I'm fairly open to other platforms/OS', even considered getting a innovative Blackberry, but the lack of BIS with their new lineup and middling hardware made it not worth it. From Apple I don't specially like the iPhone, either, because while it has nice appliance, the OS is pretty uninspiring now. It's basically an app launcher, and that is fine (it does have the many number of exclusive apps), even so like to have some compelling capabilities in the OS. at least Master of science has Zune (Xbox Popular music) and Office, and Android os is much more customizable and start. Siri does nothing for me, I can't talk to my phone, even if WP actually had built-in voice commands before iOS did.

This isn't an contra - Apple rant, though, I do like the look/feel of their phones, it is just good to see HTC in addition to Nokia pushing truly advanced feeling phones, as well. Possibly they can get Apple so that you can innovate more, which would be good for everyone. Competition is good.

Samsung SGS4 has never captured attention like the The brand new htc Nike Air Force 1 Uk One has, it seems more of a strong incremental upgrade, a l . a . what Apple does because of their iPhones, than a new cell phone. Disregarding all the shiny completely new gimmicky features, of course. Samsung is nice at cramming features and higher paper specs into their phones, though, and their displays are generally higher end than most. Need to give credit where credit history is due, lest I be accused of being a "fanboy" (I hate which term, by the way :D).

I need to post a mini evaluation after I play around with the One when I get it.

04 19 2013, 09:01 PM

Large warning, HTC decided to make use of the old, easily scratchable Gorilla Glass A couple of. While GG2 does a pretty good employment, micro scratches still be visible on the screen from purely pulling and putting your own phone in your pockets constantly which is very annoying. This HTC Trophy finally quit the ghost (well, it still works) and the screen damaged after the 145th or so time I personally dropped it, coincidentally 2 weeks in the past a little time after I found out about normally the one. I think it was committing suicide in protest. :D Similar circumstance played out with my Bb Crackedberry Storm a few phones previously, though when that occurred I looked like one of the munchkins once the house landed on the incredible witch in The Wizard connected with Oz. :P

Does damage resistance = crack level of resistance as well? I don't keep this phone in the same bank account as my keys, and I wear dress pants with plenty pocket room. maybe this is exactly why I have less problems with uncovering, who knows. I don't ever apply screen protectors, though perhaps a case would help protect coming from shattering. But I don't want to uglify normally the one.

04 20 2013, 10:46 AM

Big word of caution, HTC decided to use the older, easily scratchable Gorilla Glass 2. Though GG2 does a pretty good job, minuscule scratches still appear on the screen from simply dragging and putting your phone in your pockets constantly which is very annoying. It goes within the pocket daily. so I picture I'll be fine with GG2.

the 1080p screen is great, viewing tiny textual content on a non mobile seo'ed site was Louis Vuitton Auckland amazingly sharpened. The same text on my Take note 2 was slightly pixelated, even though I could see much more of the site on the watch's screen.

the screen wasn't as dazzling as I expected it to be. Set at max brightness, it did not seem very bright on hand, and not as bright because the Note 2 screen at less than max brightness. Still a lot more than sufficient however.

build quality has been nice, but didn't get the wow factor I expected. Individually I thought the iPhone 5 felt more premium the rubbery lines on the back vs. the graceful iPhone aluminum back seemed to be markedly different. However, nonetheless the class of the field for Android, but not the visceral feeling of wow I expected. My partner (who is stuck on Apple company and has the 5), laughed indicating this company was copying Apple inc so hard.

still not a fan with the viewable screen size to phone size. The actual amount of screen real estate where content would seem in a browser was meh excessive wasted space between the frame, black borders around the display, and button row. Believed worse than the S3, which is more intense than the Note 2, which in turn looks to be worse as opposed to S4 (and I assume the Note 3).

Darragh Donnelly Nike Free 5.0 Nz

THE LATE Con Houlihan generally wrote about the gremlins at work when something didn turn out as he had planned. I am afraid that the gremlins had been at work last week when my personal column failed to make it to their destination.

I sent them by e mail on Weekend morning, or at least I thought that left my computer, although somewhere in the cyber space among Athlone and Boyle it went missing out on. Sorry about that. Maybe it was equally efficiently. While I gave St. Brigid any hesitatnt vote to beat the Galway winners Salthill Knocknacarra, you could read between the traces that I wasn Nike Free 5.0 Nz all that confident. It seemed like they would struggle against a properly rated Galway city side. Also me of little faith.

St. Brigid produced a brilliant screen to win by a persuading 0 15 to 1 8 margin in Pearse Ground on Sunday last. There was no doubting the Kiltoom men efficiency. It was always going to be a hard ask for the Roscommon champions. A seaside club has only won 3 Galway senior championships, but about the last two occasions, 1990 and also 2005, they went on to be able to win the provincial title along with contested the All Ireland final on each occasion.

In 2005, not only did they win the Connacht title, they went on to win the All Ireland title as well. They were confident of getting it it a third time period lucky by adding another provincial title to the county title acquired a few weeks ago.

I proposed that St. Brigid wouldn need to have a very first 20 minutes like they had towards Melvin Gaels. For the first 20 minutes of the game, the champions appeared anything but comfortable. The support was all over the place against some sort of forward line that had just one Vibram Kso Uk single real quality forward, Emlyn Mulligan. The actual mental approach in Pearse Arena was totally different. It was apparent from the throw in that the Kiltoom adult men meant business.

The level of the St. Brigid tackling appeared to be something that the Galway champions couldn deal with. The losers were made to line out without past Roscommon star Gary Cox and lost corner forward Seamie Crowe who appeared to go over on his foot and had to retire injured after only six minutes.

The outlet exchanges were very besides the sides level on a few occasions in the first 10-20 minutes. Then St. Brigid began Vibram Five Fingers Uk to hop on top at midfield where Karol Mannion ended up being having one of his best games ever in the green and red jersey. With Senan Kil bride and also Frankie Dolan finding the range from frees, E. Brigid added five points prior to half time to lead by 4 8 to 0 Five at the break.

The opening min's of the second half were going to be vital as Saint. Brigid faced the wind. Frankie, Senan and also Damien Kelleher hit points to stretch control and set the Kiltoom men on the way to victory. The winners competed some brilliant football because the second half progressed and the screen was adorned by some enjoyable points from frees by way of Frankie Dolan.

Overall, it was a very good enough display and St. Brigid is now able to look forward to a final against Ballaghaderreen. There was clearly some outstanding displays out of Ronan Stack, Peter Domican, Darragh Donnelly, Damien Kelleher, Karol Mannion, Garvan Dolan, Niall Grehan, Frankie Dolan and Dude of the Match Senan Kilbride. The final really should revive memories for Senan daddy, Sen, as he played for Ballaghaderreen for a variety of years before returning to enjoy Roscommon.

I don drink alcohol having said that i tend to be very accident subject. I suffer from serious vertigo, which will doesn always make it easy to continue in an upright position.

As I left the top row of seats in Pearse Stadium after previous Sunday game, I had missed a step and landed to the broad of my rear between two rows regarding seats. I wasn injured even so was badly shaken.

I really understand the concern of those who witnessed the fall and who rushed to be able to my assistance. One gentleman was particulary kind when he offered to walk back to my motor vehicle with me. If I didn thank the pup sufficiently at the time it is because I was still shaken after the fall. In case he is reading this I want to provide him that I really liked his kindness.

Maybe it will take a tumble at times to aid us realise how type people are and how much true goodness there is out there.

The time has come of the year when some players are considering whether to call occasion on their playing careers or even to continue for another season at the least. It is not an easy decision for any player to make but time waits for no one.

A fortnight ago, Tyrone Ryan Mc Menamin chosen to call time on her playing days with the Red-colored Hand. as he was acknowledged, was a tough, uncompromising defender that often crossed the line for a few occasions and observed more than one straight red unit card in his time. In a 14 year career with Tyrone, he / she won three All Eire medals and an All Superstar award in 2005. This individual owes nothing to the game or to Tyrone. He may continue to play with his membership Dromore.

His retirement means that simply Sen Cavanagh, John Devine, Conor Gormley and Owen Mulligan remain on the team that brought the primary All Ireland title on the county in 2003. Mickey Harte with his fantastic selectors will have to start team building again but they have plenty of young skills in the county.

Cork celebrated dual star Sen g hAilpn has also thought i would quit the intercounty scene. In 35, he has given the all to the Cork cause particulary inside hurling. Born in Figi of your Figian mother and a Fermanagh father (or hurling strongholds as the legendary Michel Muircheartaig as soon as famously said), he earned three All Ireland throwing medals with Cork in 2004, 2004 and 2005. Actually is well liked played in the All Ireland in europe football final of 2007. He was named of the year in 2004. He is some sort of fluent Irish speaker and he sent a memorable victory speech while Gaeilge when accepting the Liam McCarthy Cup as captain in 2006. This is his second time for it to retire, but this time he says it for good. The game will miss him. He appeared to be one of the great characters within Irish sport.

Dublin Toms Quinn is another player who may have decided to hang up the boots immediately after ten years playing for Dublin. I often felt that the talented St. Vincent forward never actually realised his full likely with Dublin. A brilliant free taker, they didn always star in wide open play but on his or her day he could be very exciting to watch. His Dublin team mate Robert Casey also called time on her intercounty career during the week. Cheap Ray Bans Galway Person Bergin is another player who has made a decision to call it a day. He gave sterling prefer to Galway down through the years but some injuries in recent years have constrained his influence somewhat. Are experts in missed by Galway.

After a lengthy silence that was beginning to thwart followers of the game during the entire county, there was smoke upon Thursday last when the scheduled appointment of John Evans was proclaimed.

The Killorglin native becomes the first ever Kerry man to manage some sort of Roscommon team. The last Kerryman to manage a new Connacht team was Jack U when he managed Mayo with 1992 and believe you can find widespread welcome for the visit of John Evans. He is excited about the game and had a fair deal of success with Tipperary. He brought about the Premier County for you to back to back league titles in 2008 and and actually brought them to Division Two. He previously had previously led his golf club Laune Rangers to All Ireland pub success in 1996.

He was involved in an advisory capacity together with the Tipperary minor team that gained the All Ireland a year ago and, ironically, beat Roscommon from the semi final. He warrants the support of everyone inside the county.

How To Write A Powerful Resume Part 2 Cheap Hollister Uk

It was amazing to see how Cheap Hollister Uk nothing had changed a great deal in the five years between classes. The vast majority of people who came to the new workshops had resumes that ranged from not bad so that you can non existent.

When it comes to finding a job or changing careers you dont want to settle for a not bad resume. All the term means is because get paid to provide the assistance. That is not a guarantee that Hollister Hoodies Uk they are anything good at it.

It was once said to everyone, "Anyone can write a resume." That is the same as declaring, "Anyone can dance." We all know that some people do a superior job Longchamp Bag at it than others.

Mentionened above previously in an earlier article of the thousands of students who arrived through my workshops I'll easily say that the majority of them have very common mistakes. These problems seriously prevented their cv having any real power. When I teach a class on how to write a powerful curriculum vitae I like to start Nike Roshe Run Women with a numerous choice quiz so that I am able to begin to lay the foundation for changing how the jobseeker strategies writing a resume. As a facilitator is is important to commence where the participant is already making sure that I can add to what they know.

Cape Delmar III. Cheap Nike Free Run 2

Cape Henlopen took care of business in Thursday night, rolling past Polytech Several 0 in the semifinals of the DIAA Discipline Hockey Tournament at Dover High.

Now comes the big one particular: Cape Delmar III.

The Vikings surrounded Delmar 3 2 for the 2011 state title. They stowed past the Wildcats 2 1 for your 2012 state title. from Dover High.

"Good. We want to play Delmar,Inch said the Vikings' Sam Broadhurst, whom scored two goals against Polytech. "We want the competition. We're enthusiastic for it."

Cape Henlopen (Seventeen 0 1) outshot Polytech 11 3 in running its inside state winning streak to be able to 50 games. The fifth seeded Panthers accomplished their season at 20 Cheap Nike Free Run 2 3.

"We knew they were about to come out strong," Vikings mentor Kate Windett said of Polytech. "They don't have anything to lose. They're a young staff, and they know they will be returning here the next couple of years. We have Nike Free Run Sale Uk a lot of seniors, so we knew we had to come out and get that done."

Broadhurst got Cape on the board in the 16th minute, taking a short complete from Marlee McGinness and chipping the idea over the goalkeeper.

"For us, the primary goal is huge," Broadhurst said. "And to get it in the first one half, for us Nike Free Run 2 Uk it's all downhill. As we start rolling, it just almost all clicks."

McGinness made it Two 0 seven minutes in the future, raking in a shot with the side of the cage.

"It had been a tricky shot," McGinness reported. "It went between [the goalkeeper's] legs, however it still made it. I think it is precisely what gave us the confidence along with pushed us forward.In .

Allie Yeager, Broadhurst and Alexis Gooch scored for the Vikings from the second half to clinch the game everyone would like to see. Cape has also earned three straight regular period games over Delmar in the last three years and is the only team so that you can defeat the Wildcats in that period of time. The Vikings won at Delmar 3 Adidas Jeremy Scott Wings Uk 1 in overtime on August. 1, and both competitors have been shooting for the subject game ever since.

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