en>Yobot: /* Primality */clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

2012-12-05T18:43:54Z

Primality: clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

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	~~Do you want legal services? Are you searching~~ for ~~somebody~~ to ~~stand in your area~~ and ~~battle for yourself? Then~~, ~~you almost certainly have to meet with a legal representative~~. ~~Deciding on a great lawyer and working together with 1 you rely on isn't always easy~~. ~~But with the tips discussed on this page you should certainly get a attorney to suit your requires~~.<br><br>A ~~great hint to remember when looking to use a lawyer is to ensure that you get a attorney that has the necessary practical experience that you're searching for~~. ~~If you're going to judge soon~~ for ~~a felony scenario~~, ~~it wouldn't appear sensible to take over a divorce lawyer~~, ~~you'll require~~ a ~~criminal attorney.~~<br><br>~~When looking for a great attorney~~, ~~make sure you get private recommendations. Conversing with the regional community which have experienced concerns much like you. For example~~, ~~if you're a victim of sexual harassment, consult with a women's group~~. ~~Ask them about~~ the ~~legal professionals that they had~~ and ~~what their activities have been like.~~<br><br>~~You should establish a price range before you start looking for a attorney. Go over your financial situation~~ and ~~evaluate what you can afford to spend on a lawyer~~. ~~You must not permit legal professionals find out about your financial allowance when you request estimates but~~ this ~~is an excellent approach to define your www.youtube.com/watch?v~~=~~TOFAmHkvdkM ([http:~~/~~/www~~.~~youtube.com/watch?v~~=~~Obn~~-~~lfHhI4g recommended you read]~~) ~~research.~~<br><br>~~Check with your attorney for~~ the ~~estimate if you decide they~~'re the ~~one for you~~. ~~When they refuse~~, ~~move on quickly. Even if they merely supply you~~ with ~~a collection and explain exactly what makes a case pretty much high-priced, that may be suitable~~. ~~Ensure~~ [~~http:~~//~~www~~.~~youtube.com~~/~~watch?v~~=~~TOFAmHkvdkM Personal Injury And Car Accident Lawyers In Lafayette Louisiana] that you buy your payment arrangement on paper!~~<br><br>~~Should you need lawful support, don~~'~~t necessarily utilize~~ the ~~attorney you~~ have ~~worked with in~~ the ~~past~~. ~~You may need~~ a ~~certain sort~~ of ~~attorney, somebody who specializes in~~ the ~~kind~~ of ~~case you are involved in~~. ~~Don't get worried, although. A specialist isn't automatically pricey, so that you must be able~~ to ~~manage what you require~~.<br><br>~~Funds~~ is a [~~http://Www~~.~~Concern~~.~~com/ concern] for many people today~~, and ~~thinking about investing~~ in ~~substantial legal costs can seem~~ to ~~be utterly alarming! Check with your attorney if she or he may think about enabling you~~ to ~~pay off your costs using a payment plan it's possible that this might be an alternative for yourself~~.<br><br>~~Make sure you comprehend~~ the ~~approach included in firing your attorney~~. ~~There are lots of legal representatives that still need you~~ to ~~pay out a percentage of your respective costs even if you fireplace them~~. ~~Go through over-all commitments meticulously~~, ~~so there are no unexpected situations later when you~~ should ~~search~~ for ~~a new legal representative~~.<br><br>~~If you need to work with a lawyer but don't know how to begin, try looking in a legal professional directory like Nolo. This website directory delivers a comprehensive profile~~ of ~~legal professionals~~. ~~The information gives information for example his training~~, ~~expertise~~, ~~service fees as well as~~ the ~~lawyer's simple philosophy of rules. These details can be crucial in selecting a good legal representative~~.<br><br>~~If you believe just like your legal representative is just not being~~ in ~~advance together with you about all of~~ the ~~service fees concerned~~, ~~usually do not reconsider hiring somebody else~~. ~~Typically when lawyers~~ are ~~shifty about financial details this means that they have one thing to cover. Make it easier on oneself and get this question right away.~~<br><br>~~You might be spending your legal representative, so tend not to think that you will be tied to them~~. ~~Should you not feel as if these are doing a very good work~~, ~~offering you helpful advice or maybe you simply do not like them, don't wait to blaze them. There are many other legal professionals~~ that ~~will workout better for you.~~<br><br>~~Ideally you might have acquired a lot of things about attorneys out of this post. Employing a excellent legal professional can certainly make your lifestyle easier. Take advantage of~~ the ~~ideas you acquired right here and you could locate and make use of a reliable lawyer~~ that is ~~perfect~~ for ~~you~~ and ~~help you along with whatever you are coping with~~.		{{Distinguish\|Isochrone}}
			{{for\|the drug with trade name Isochron\|Isosorbide dinitrate }}

			In the mathematical theory of [[dynamical systems]], an '''isochron''' is a set of initial conditions for the system that all lead to the same long-term behaviour.<ref>J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol., 1:259–273 (1975)</ref><ref>S.M. Cox and A.J. Roberts, Initial conditions for models of dynamical systems, Physica D, 85:126–141 (1995)</ref>

			==Mathematical isochron==
			===An introductory example===

			Consider the [[ordinary differential equation]] for a solution <math>y(t)</math> evolving in time:

			:<math> \frac{d^2y}{dt^2} + \frac{dy}{dt} = 1</math>

			This [[ordinary differential equation]] (ODE) needs two [[initial conditions]] at, say, time <math>t=0</math>. Denote the [[initial conditions]] by <math>y(0)=y_0</math> and <math>dy/dt(0)=y'_0</math> where <math>y_0</math> and <math>y'_0</math> are some parameters. The following argument shows that the isochrons for this system are here the straight lines <math>y_0+y'_0=\mbox{constant}</math>.

			The general solution of the above ODE is

			:<math>y=t+A+B\exp(-t) \, </math>

			Now, as time increases, <math>t\to\infty</math>, the exponential terms decays very quickly to zero ([[exponential decay]]). Thus ''all'' solutions of the ODE quickly approach <math>y\to t+A</math>. That is, ''all'' solutions with the same <math>A</math> have the same long term evolution. The [[exponential decay]] of the <math>B\exp(-t)</math> term brings together a host of solutions to share the same long term evolution. Find the isochrons by answering which initial conditions have the same <math>A</math>.

			At the initial time <math>t=0</math> we have <math>y_0=A+B</math> and <math>y'_0=1-B</math>. Algebraically eliminate the immaterial constant <math>B</math> from these two equations to deduce that all initial conditions <math>y_0+y'_0=1+A</math> have the same <math>A</math>, hence the same long term evolution, and hence form an isochron.

			===Accurate forecasting requires isochrons===

			Let's turn to a more interesting application of the notion of isochrons. Isochrons arise when trying to forecast predictions from models of dynamical systems. Consider the toy system of two coupled [[ordinary differential equations]]

			:<math> \frac{dx}{dt} = -xy \text{ and } \frac{dy}{dt} = -y+x^2 - 2y^2</math>

			A marvellous mathematical trick is the [[normal form (mathematics)]] transformation.<ref>A.J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, ''Physica A: Statistical Mechanics and its Applications'' 387:12–38 (2008)</ref> Here the coordinate transformation near the origin

			:<math> x=X+XY+\cdots \text{ and } y=Y+2Y^2+X^2+\cdots</math>

			to new variables <math>(X,Y)</math> transforms the dynamics to the separated form

			:<math> \frac{dX}{dt} = -X^3+ \cdots \text{ and } \frac{dY}{dt} = (-1-2X^2+\cdots)Y</math>

			Hence, near the origin, <math>Y</math> decays to zero exponentially quickly as its equation is <math>dY/dt= (\text{negative})Y</math>. So the long term evolution is determined solely by <math>X</math>: the <math>X</math> equation is the model.

			Let us use the <math>X</math> equation to predict the future. Given some initial values <math>(x_0,y_0)</math> of the original variables: what initial value should we use for <math>X(0)</math>? Answer: the <math>X_0</math> that has the same long term evolution. In the normal form above, <math>X</math> evolves independently of <math>Y</math>. So all initial conditions with the same <math>X</math>, but different <math>Y</math>, have the same long term evolution. Fix <math>X</math> and vary <math>Y</math> gives the curving isochrons in the <math>(x,y)</math> plane. For example, very near the origin the isochrons of the above system are approximately the lines <math>x-Xy=X-X^3</math>. Find which isochron the initial values <math>(x_0,y_0)</math> lie on: that isochron is characterised by some <math>X_0</math>; the initial condition that gives the correct forecast from the model for all time is then <math>X(0)=X_0</math>.

			You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.[http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.html]

			==References==
			<references/>

			[[Category:Dynamical systems]]

en>Bender235: /* References */

2010-01-29T00:53:17Z

References

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en>Yobot: /* Primality */clean up, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (8748)

en>Bender235: /* References */