Talk:Green's function

2003:

equally famous (and well respected) texts that use the term Green function (Jackson, Cohen-Tanoudji, Eygers, etc.). Julian Schwainger (sp?) flips back and forth indiscriminantly in his essay on Greening Quantum Theory. A 2003 Physics Today article (The Green of Green Functions) by Lawrie Challis (professor emeritus of Notingham Univ. Physics Dept., and George Green historian) uses the term Green, but acknowledges that others use Green's, without explaining which it SHOULD be. This issue is very annoying to me (and apparently others here) but will not likely be resolved by simply stating that "so-and-so" uses "Green" or "Dr. whoever" told me that it is "Green's". The most appropriate usage would be whatever was first used. That begs the questions, who was the first to coin the term, where is it published, and what form did they use? I have a feeling that it was first used by Lord Kelvin in the 1850's, but I can't prove it, as I don't have any of his documents. Can anyone here answer those questions definitively and with citations?

253: 243: 222: 4542:. The article is also currently inconsistent about whether or not the Green's function should have a 1/(4 pi). I understand that in physics it can be normal to take the Green's function without the 4 pi, but then it's really the Green's function for 1/(4 pi) times the Laplacian. These choices of whether or not there is a 4 pi, and whether the Laplacian is positive or negative are not consistent through out the article ... . 2158:

equation. If all these articles were to be written and merged into this one, this page would get very long. It seems better to have this page be an introduction to Green's function theory with links to more specific pages on Green's functions. Just as there are different pages for the various partial differential equations, so should there be, for their corresponding Green's functions.

191: 2025:

going back and forth about who uses Green and who uses Green`s is not all that useful in finding the correct usage. I do find it odd that of all the functions that I know, the Green function is the only one that people feel the need to make possessive, (e.g., it is a Bessel function NOT Bessel`s function.)

847:

You've renamed the page, but left the references to Green's function in the article. Not very thorough. In my experience, most of the standard texts (e.g., Morse and Feshbach) refer to it as Green's function. When I studied the Green's function, the professor specifically mentioned the unusual use of

4312:

Edit: well, with boundary conditions, G will be more complicated. But for example, with Neumann boundary conditions, it is clear to see that the remaining surface integral should not be negative (by comparing with the wikipedia page for Helmholtz decomposition). It should have the opposite sign. The

2116:

It's "the" because it's "the Green's function ". An analogy: When we say "the Adam's apple", we refer to a particular instance of the protuberance in the human neck, rather than a specific entity belonging to a person named Adam. Just like "the Green's function" refers to a specific Green's function

1985:

The article calls it "Green's Function" throughout (aside from a rather obscurely-written lead, which I've now changed), and all the sources given call it "Green's Function". There seems to be no good reason for us not to call it by that name. (Googling supports the "Green's" versus "Green" name.)

1870:

Let u be the electrostatic potential of an arbitrary charge distribution f. The Green's function is the solution of a point charge (delta function). Thus the total potential u can be obtained simply by convolution of the Green's function with the charge distribution f. Spacially it can be thought of

5611:

What is a inhomogeneous linear differential operator? There may be non-linear ones, or there may be linear ones in an inhomogeneous linear differential equation...but the operator itself is not inhomogeneous. At least there is no mention of such thing in the Wiki article "Differential operator" and

2069:"Of minor note is the change from Maxwell's equations and a Green's function to the Maxwell equations and a Green function. The latter boggles some minds, but is in conformity with other usage (Bessel function, for example). It is still Green's theorem, however, because that's whose theorem it is." 2024:

Jackson in his text on Electrodynamics (possibly THE most famous and widely used text book in graduate physics course on E & M), as well as the Nobel laureate Cohen-Tanoudji (in his textbook on Quantum mechanics ... another extremely popular graduate level text) both use GREEN. As stated above

2002:

Actually, Googling supports either the use of Green or Green's (and google results alone should not be the basis on which to settle this issue). Many famous physics text books (e.g., Morse and Fechbach, Fetter and Walecka, etc.) use the term Green's consistently, however, it is just as easy to find

909:

You raise a good point about the Green's being called stricly a distribution, and not a 'true' function. What of the case of a dimensionless operator L=1? Would not the Green's be identified with the Dirac Delta (assuming infinitesimal integration)? I.E. - it is only a function in the sense that it

800:

in number theory. Bessel functions are also solutions to Bessel's differential equation. i agree the form without the possessive is more common overall (not for Green's functions though), but there are many well-established counterexamples, which tend to lead to these discussions... see the preface

2269:

The derivation leading to (3) is not quite correct. The solution to the preceding equation can include a homogeneous part i.e. solving Lu=0 which should not be allowed here, since G has presumably been chosen so that the boundary conditions are already satisfied - i.e. without an extra homogeneous

880:

The article says "Green's functions are distributions in general, not functions, meaning they can have discontinuities.". This cannot possibly be correct, can it? I mean, a function having discontinuities sure doesn't make it any less of a function. The author must have had something else in mind.

2050:

Actually Jackson changed his usage from “Green’s function” to the (IMO correct) “Green function” between revisions of his classic textbook. I don’t have the 2nd edition at hand, so I cannot reproduce the reasoning from his foreword, but if anything, he makes a good witness for “Green’s function”

2157:

Well, you know there are many different Green's functions for partial differential equations. Specifically in 3-dimensions there is the Green's function for the Helmholtz equation, for the diffusion (heat) equation, and for the wave equation. There are even Green's functions for the biharmonic

3369:

This page has some major errors. The definition of the Green's function given is only for self-adjoint operators (fortunately all of the examples have self-adjoint operators), and will not work for other operators. It should be the adjoint of the operator that appears in the definition of the

1463:

I think we all know the best way to learn stuff is Examples, Like actual problems. I think it would be great if we could get many examples from some text books or somewhere, to give readers (and hopless students) an idea of how to use the this function. this is a more General statement to the

1478:

I think there's a little confusion with regards to this section. Until someone disagrees (Andywall?), I'd like to replace it with a section explaining the connections and differences between this kind of Green('s) function — the inverse of a linear operator — and what's meant by the term in

979:

If you are going to refer to the solution to electrostatic problems them you will need to insert the proper constants to make the units correct. The sign depends on the choice of the Poisson problem that you are trying to solve. In the case of electrostatics in the CGS system you have

864:

I don't care whether it is Green or Green's, all I want is to know which it SHOULD be. By that I mean, we should use whatever form was first introduced. Who introduced the term and when? Skip down to the discussion on the "title" (on this page) for more about who uses which form.

2749:

This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over δ(x − s)) and a superposition of the solution on each projection. Such an integral is known as a Fredholm integral equation, the study of which constitutes Fredholm theory."

5570:

In other words, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.

3464:

The article's structure seems messy to me. Could someone (native speaker) try to restructure the whole article? (And help me, where my table of Green's functions fits the best. I added it because I suppose that's the kind of information many people are looking for.) Thanks,

1424: 5225:

Yes, I would endorse that change. Perhaps also a comment should be added explaining that there are differing conventions on whether or not Poissson's equation has a negative sign. Also, I think the Green's function later down in the same section should have the 4

2186:

The 'example' section on this page refers to "demand-2","demand-3",and "demand-4" without any actual reference to what these entities are. Have I missed something here? Is it referring to the constraints? In any case I think it should be made more clear. Gazzo

3370:

Green's function. Also taking the operator outside the integral is not ok with the assumptions given. This should be done by repeated "integration by parts" (or application of the divergence theorem) from which the definition of the adjoint operator arises.

2667:"Convolving with a Green's function gives solutions to inhomogeneous differential-integral equations, most commonly a Sturm-Liouville problem. If G is the Green's function of an operator L, then the solution for u of the equation Lu = f is given by 4291:(you can look on the wikipedia page for Helmholtz Decomposition for why it must be this way around). Anyway, this surface integral term needs a negative sign. I thought I would write this, in case anyone can think of an objection to changing it. 5220: 4129:(or without the 4pi, it doesn't matter). But crucially, G must be positive here, because of the volume integral on the right-hand side of the equation. But, since G is positive, the surface integral term on the right hand side must instead be: 4037: 4970: 1631:

For example, suppose we have a horizontal string with distributed and point weights on it and we want to know its deflection along its length. Assume small deflection (so that the problem is linear). We could hang a unit mass at every point,

2117:

in a given context, "the Adam's apple" refers to the particular Adam's apple of a given individual. In both cases, the phrase can also be used with "a" instead of "the" to refer to the general concept (a Green's function, an Adam's apple).

3149:

Third, I know it's common to say that distributions are not properly functions, but are they not linear functionals over a vector subspace of infinitely differentiable functions? So technically, they are functions, just not functions from

1291: 965:

shouldn't there be a minus sign in the Laplacian Green's function 1/|x-x'| (and in the expressionwith the charge density \rho)? maybe I'm confused at the moment, but I'm pretty sure there should be... If no one answers I'll change it.

1966:(due to its role in its main usage - solving boundary value problems). The two concepts are distinct (though both are related to linear transformations). However, the wonders of hypertext should help the reader clear things up :-) -- 1813:

Currently "kernel" links to the "kernel (algebra)" article i.e. the article on the kernel of a homomorphism. I don't see the connection between the two uses of the word "kernel," and I'm wondering if the link should be removed.

2100:

Since both uses are common, I don't think it matters which is "correct". Both are. Except, and by this I'm really annoyed: if it's Green's function, then how come THE Green's function?! I ate John's apple, not the John's apple.

153: 1866:

Clarified confusion: In physics, Green's functions are often taught as a solution to a specific differential equation without much accompanying mathematical formalism. The classic example is that of a point charge.

5284:

I feel that the Green's function for the 1 dimensional laplace operator should be mentioned. This is important for problems such finding the potential using Green's functions for an infinitely large capacitor.

3346:

The article starts well givng general statements, but in the Framework subsection, it restricts to second order diferential equations, is there any reason for that? It exists Green's function for higher order?

3614: 1116: 848:

the possessive. His explanation was that to call it the 'Green function' sounded as if you were talking about a function that was colored green in some way, which tended to confuse people, especially students.

1772: 2544: 2753:

First of all, 'convolution' should be referenced. Second, it is appropriate here only if the operator L(x) is invariant under displacement - i.e. does not include an explicit reference to x. If so then

4289: 348:

Actually after checking some sources I see there is some disagreement over whether the definition of a Green's function should use the adjoint or the original operator. That is whether it should be

1185:

is the permittivity of free space given in terms of Farads per meter. In both cases there is a negative sign in the Poisson equation and hence the solutions to these Poisson equations will be

5623: 2350: 1296: 4723: 4612: 4540: 4425: 309: 4666: 4483: 4368: 1023: 3658: 3616:) appears to be wrong, according to this source, doing out myself, and basic logic e.g. (for physicists) it makes it look like an election and a proton will attract in 3D but repel in 2D. 2134: 2743: 5583:

I understand that jargon sometimes bends the rules, but this seems a little too bent, especially as we shouldn't be assuming that we're preaching to a choir of pre-existing jargon fu. —

679: 588: 420: 4127: 3002: 2417: 1943: 2614: 497: 1677: 1583: 5665: 3373:

I also can't see why there should be any linear operator without a Green's function (it might not be found analytically, but it will still exist and could be found numerically).

3206: 3177: 1183: 1622: 334:

The use of the adjoint operator in the first equation of the article is wrong. It should be the operator itself, not the adjoint. Anyone care to defend the current version?

147: 3298: 3780: 2921: 5612:

it is anything but intuitive what such an operator would look like. It would have to contain the inverse of the function I'm trying to find in a differential equation.

4975: 3792: 3249: 3109: 4728: 3144: 3069: 2651: 1540: 1156: 1136: 1063: 1043: 5687:. This makes things confusing as one might think they are not exactly the same. I might be missing simethingv, if there is a difference can someone clarify please? 3398:

The introduction ends with: "Green's functions are studied largely from the point of view of fundamental solutions instead." Instead of what? I find this unclear.

5685: 5511: 5507: 5493: 5403: 5399: 5385: 3318: 3269: 3229: 1188: 937:

The relationship between the Schrödinger Green's function and the local density of states is essentially the Plemelj-Sokhotsky equations (for the resolvent).

3550:

I think that there is an error in the "Table of Green's Functions" section, unless I'm reading something wrong. The claimed solution for the 2D Laplacian (

2013:

Haberman's "Elementary Applied PDEs" and Evan's "PDEs" both use "Green's functions" (and not "Green functions"). This are both popular, well-known texts.

3512:

Oh thanks, I didn't notice that the references tag was missing since there was already a references section. Anyone comments on the article's structure? --

1490: 948:"we leave it to the reader to fill in the in-between steps" Is something contained in bad textbooks and bad lectures. Can anyone fill in the blanks here? 5757: 299: 79: 44: 1794:

Yes, that sounds exactly right. If you put this into the page, make sure you explicitly write the differential equation that the string satisfies.

275: 5752: 1444: 808: 85: 5711: 5298: 4298: 3405: 5729:

The current formula in the table doesn't seem to make sense: the cross product of (x - x0) with itself will always yield the zero vector.

714:

I've always heard it called a Green's function. This case seems somewhat specific, would it be possible to generalise it to a degree n? -

5479: 5730: 2188: 2052: 2032: 3716: 3327: 5627: 5349: 4668:

Since this equation is written right at the 'incorrect' step in question. Or (and I think this is better) we could keep the equation

3553: 1852:. Linear operators are group morphism (among other things) and thus we can define a kernel for them which is exactly what is done at 1068: 5489:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

266: 227: 1688: 1065:

is the charge density in esu per centimeter. However, in the MKS unit system the Poisson equation for electrostatics is given by

3448: 3430:

i.e., Green's functions give the wavefield response in one place (x) as a result of an impulse-like source in another place (s).

2428: 168: 3715:

The Table of Green's Functions uses θ(t) in several places, but as far as I can see, this notation is not defined on this page.

2162: 2152: 135: 99: 30: 509:(In both cases the operator acts in the first variable, apparently.) One should note however that in order for something like 5268: 3536: 3489: 104: 20: 3251:

in the section on Green's Theorem for the Laplacian? Is this standard in physics? It looks a bit odd and cluttered, as if

4132: 3688: 2140:

This very specific, (previously) uncategorized article doesn't stand on its own. It needs to move to this article (or to

1419:{\displaystyle \Phi (x)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (x^{\prime })}{|x-x^{\prime }|}}d^{3}x^{\prime }} 600:

to be correct the non-adjoint version should be used. If the adjoint version is used, then the previous formula should be

74: 5554: 202: 1848:

I am confused by you confusion :) So let me try to clear this up. A green function is associated to a linear operator

1426:. So the (positive) sign is correct in both unit systems and the units are correct if working in the cgs unit system. 65: 1628:

as the sum of delta functions and expect our solution to be a sum of the responses to each of those delta functions.

1483: 129: 2282: 2172:

Is it possible to give a definition without distributions? This should be helpful for the general reader, I think.

1856:. Perhaps you could describe more clearly what exactly in the text confuses you so we can try to clear things up ? 4671: 4560: 4488: 4373: 4617: 4434: 4319: 983: 777: 5510:

to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

5402:

to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

3622: 691:

At any rate, I think most references use the non-adjoint version, and I think that is what should be given here.

2673: 1448: 812: 773: 125: 5715: 5302: 4302: 3409: 1971: 608: 517: 356: 5734: 2056: 2036: 5545: 5471: 4045: 3352: 2192: 2122: 1542:. I would add my understanding to the page, but am not sure it is a complete understanding. Is this right?: 109: 2936: 2361: 1877: 3735: 3720: 3665: 3444: 3331: 2555: 440: 175: 5692: 1962:

I believe the confusion comes about because we are used to think of Green's function as the kernel of an

736: 5529:

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

5517: 5441: 5421:

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

5409: 5264: 3348: 1825:

The usage of the term kernel in the article is correct. What second use of the word, besides the one at

1643: 1549: 1465: 761: 732: 208: 5650: 5483: 5470:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 3661: 3384: 252: 5480:

https://archive.is/20130101181958/http://academicearth.com/lectures/delta-function-and-greens-function

882: 797: 785: 749: 5619: 5375: 5294: 5256: 4294: 3783: 3731: 3524: 3502: 3477: 3440: 3436: 3401: 3323: 3182: 3153: 2074: 2028: 1991: 1948:

It is good for us physics folks to be made aware of the deeper connections between the tools we use.

1440: 895: 804: 793: 5688: 2235: 885: 190: 5260: 5215:{\displaystyle \phi (x)=-\int _{V}G(x,x')\rho (x')\ d^{3}x'+\int _{S}\left\cdot d{\hat {\sigma }}'} 4032:{\displaystyle \phi (x)=\int _{V}G(x,x')\rho (x')\ d^{3}x'+\int _{S}\left\cdot d{\hat {\sigma }}'.} 2254: 2239: 2207: 2145: 2141: 2106: 2089: 1967: 1953: 789: 753: 161: 55: 5350:

http://web.archive.org/web/20160105195811/http://www.boulder.nist.gov/div853/greenfn/tutorial.html

4965:{\displaystyle \phi (x)=\int _{V}G(x,x')\rho (x')\ d^{3}x'+\int _{S}\left\cdot d{\hat {\sigma }}'} 3380: 1489:

I have added a sentence in the introduction to mention this distinction and will add something to

1161: 274:

on Knowledge. If you would like to participate, please visit the project page, where you can join

5231: 4547: 3730:

Just above the table of Green's Functions it is defined as 'θ(t) is the Heaviside step function'

3532: 3485: 3146:

is much less confusing considering that prime is overloaded and typically means differentiation.

1963: 1838:

A kernel function? I don't believe a green's function is an abelian group. I'm confused as well.

1781: 1592: 835: 724: 696: 339: 258: 70: 5514:

before doing mass systematic removals. This message is updated dynamically through the template

5406:

before doing mass systematic removals. This message is updated dynamically through the template

2070: 2004: 894:

Umh. Not totally satisfactory formulation. More discontinous than any function, so to say, like

866: 242: 221: 141: 5530: 5463: 5422: 5321: 3376:

Source: Applied Partial Differential Equations: S. Howison, A. Lacey, A. Movchan, J. Ockendon.

3274: 2763: 1817: 781: 24: 5331: 2159: 2118: 1839: 1427: 51: 5613: 3756: 5437: 3234: 3074: 3016: 1853: 1826: 920: 876:

All functions are equal, and discontinous functions aren't less equal than others, are they?

765: 757: 5537: 5429: 5353: 3114: 3039: 2625: 2088:

This article (from Nature Physics) gives a good argument for the title "Green's function";

1511: 1286:{\displaystyle \Phi (x)=\int {\frac {\rho (x^{\prime })}{|x-x^{\prime }|}}d^{3}x^{\prime }} 5339: 3498: 1987: 1871:

as the superposition of the potentials of a bunch of point charges added up (integrated).

1857: 1830: 1141: 1121: 1048: 1028: 5707:

Find the Green function for the following problem, whose Green's function number is X11:

2249:

Your absolutely right, for the moment I am changing the article remove this discrepancy.

1508:

After looking at my math text, I think I just "got" this topic, at least for the case of

5306: 1479:

quantum/statistical field theory, where it essentially just means correlation function.

5738: 5496:, "External links modified" talk page sections are no longer generated or monitored by 5388:, "External links modified" talk page sections are no longer generated or monitored by 2250: 2203: 2102: 1949: 1624:. Because the system is linear, we can apply superposition: we can represent our given 967: 5670: 5536:

If you found an error with any archives or the URLs themselves, you can fix them with

5428:

If you found an error with any archives or the URLs themselves, you can fix them with

3303: 3254: 3214: 5746: 5227: 4543: 3528: 3481: 1795: 1494: 1480: 728: 692: 335: 3424:

I would propose adding the following to the end of the "Definitions and Uses" part:

928:

Green Functions for Ordered and Disordered Systems (Studies in Mathematical Physics)

4313:

other part of the surface integral should also be the negative of what is written.

2664:

The paragraph mentioning convolution is not quite right either. Presently it reads

899: 852: 769: 940:

Anyway, it would be nice to have a better expert than I to fill in these details.

2202:

I think the equation numbering got messed up. Does someone know how to fix this?

5594: 5584: 5503: 5395: 3033:

First, why does the definition begin with "In addition..." In addition to what?

3012: 952: 934:

by Eleftherios N. Economou (# ISBN-10: 3540288384 or # ISBN-13: 978-3540288381)

271: 5502:. No special action is required regarding these talk page notices, other than 5394:. No special action is required regarding these talk page notices, other than 2173: 2149: 2092: 2014: 715: 248: 5719: 5696: 5597: 5587: 5559: 5449: 5272: 5235: 4551: 4306: 3739: 3724: 3669: 3506: 3452: 3413: 3388: 3356: 3335: 3020: 2258: 2243: 2211: 2196: 2176: 2126: 2110: 2095: 2078: 2060: 2040: 2017: 2007: 1997: 1975: 1957: 1860: 1842: 1833: 1820: 1798: 1784: 1497: 1468: 1452: 1430: 1045:

is the electrostatic potential electrostatic units (esu) per centimeter and

970: 955: 902: 869: 855: 838: 816: 739: 700: 343: 5593:

perhaps the fix looks like this: ... and in doing so, recognize that ... —

930:

by Antonios Gonis (# ISBN-10: 0444889868 or # ISBN-13: 978-0444889867) and

710:

People, let's call it a Green function. This use of Green's is killing me!

5639:

The operator L occurs in this article in 4 different formatting styles:

4431:

I agree that the mistake has entered from a confusion about whether it's

3619:

Unless I hear someone with an argument not to, I'm going to change it to

3517: 3470: 3208:. I think this perhaps could be clarified since technically it's wrong. 1437:

I agree with the change suggested by HowiAuckland, his point is correct

5336:

after the link to keep me from modifying it. Alternatively, you can add

831: 926:

A couple of references I would suggest mentioning here are the books

5484:

http://academicearth.com/lectures/delta-function-and-greens-function

3030:

There are a couple of somewhat confusing things in this article.

792:. Also, the Legendre polynomials are often defined as solutions to 5344:

to keep me off the page altogether. I made the following changes:

3609:{\displaystyle \Delta _{2D}=\partial _{x}^{2}+\partial _{y}^{2}} 2132: 1111:{\displaystyle \nabla ^{2}\Phi =-{\frac {\rho }{\epsilon _{0}}}} 3513: 3466: 1636:, along the string and record the deflection as a function of 1585:, can be thought of as the response of the solution variable, 184: 15: 1767:{\displaystyle u(x)=\int _{a}^{b}{\mathcal {G}}(x,y)f(y)\,dy} 5359:

When you have finished reviewing my changes, please set the

3750:

The mistake is about halfway down the page, and this is it:

2539:{\displaystyle f(x)=\int LG(x,s)f(s)ds=L\int G(x,s)f(s)ds\,} 1724: 1649: 1555: 1411: 1383: 1357: 1278: 1250: 1224: 919:

I'd love to see an elaboration of the applications to QM. --

2051:

being widespread jargon and “Green function” being correct.

801:

to the 3rd ed of Jackson's Electrodynamics for example...

5474:

for additional information. I made the following changes:

3007:

I'll change it to this on 3rd December if no one objects.

2656:

I'll change it to this on 3rd December if no one objects.

5579:

One can solve and realizing that the solution is a sum.

5354:

http://www.boulder.nist.gov/div853/greenfn/tutorial.html

5320:

I have just added archive links to one external link on

2135:

Green's function for the three variable Laplace equation

5467: 5325: 1683:

shape centered on the point). The the solution will be

4284:{\displaystyle \int _{S}\left\cdot d{\hat {\sigma }}'} 951:

That sentence has been replaced by in-between steps --

160: 5673: 5653: 4978: 4731: 4674: 4620: 4563: 4491: 4437: 4376: 4322: 4135: 4048: 3795: 3759: 3625: 3556: 3497:

Using address 125.202.66.200 I corrected the error.--

3306: 3277: 3257: 3237: 3217: 3185: 3156: 3117: 3077: 3042: 2939: 2769: 2676: 2628: 2558: 2431: 2364: 2285: 1880: 1691: 1646: 1595: 1552: 1514: 1473: 1299: 1191: 1164: 1144: 1124: 1071: 1051: 1031: 986: 611: 520: 443: 359: 5710:

What does X11 mean? there's no link or explanations

4316:

edit again: or maybe the mistake is caused by using

270:, a collaborative effort to improve the coverage of 5506:using the archive tool instructions below. Editors 5398:using the archive tool instructions below. Editors 1158:is given in terms of Coulombs per cubic meter, and 826:I'm going to rename the page accordingly. AMS 108, 5679: 5659: 5214: 4964: 4717: 4660: 4606: 4534: 4477: 4419: 4362: 4283: 4121: 4031: 3774: 3652: 3608: 3312: 3292: 3263: 3243: 3223: 3200: 3171: 3138: 3103: 3063: 2996: 2912: 2737: 2645: 2608: 2538: 2411: 2344: 1937: 1766: 1671: 1616: 1577: 1534: 1418: 1285: 1177: 1150: 1130: 1110: 1057: 1037: 1017: 673: 582: 491: 414: 875: 33:for general discussion of the article's subject. 4557:ah, OK. So maybe we should change the equation 5492:This message was posted before February 2018. 5384:This message was posted before February 2018. 2345:{\displaystyle f(x)=\int \delta (x-s)f(s)ds\,} 4718:{\displaystyle \nabla ^{2}\phi (x)=-\rho (x)} 4607:{\displaystyle \nabla ^{2}\phi (x)=-\rho (x)} 4535:{\displaystyle \nabla ^{2}\phi (x)=-\rho (x)} 4420:{\displaystyle \nabla ^{2}\phi (x)=-\rho (x)} 4370:in the calculation, when it should really be 3546:Another missing minus sign (or just an error) 1474:Green's functions in condensed matter physics 174: 8: 4661:{\displaystyle \nabla ^{2}\phi (x)=\rho (x)} 4478:{\displaystyle \nabla ^{2}\phi (x)=\rho (x)} 4363:{\displaystyle \nabla ^{2}\phi (x)=\rho (x)} 3379:I will try and make changes when I have time 1018:{\displaystyle \nabla ^{2}\Phi =-4\pi \rho } 3653:{\displaystyle {\frac {1}{2\pi }}\ln \rho } 2065:Jackson writes in the 2nd edition foreword: 1491:Correlation function (quantum field theory) 5617: 5607:Inhomogeneous linear differential operator 5462:I have just modified one external link on 5292: 2738:{\displaystyle u(x)=\int {f(s)G(x,s)\,ds}} 1464:applications to Quantum Mechanics Section. 802: 216: 5672: 5652: 5197: 5196: 5074: 5053: 5001: 4977: 4947: 4946: 4824: 4803: 4751: 4730: 4679: 4673: 4625: 4619: 4568: 4562: 4496: 4490: 4442: 4436: 4381: 4375: 4327: 4321: 4266: 4265: 4140: 4134: 4111: 4092: 4078: 4047: 4011: 4010: 3888: 3867: 3815: 3794: 3758: 3626: 3624: 3600: 3595: 3582: 3577: 3561: 3555: 3305: 3279: 3278: 3276: 3256: 3236: 3216: 3192: 3188: 3187: 3184: 3163: 3159: 3158: 3155: 3116: 3076: 3041: 2938: 2768: 2695: 2675: 2627: 2557: 2430: 2363: 2284: 1879: 1723: 1722: 1716: 1711: 1690: 1648: 1647: 1645: 1594: 1554: 1553: 1551: 1513: 1410: 1400: 1388: 1382: 1367: 1356: 1343: 1331: 1315: 1298: 1277: 1267: 1255: 1249: 1234: 1223: 1210: 1190: 1169: 1163: 1143: 1138:is given in terms of Coulombs per meter, 1123: 1100: 1091: 1076: 1070: 1050: 1030: 991: 985: 674:{\displaystyle u(x)=\int G(s,x)f(s)\,ds.} 610: 583:{\displaystyle u(x)=\int G(x,s)f(s)\,ds.} 519: 442: 415:{\displaystyle L^{*}G(x,s)=\delta (x-s),} 364: 358: 2930:and now u is a convolution of G with f: 4122:{\displaystyle G(x,x')=1/(4\pi |x-x'|)} 3679: 2908: 2726: 2641: 2604: 2534: 2407: 2340: 1756: 723:It should be 'Green function': compare 660: 569: 218: 188: 2997:{\displaystyle u(x)=\int G(x-s)f(s)ds} 2412:{\displaystyle LG(x,s)=\delta (x-s)\,} 2234:, which is also used, makes no sense. 1938:{\displaystyle u(x)=\int G(x,y)f(y)dy} 5624:2A02:8070:BAB:DB00:9D2F:17C:B78D:9306 5371:to let others know (documentation at 4042:This must be incorrect, because here 3211:Fourth, what is with the hats on the 2609:{\displaystyle u=\int G(x,s)f(s)ds\,} 2230:are used as points on a manifold, so 492:{\displaystyle LG(x,s)=\delta (x-s).} 7: 5246:In the subsection Further examples: 3782:due to the defining property of the 932:Green's Functions in Quantum Physics 796:, which should not be confused with 602: 511: 434: 350: 264:This article is within the scope of 5635:Usage of different formatting for L 1672:{\displaystyle {\mathcal {G}}(x,y)} 1578:{\displaystyle {\mathcal {G}}(x,y)} 207:It is of interest to the following 23:for discussing improvements to the 5660:{\displaystyle \operatorname {L} } 5654: 5160: 5103: 4910: 4853: 4676: 4622: 4565: 4493: 4439: 4378: 4324: 4229: 4172: 3974: 3917: 3592: 3574: 3558: 1300: 1192: 1125: 1082: 1073: 1032: 997: 988: 14: 5758:Mid-priority mathematics articles 5466:. Please take a moment to review 5324:. Please take a moment to review 1854:Kernel_(algebra)#Linear_operators 915:Applications to Quantum Mechanics 284:Knowledge:WikiProject Mathematics 5575:Stripping away lower branches: 3201:{\displaystyle \mathbb {R} ^{n}} 3172:{\displaystyle \mathbb {R} ^{m}} 2913:{\displaystyle L(x)=L(x-s)=: --> 794:Legendre's differential equation 287:Template:WikiProject Mathematics 251: 241: 220: 189: 45:Click here to start a new topic. 5242:Laplace operator defined wrong? 3271:was defined somewhere else and 2922:L(x-s)G(x,s)=\delta (x-s)=: --> 2914:L(x-s)G(x,s)=\delta (x-s)=: --> 2764:L(x-s)G(x,s)=\delta (x-s)=: --> 1679:. (Obviously we would expect a 304:This article has been rated as 5725:Curl operator Green's function 5307:15:44, 25 September 2015 (UTC) 5252:Shouldn't that be d^2/dx^2 ? 5202: 5181: 5170: 5156: 5139: 5130: 5113: 5099: 5088: 5044: 5033: 5027: 5010: 4988: 4982: 4952: 4931: 4920: 4906: 4889: 4880: 4863: 4849: 4838: 4794: 4783: 4777: 4760: 4741: 4735: 4712: 4706: 4694: 4688: 4655: 4649: 4640: 4634: 4601: 4595: 4583: 4577: 4529: 4523: 4511: 4505: 4472: 4466: 4457: 4451: 4427:as it is written in the text. 4414: 4408: 4396: 4390: 4357: 4351: 4342: 4336: 4271: 4250: 4239: 4225: 4208: 4199: 4182: 4168: 4157: 4116: 4112: 4093: 4083: 4069: 4052: 4016: 3995: 3984: 3970: 3953: 3944: 3927: 3913: 3902: 3858: 3847: 3841: 3824: 3805: 3799: 3769: 3763: 3284: 3133: 3121: 3098: 3081: 3058: 3046: 2985: 2979: 2973: 2961: 2949: 2943: 2905: 2893: 2881: 2869: 2860: 2848: 2839: 2827: 2821: 2809: 2800: 2788: 2779: 2773: 2723: 2711: 2705: 2699: 2686: 2680: 2595: 2589: 2583: 2571: 2525: 2519: 2513: 2501: 2480: 2474: 2468: 2456: 2441: 2435: 2404: 2392: 2383: 2371: 2331: 2325: 2319: 2307: 2295: 2289: 2259:18:52, 17 September 2008 (UTC) 2244:15:42, 16 September 2008 (UTC) 2163:02:15, 16 September 2007 (UTC) 2041:10:00, 17 September 2007 (UTC) 1926: 1920: 1914: 1902: 1890: 1884: 1753: 1747: 1741: 1729: 1701: 1695: 1666: 1654: 1611: 1599: 1572: 1560: 1431:00:20, 11 September 2007 (UTC) 1389: 1368: 1362: 1349: 1309: 1303: 1256: 1235: 1229: 1216: 1201: 1195: 910:acts locally on said operator. 903:23:01, 27 September 2005 (UTC) 886:21:36, 27 September 2005 (UTC) 657: 651: 645: 633: 621: 615: 566: 560: 554: 542: 530: 524: 483: 471: 462: 450: 406: 394: 385: 373: 1: 5560:19:49, 3 September 2017 (UTC) 5450:06:09, 28 February 2016 (UTC) 3689:"Method of Green's Functions" 3520:) 21:42, 2 April 2013 (UTC) 3473:) 00:29, 2 April 2013 (UTC) 1976:15:45, 27 November 2008 (UTC) 1799:21:27, 15 November 2006 (UTC) 1785:00:16, 13 November 2006 (UTC) 1498:01:16, 10 November 2006 (UTC) 1178:{\displaystyle \epsilon _{0}} 956:04:53, 5 September 2007 (UTC) 839:00:17, 13 November 2006 (UTC) 278:and see a list of open tasks. 42:Put new text under old text. 5753:B-Class mathematics articles 5739:18:03, 10 October 2023 (UTC) 5598:16:28, 30 January 2019 (UTC) 5588:16:27, 30 January 2019 (UTC) 3740:17:09, 24 October 2022 (UTC) 3725:16:07, 24 October 2013 (UTC) 3414:02:41, 24 January 2012 (UTC) 3389:16:25, 21 January 2011 (UTC) 3357:00:11, 13 January 2011 (UTC) 3036:Second, why do we move from 3021:22:04, 30 October 2008 (UTC) 2273:The proper steps should be: 2177:15:19, 8 November 2007 (UTC) 2096:15:23, 8 November 2007 (UTC) 1834:18:46, 7 February 2007 (UTC) 1821:01:33, 29 January 2007 (UTC) 1617:{\displaystyle \delta (x-y)} 1484:03:15, 5 November 2006 (UTC) 1469:04:48, 24 October 2006 (UTC) 971:16:35, 10 October 2006 (UTC) 898:, which isn't a function. -- 856:16:22, 8 February 2007 (UTC) 701:21:44, 5 February 2014 (UTC) 344:21:04, 5 February 2014 (UTC) 5720:16:10, 23 August 2023 (UTC) 5616:14:33, 17 June 2020 (UTC) 3336:17:09, 16 August 2009 (UTC) 2422:Then combine and to give 834:calls it "Green function". 828:Applied Functional Analysis 50:New to Knowledge? Welcome! 5774: 5523:(last update: 5 June 2024) 5459:Hello fellow Wikipedians, 5415:(last update: 5 June 2024) 5342:|deny=InternetArchiveBot}} 5317:Hello fellow Wikipedians, 5280:Table of Green's functions 5236:09:35, 19 March 2014 (UTC) 4552:10:38, 17 March 2014 (UTC) 4307:12:56, 16 March 2014 (UTC) 3746:another missing minus sign 3453:10:47, 26 March 2012 (UTC) 3293:{\displaystyle {\hat {A}}} 2153:03:50, 8 August 2007 (UTC) 2018:03:50, 8 August 2007 (UTC) 1861:15:04, 21 March 2007 (UTC) 817:01:49, 21 April 2022 (UTC) 5697:14:57, 15 July 2020 (UTC) 5279: 3670:16:04, 15 July 2013 (UTC) 3507:13:53, 2 April 2013 (UTC) 2212:05:12, 20 June 2008 (UTC) 2127:16:44, 23 July 2023 (UTC) 2111:09:58, 29 July 2012 (UTC) 2079:11:11, 27 July 2023 (UTC) 2061:09:29, 27 July 2021 (UTC) 1998:09:23, 1 March 2007 (UTC) 1843:05:17, 1 March 2007 (UTC) 1829:, are you referring to ? 303: 236: 215: 80:Be welcoming to newcomers 5724: 5291:G = 1/2 |x| + Constant 5273:08:57, 4 July 2014 (UTC) 4725:And instead change from 3775:{\displaystyle \phi (x)} 3300:is some modification of 2197:17:43, 3 June 2008 (UTC) 2008:19:38, 24 May 2007 (UTC) 1958:19:33, 28 May 2008 (UTC) 1453:20:24, 8 July 2009 (UTC) 870:19:45, 24 May 2007 (UTC) 778:Euler's totient function 740:11:05, 9 July 2006 (UTC) 718:01:23, 20 Jun 2005 (UTC) 310:project's priority scale 5565:Strange sentence kernel 5455:External links modified 5313:External links modified 3420:Add use in wave physics 3244:{\displaystyle \sigma } 3104:{\displaystyle G(x,x')} 2222:Throughout the article 2168:Change of introduction? 1589:, to a delta function, 774:Fick's law of diffusion 267:WikiProject Mathematics 5681: 5661: 5581: 5573: 5288:I think it should be: 5216: 4966: 4719: 4662: 4608: 4536: 4479: 4421: 4364: 4285: 4123: 4033: 3776: 3654: 3610: 3314: 3294: 3265: 3245: 3225: 3202: 3173: 3140: 3139:{\displaystyle G(x,s)} 3105: 3065: 3064:{\displaystyle G(x,s)} 2998: 2917: 2739: 2647: 2646:{\displaystyle Lu=f\,} 2610: 2540: 2413: 2346: 1939: 1768: 1673: 1618: 1579: 1536: 1535:{\displaystyle -u''=f} 1420: 1287: 1179: 1152: 1132: 1112: 1059: 1039: 1019: 675: 584: 493: 416: 197:This article is rated 75:avoid personal attacks 5682: 5662: 5577: 5568: 5217: 4967: 4720: 4663: 4609: 4537: 4480: 4422: 4365: 4286: 4124: 4034: 3777: 3655: 3611: 3365:Only for self-adjoint 3315: 3295: 3266: 3246: 3226: 3203: 3174: 3141: 3106: 3066: 2999: 2918: 2740: 2648: 2611: 2541: 2414: 2347: 2091:doi:10.1038/nphys411 1940: 1780:Is that about right? 1769: 1674: 1619: 1580: 1537: 1421: 1288: 1180: 1153: 1151:{\displaystyle \rho } 1133: 1131:{\displaystyle \Phi } 1113: 1060: 1058:{\displaystyle \rho } 1040: 1038:{\displaystyle \Phi } 1020: 733:Grothendieck topology 676: 585: 494: 417: 100:Neutral point of view 5671: 5651: 5504:regular verification 5396:regular verification 5328:. If necessary, add 4976: 4729: 4672: 4618: 4561: 4489: 4435: 4374: 4320: 4133: 4046: 3793: 3784:Dirac delta function 3757: 3687:Hancock, Matthew J. 3623: 3554: 3304: 3275: 3255: 3235: 3215: 3183: 3154: 3115: 3075: 3040: 3026:Notation And Clarity 2937: 2767: 2674: 2626: 2556: 2429: 2362: 2355:So find G such that 2283: 1878: 1689: 1644: 1640:. This would be our 1593: 1550: 1546:The Green function, 1512: 1297: 1189: 1162: 1142: 1122: 1069: 1049: 1029: 984: 896:Dirac delta function 762:Hamilton's principle 609: 518: 441: 357: 290:mathematics articles 105:No original research 5494:After February 2018 5386:After February 2018 5363:parameter below to 3605: 3587: 1721: 798:Legendre's equation 786:Bessel's correction 750:Maxwell's equations 5677: 5657: 5548:InternetArchiveBot 5499:InternetArchiveBot 5391:InternetArchiveBot 5212: 4962: 4715: 4658: 4604: 4532: 4475: 4417: 4360: 4281: 4119: 4029: 3772: 3753:reduces to simply 3650: 3606: 3591: 3573: 3310: 3290: 3261: 3241: 3221: 3198: 3169: 3136: 3101: 3061: 2994: 2910: 2909: 2735: 2727: 2643: 2642: 2606: 2605: 2536: 2535: 2409: 2408: 2342: 2341: 2146:Poisson's equation 2142:Laplace's equation 1964:Integral transform 1935: 1764: 1757: 1707: 1669: 1614: 1575: 1532: 1416: 1283: 1175: 1148: 1128: 1108: 1055: 1035: 1015: 961:Minus sign missing 790:Fermat's principle 754:Laplace's equation 725:Hermite polynomial 671: 661: 580: 570: 489: 412: 259:Mathematics portal 203:content assessment 86:dispute resolution 47: 5680:{\displaystyle L} 5631: 5622:comment added by 5524: 5448: 5416: 5309: 5297:comment added by 5276: 5259:comment added by 5205: 5048: 4955: 4798: 4297:comment added by 4274: 4019: 3862: 3639: 3541: 3527:comment added by 3494: 3480:comment added by 3456: 3439:comment added by 3404:comment added by 3326:comment added by 3313:{\displaystyle A} 3287: 3264:{\displaystyle A} 3224:{\displaystyle A} 2924:G(x-s)\,}" /: --> 2043: 2031:comment added by 1443:comment added by 1394: 1338: 1261: 1106: 881:Please rephrase. 819: 807:comment added by 689: 688: 598: 597: 507: 506: 430: 429: 324: 323: 320: 319: 316: 315: 183: 182: 66:Assume good faith 43: 5765: 5686: 5684: 5683: 5678: 5666: 5664: 5663: 5658: 5646: 5558: 5549: 5522: 5521: 5500: 5464:Green's function 5444: 5443:Talk to my owner 5439: 5414: 5413: 5392: 5380: 5374: 5343: 5335: 5322:Green's function 5275: 5253: 5249:"Let L be d/dx" 5221: 5219: 5218: 5213: 5211: 5207: 5206: 5198: 5188: 5184: 5180: 5166: 5155: 5129: 5109: 5098: 5079: 5078: 5066: 5058: 5057: 5047: 5043: 5026: 5006: 5005: 4971: 4969: 4968: 4963: 4961: 4957: 4956: 4948: 4938: 4934: 4930: 4916: 4905: 4879: 4859: 4848: 4829: 4828: 4816: 4808: 4807: 4797: 4793: 4776: 4756: 4755: 4724: 4722: 4721: 4716: 4684: 4683: 4667: 4665: 4664: 4659: 4630: 4629: 4613: 4611: 4610: 4605: 4573: 4572: 4541: 4539: 4538: 4533: 4501: 4500: 4484: 4482: 4481: 4476: 4447: 4446: 4426: 4424: 4423: 4418: 4386: 4385: 4369: 4367: 4366: 4361: 4332: 4331: 4309: 4290: 4288: 4287: 4282: 4280: 4276: 4275: 4267: 4257: 4253: 4249: 4235: 4224: 4198: 4178: 4167: 4145: 4144: 4128: 4126: 4125: 4120: 4115: 4110: 4096: 4082: 4068: 4038: 4036: 4035: 4030: 4025: 4021: 4020: 4012: 4002: 3998: 3994: 3980: 3969: 3943: 3923: 3912: 3893: 3892: 3880: 3872: 3871: 3861: 3857: 3840: 3820: 3819: 3781: 3779: 3778: 3773: 3703: 3702: 3700: 3698: 3693: 3684: 3659: 3657: 3656: 3651: 3640: 3638: 3627: 3615: 3613: 3612: 3607: 3604: 3599: 3586: 3581: 3569: 3568: 3540: 3521: 3493: 3474: 3455: 3433: 3416: 3394:Instead of what? 3338: 3319: 3317: 3316: 3311: 3299: 3297: 3296: 3291: 3289: 3288: 3280: 3270: 3268: 3267: 3262: 3250: 3248: 3247: 3242: 3230: 3228: 3227: 3222: 3207: 3205: 3204: 3199: 3197: 3196: 3191: 3178: 3176: 3175: 3170: 3168: 3167: 3162: 3145: 3143: 3142: 3137: 3111:? The notation 3110: 3108: 3107: 3102: 3097: 3070: 3068: 3067: 3062: 3003: 3001: 3000: 2995: 2925: 2920: 2919: 2911: 2744: 2742: 2741: 2736: 2734: 2652: 2650: 2649: 2644: 2615: 2613: 2612: 2607: 2545: 2543: 2542: 2537: 2418: 2416: 2415: 2410: 2351: 2349: 2348: 2343: 2218:Manifolds vs. R3 2026: 1994: 1944: 1942: 1941: 1936: 1827:kernel (algebra) 1773: 1771: 1770: 1765: 1728: 1727: 1720: 1715: 1678: 1676: 1675: 1670: 1653: 1652: 1623: 1621: 1620: 1615: 1584: 1582: 1581: 1576: 1559: 1558: 1541: 1539: 1538: 1533: 1525: 1455: 1425: 1423: 1422: 1417: 1415: 1414: 1405: 1404: 1395: 1393: 1392: 1387: 1386: 1371: 1365: 1361: 1360: 1344: 1339: 1337: 1336: 1335: 1316: 1292: 1290: 1289: 1284: 1282: 1281: 1272: 1271: 1262: 1260: 1259: 1254: 1253: 1238: 1232: 1228: 1227: 1211: 1184: 1182: 1181: 1176: 1174: 1173: 1157: 1155: 1154: 1149: 1137: 1135: 1134: 1129: 1117: 1115: 1114: 1109: 1107: 1105: 1104: 1092: 1081: 1080: 1064: 1062: 1061: 1056: 1044: 1042: 1041: 1036: 1024: 1022: 1021: 1016: 996: 995: 921:User:TobinFricke 782:Euler's constant 680: 678: 677: 672: 603: 589: 587: 586: 581: 512: 498: 496: 495: 490: 435: 421: 419: 418: 413: 369: 368: 351: 292: 291: 288: 285: 282: 261: 256: 255: 245: 238: 237: 232: 224: 217: 200: 194: 193: 185: 179: 178: 164: 95:Article policies 25:Green's function 16: 5773: 5772: 5768: 5767: 5766: 5764: 5763: 5762: 5743: 5742: 5727: 5704: 5669: 5668: 5649: 5648: 5644: 5637: 5609: 5567: 5552: 5547: 5515: 5508:have permission 5498: 5472:this simple FaQ 5457: 5447: 5442: 5407: 5400:have permission 5390: 5378: 5372: 5337: 5329: 5315: 5282: 5254: 5244: 5195: 5173: 5159: 5148: 5122: 5102: 5091: 5084: 5080: 5070: 5059: 5049: 5036: 5019: 4997: 4974: 4973: 4945: 4923: 4909: 4898: 4872: 4852: 4841: 4834: 4830: 4820: 4809: 4799: 4786: 4769: 4747: 4727: 4726: 4675: 4670: 4669: 4621: 4616: 4615: 4564: 4559: 4558: 4492: 4487: 4486: 4438: 4433: 4432: 4377: 4372: 4371: 4323: 4318: 4317: 4292: 4264: 4242: 4228: 4217: 4191: 4171: 4160: 4150: 4146: 4136: 4131: 4130: 4103: 4061: 4044: 4043: 4009: 3987: 3973: 3962: 3936: 3916: 3905: 3898: 3894: 3884: 3873: 3863: 3850: 3833: 3811: 3791: 3790: 3755: 3754: 3748: 3713: 3708: 3707: 3706: 3696: 3694: 3691: 3686: 3685: 3681: 3660:in a few days. 3631: 3621: 3620: 3557: 3552: 3551: 3548: 3522: 3475: 3462: 3460:Messy structure 3434: 3422: 3399: 3396: 3367: 3344: 3321: 3302: 3301: 3273: 3272: 3253: 3252: 3233: 3232: 3213: 3212: 3186: 3181: 3180: 3157: 3152: 3151: 3113: 3112: 3090: 3073: 3072: 3038: 3037: 3028: 2935: 2934: 2766:G(x-s)\,}": --> 2762: 2761: 2672: 2671: 2662: 2624: 2623: 2554: 2553: 2427: 2426: 2360: 2359: 2281: 2280: 2267: 2220: 2184: 2170: 2138: 1992: 1983: 1876: 1875: 1811: 1782:—Ben FrantzDale 1687: 1686: 1642: 1641: 1591: 1590: 1548: 1547: 1518: 1510: 1509: 1506: 1476: 1461: 1445:190.244.173.152 1438: 1406: 1396: 1378: 1366: 1352: 1345: 1327: 1320: 1295: 1294: 1273: 1263: 1245: 1233: 1219: 1212: 1187: 1186: 1165: 1160: 1159: 1140: 1139: 1120: 1119: 1096: 1072: 1067: 1066: 1047: 1046: 1027: 1026: 987: 982: 981: 963: 946: 917: 878: 836:—Ben FrantzDale 809:179.181.146.173 766:Newton's method 758:Green's theorem 708: 607: 606: 516: 515: 439: 438: 360: 355: 354: 332: 289: 286: 283: 280: 279: 257: 250: 230: 201:on Knowledge's 198: 121: 116: 115: 114: 91: 61: 12: 11: 5: 5771: 5769: 5761: 5760: 5755: 5745: 5744: 5726: 5723: 5712:132.204.27.207 5703: 5700: 5676: 5656: 5636: 5633: 5608: 5605: 5603: 5601: 5600: 5566: 5563: 5542: 5541: 5534: 5487: 5486: 5478:Added archive 5456: 5453: 5440: 5434: 5433: 5426: 5357: 5356: 5348:Added archive 5314: 5311: 5299:84.148.212.147 5281: 5278: 5243: 5240: 5239: 5238: 5210: 5204: 5201: 5194: 5191: 5187: 5183: 5179: 5176: 5172: 5169: 5165: 5162: 5158: 5154: 5151: 5147: 5144: 5141: 5138: 5135: 5132: 5128: 5125: 5121: 5118: 5115: 5112: 5108: 5105: 5101: 5097: 5094: 5090: 5087: 5083: 5077: 5073: 5069: 5065: 5062: 5056: 5052: 5046: 5042: 5039: 5035: 5032: 5029: 5025: 5022: 5018: 5015: 5012: 5009: 5004: 5000: 4996: 4993: 4990: 4987: 4984: 4981: 4960: 4954: 4951: 4944: 4941: 4937: 4933: 4929: 4926: 4922: 4919: 4915: 4912: 4908: 4904: 4901: 4897: 4894: 4891: 4888: 4885: 4882: 4878: 4875: 4871: 4868: 4865: 4862: 4858: 4855: 4851: 4847: 4844: 4840: 4837: 4833: 4827: 4823: 4819: 4815: 4812: 4806: 4802: 4796: 4792: 4789: 4785: 4782: 4779: 4775: 4772: 4768: 4765: 4762: 4759: 4754: 4750: 4746: 4743: 4740: 4737: 4734: 4714: 4711: 4708: 4705: 4702: 4699: 4696: 4693: 4690: 4687: 4682: 4678: 4657: 4654: 4651: 4648: 4645: 4642: 4639: 4636: 4633: 4628: 4624: 4603: 4600: 4597: 4594: 4591: 4588: 4585: 4582: 4579: 4576: 4571: 4567: 4555: 4554: 4531: 4528: 4525: 4522: 4519: 4516: 4513: 4510: 4507: 4504: 4499: 4495: 4474: 4471: 4468: 4465: 4462: 4459: 4456: 4453: 4450: 4445: 4441: 4416: 4413: 4410: 4407: 4404: 4401: 4398: 4395: 4392: 4389: 4384: 4380: 4359: 4356: 4353: 4350: 4347: 4344: 4341: 4338: 4335: 4330: 4326: 4299:151.228.183.81 4279: 4273: 4270: 4263: 4260: 4256: 4252: 4248: 4245: 4241: 4238: 4234: 4231: 4227: 4223: 4220: 4216: 4213: 4210: 4207: 4204: 4201: 4197: 4194: 4190: 4187: 4184: 4181: 4177: 4174: 4170: 4166: 4163: 4159: 4156: 4153: 4149: 4143: 4139: 4118: 4114: 4109: 4106: 4102: 4099: 4095: 4091: 4088: 4085: 4081: 4077: 4074: 4071: 4067: 4064: 4060: 4057: 4054: 4051: 4040: 4039: 4028: 4024: 4018: 4015: 4008: 4005: 4001: 3997: 3993: 3990: 3986: 3983: 3979: 3976: 3972: 3968: 3965: 3961: 3958: 3955: 3952: 3949: 3946: 3942: 3939: 3935: 3932: 3929: 3926: 3922: 3919: 3915: 3911: 3908: 3904: 3901: 3897: 3891: 3887: 3883: 3879: 3876: 3870: 3866: 3860: 3856: 3853: 3849: 3846: 3843: 3839: 3836: 3832: 3829: 3826: 3823: 3818: 3814: 3810: 3807: 3804: 3801: 3798: 3771: 3768: 3765: 3762: 3747: 3744: 3743: 3742: 3712: 3709: 3705: 3704: 3678: 3677: 3673: 3649: 3646: 3643: 3637: 3634: 3630: 3603: 3598: 3594: 3590: 3585: 3580: 3576: 3572: 3567: 3564: 3560: 3547: 3544: 3543: 3542: 3461: 3458: 3421: 3418: 3406:171.64.163.233 3395: 3392: 3366: 3363: 3361: 3343: 3340: 3309: 3286: 3283: 3260: 3240: 3220: 3195: 3190: 3166: 3161: 3135: 3132: 3129: 3126: 3123: 3120: 3100: 3096: 3093: 3089: 3086: 3083: 3080: 3060: 3057: 3054: 3051: 3048: 3045: 3027: 3024: 3010: 3005: 3004: 2993: 2990: 2987: 2984: 2981: 2978: 2975: 2972: 2969: 2966: 2963: 2960: 2957: 2954: 2951: 2948: 2945: 2942: 2928: 2927: 2907: 2904: 2901: 2898: 2895: 2892: 2889: 2886: 2883: 2880: 2877: 2874: 2871: 2868: 2865: 2862: 2859: 2856: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2826: 2823: 2820: 2817: 2814: 2811: 2808: 2805: 2802: 2799: 2796: 2793: 2790: 2787: 2784: 2781: 2778: 2775: 2772: 2757: 2747: 2746: 2733: 2730: 2725: 2722: 2719: 2716: 2713: 2710: 2707: 2704: 2701: 2698: 2694: 2691: 2688: 2685: 2682: 2679: 2661: 2658: 2654: 2653: 2640: 2637: 2634: 2631: 2617: 2616: 2603: 2600: 2597: 2594: 2591: 2588: 2585: 2582: 2579: 2576: 2573: 2570: 2567: 2564: 2561: 2547: 2546: 2533: 2530: 2527: 2524: 2521: 2518: 2515: 2512: 2509: 2506: 2503: 2500: 2497: 2494: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2470: 2467: 2464: 2461: 2458: 2455: 2452: 2449: 2446: 2443: 2440: 2437: 2434: 2420: 2419: 2406: 2403: 2400: 2397: 2394: 2391: 2388: 2385: 2382: 2379: 2376: 2373: 2370: 2367: 2353: 2352: 2339: 2336: 2333: 2330: 2327: 2324: 2321: 2318: 2315: 2312: 2309: 2306: 2303: 2300: 2297: 2294: 2291: 2288: 2266: 2263: 2262: 2261: 2219: 2216: 2215: 2214: 2183: 2180: 2169: 2166: 2137: 2131: 2130: 2129: 2086: 2085: 2084: 2083: 2082: 2081: 2066: 2045: 2044: 2021: 2020: 1982: 1979: 1968:AmitAronovitch 1946: 1945: 1934: 1931: 1928: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1904: 1901: 1898: 1895: 1892: 1889: 1886: 1883: 1864: 1863: 1810: 1807: 1806: 1805: 1804: 1803: 1802: 1801: 1778: 1777: 1776: 1775: 1763: 1760: 1755: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1731: 1726: 1719: 1714: 1710: 1706: 1703: 1700: 1697: 1694: 1668: 1665: 1662: 1659: 1656: 1651: 1629: 1613: 1610: 1607: 1604: 1601: 1598: 1574: 1571: 1568: 1565: 1562: 1557: 1531: 1528: 1524: 1521: 1517: 1505: 1502: 1501: 1500: 1475: 1472: 1460: 1457: 1436: 1434: 1433: 1413: 1409: 1403: 1399: 1391: 1385: 1381: 1377: 1374: 1370: 1364: 1359: 1355: 1351: 1348: 1342: 1334: 1330: 1326: 1323: 1319: 1314: 1311: 1308: 1305: 1302: 1280: 1276: 1270: 1266: 1258: 1252: 1248: 1244: 1241: 1237: 1231: 1226: 1222: 1218: 1215: 1209: 1206: 1203: 1200: 1197: 1194: 1172: 1168: 1147: 1127: 1103: 1099: 1095: 1090: 1087: 1084: 1079: 1075: 1054: 1034: 1014: 1011: 1008: 1005: 1002: 999: 994: 990: 975: 962: 959: 945: 942: 925: 916: 913: 912: 911: 906: 905: 890: 877: 874: 873: 872: 861: 860: 859: 858: 849: 842: 841: 823: 822: 821: 820: 743: 742: 737:88.104.215.243 720: 719: 707: 704: 687: 686: 683: 681: 670: 667: 664: 659: 656: 653: 650: 647: 644: 641: 638: 635: 632: 629: 626: 623: 620: 617: 614: 596: 595: 592: 590: 579: 576: 573: 568: 565: 562: 559: 556: 553: 550: 547: 544: 541: 538: 535: 532: 529: 526: 523: 505: 504: 501: 499: 488: 485: 482: 479: 476: 473: 470: 467: 464: 461: 458: 455: 452: 449: 446: 428: 427: 424: 422: 411: 408: 405: 402: 399: 396: 393: 390: 387: 384: 381: 378: 375: 372: 367: 363: 331: 328: 326: 322: 321: 318: 317: 314: 313: 302: 296: 295: 293: 276:the discussion 263: 262: 246: 234: 233: 225: 213: 212: 206: 195: 181: 180: 118: 117: 113: 112: 107: 102: 93: 92: 90: 89: 82: 77: 68: 62: 60: 59: 48: 39: 38: 35: 34: 28: 13: 10: 9: 6: 4: 3: 2: 5770: 5759: 5756: 5754: 5751: 5750: 5748: 5741: 5740: 5736: 5732: 5731:128.83.139.11 5722: 5721: 5717: 5713: 5708: 5701: 5699: 5698: 5694: 5690: 5674: 5642: 5634: 5632: 5629: 5625: 5621: 5615: 5606: 5604: 5599: 5596: 5592: 5591: 5590: 5589: 5586: 5580: 5576: 5572: 5564: 5562: 5561: 5556: 5551: 5550: 5539: 5535: 5532: 5528: 5527: 5526: 5519: 5513: 5509: 5505: 5501: 5495: 5490: 5485: 5481: 5477: 5476: 5475: 5473: 5469: 5465: 5460: 5454: 5452: 5451: 5445: 5438: 5431: 5427: 5424: 5420: 5419: 5418: 5411: 5405: 5401: 5397: 5393: 5387: 5382: 5377: 5370: 5366: 5362: 5355: 5351: 5347: 5346: 5345: 5341: 5333: 5327: 5323: 5318: 5312: 5310: 5308: 5304: 5300: 5296: 5289: 5286: 5277: 5274: 5270: 5266: 5262: 5258: 5250: 5247: 5241: 5237: 5233: 5229: 5224: 5223: 5222: 5208: 5199: 5192: 5189: 5185: 5177: 5174: 5167: 5163: 5152: 5149: 5145: 5142: 5136: 5133: 5126: 5123: 5119: 5116: 5110: 5106: 5095: 5092: 5085: 5081: 5075: 5071: 5067: 5063: 5060: 5054: 5050: 5040: 5037: 5030: 5023: 5020: 5016: 5013: 5007: 5002: 4998: 4994: 4991: 4985: 4979: 4958: 4949: 4942: 4939: 4935: 4927: 4924: 4917: 4913: 4902: 4899: 4895: 4892: 4886: 4883: 4876: 4873: 4869: 4866: 4860: 4856: 4845: 4842: 4835: 4831: 4825: 4821: 4817: 4813: 4810: 4804: 4800: 4790: 4787: 4780: 4773: 4770: 4766: 4763: 4757: 4752: 4748: 4744: 4738: 4732: 4709: 4703: 4700: 4697: 4691: 4685: 4680: 4652: 4646: 4643: 4637: 4631: 4626: 4598: 4592: 4589: 4586: 4580: 4574: 4569: 4553: 4549: 4545: 4526: 4520: 4517: 4514: 4508: 4502: 4497: 4469: 4463: 4460: 4454: 4448: 4443: 4430: 4429: 4428: 4411: 4405: 4402: 4399: 4393: 4387: 4382: 4354: 4348: 4345: 4339: 4333: 4328: 4314: 4310: 4308: 4304: 4300: 4296: 4277: 4268: 4261: 4258: 4254: 4246: 4243: 4236: 4232: 4221: 4218: 4214: 4211: 4205: 4202: 4195: 4192: 4188: 4185: 4179: 4175: 4164: 4161: 4154: 4151: 4147: 4141: 4137: 4107: 4104: 4100: 4097: 4089: 4086: 4079: 4075: 4072: 4065: 4062: 4058: 4055: 4049: 4026: 4022: 4013: 4006: 4003: 3999: 3991: 3988: 3981: 3977: 3966: 3963: 3959: 3956: 3950: 3947: 3940: 3937: 3933: 3930: 3924: 3920: 3909: 3906: 3899: 3895: 3889: 3885: 3881: 3877: 3874: 3868: 3864: 3854: 3851: 3844: 3837: 3834: 3830: 3827: 3821: 3816: 3812: 3808: 3802: 3796: 3789: 3788: 3787: 3786:and we have: 3785: 3766: 3760: 3751: 3745: 3741: 3737: 3733: 3729: 3728: 3727: 3726: 3722: 3718: 3711:What is θ(t)? 3710: 3690: 3683: 3680: 3676: 3672: 3671: 3667: 3663: 3647: 3644: 3641: 3635: 3632: 3628: 3617: 3601: 3596: 3588: 3583: 3578: 3570: 3565: 3562: 3545: 3538: 3534: 3530: 3526: 3519: 3515: 3511: 3510: 3509: 3508: 3504: 3500: 3495: 3491: 3487: 3483: 3479: 3472: 3468: 3459: 3457: 3454: 3450: 3446: 3442: 3438: 3431: 3428: 3425: 3419: 3417: 3415: 3411: 3407: 3403: 3393: 3391: 3390: 3386: 3382: 3377: 3374: 3371: 3364: 3362: 3359: 3358: 3354: 3350: 3349:Paranoidhuman 3341: 3339: 3337: 3333: 3329: 3325: 3307: 3281: 3258: 3238: 3218: 3209: 3193: 3164: 3147: 3130: 3127: 3124: 3118: 3094: 3091: 3087: 3084: 3078: 3055: 3052: 3049: 3043: 3034: 3031: 3025: 3023: 3022: 3018: 3014: 3008: 2991: 2988: 2982: 2976: 2970: 2967: 2964: 2958: 2955: 2952: 2946: 2940: 2933: 2932: 2931: 2902: 2899: 2896: 2890: 2887: 2884: 2878: 2875: 2872: 2866: 2863: 2857: 2854: 2851: 2845: 2842: 2836: 2833: 2830: 2824: 2818: 2815: 2812: 2806: 2803: 2797: 2794: 2791: 2785: 2782: 2776: 2770: 2760: 2759: 2758: 2755: 2751: 2731: 2728: 2720: 2717: 2714: 2708: 2702: 2696: 2692: 2689: 2683: 2677: 2670: 2669: 2668: 2665: 2659: 2657: 2638: 2635: 2632: 2629: 2622: 2621: 2620: 2601: 2598: 2592: 2586: 2580: 2577: 2574: 2568: 2565: 2562: 2559: 2552: 2551: 2550: 2531: 2528: 2522: 2516: 2510: 2507: 2504: 2498: 2495: 2492: 2489: 2486: 2483: 2477: 2471: 2465: 2462: 2459: 2453: 2450: 2447: 2444: 2438: 2432: 2425: 2424: 2423: 2401: 2398: 2395: 2389: 2386: 2380: 2377: 2374: 2368: 2365: 2358: 2357: 2356: 2337: 2334: 2328: 2322: 2316: 2313: 2310: 2304: 2301: 2298: 2292: 2286: 2279: 2278: 2277: 2274: 2271: 2264: 2260: 2256: 2252: 2248: 2247: 2246: 2245: 2241: 2237: 2233: 2229: 2225: 2217: 2213: 2209: 2205: 2201: 2200: 2199: 2198: 2194: 2190: 2189:144.32.126.14 2181: 2179: 2178: 2175: 2167: 2165: 2164: 2161: 2155: 2154: 2151: 2147: 2143: 2136: 2128: 2124: 2120: 2115: 2114: 2113: 2112: 2108: 2104: 2098: 2097: 2094: 2090: 2080: 2076: 2072: 2067: 2064: 2063: 2062: 2058: 2054: 2053:61.213.72.164 2049: 2048: 2047: 2046: 2042: 2038: 2034: 2033:134.157.226.1 2030: 2023: 2022: 2019: 2016: 2012: 2011: 2010: 2009: 2006: 2000: 1999: 1995: 1989: 1980: 1978: 1977: 1973: 1969: 1965: 1960: 1959: 1955: 1951: 1932: 1929: 1923: 1917: 1911: 1908: 1905: 1899: 1896: 1893: 1887: 1881: 1874: 1873: 1872: 1868: 1862: 1859: 1855: 1851: 1847: 1846: 1845: 1844: 1841: 1836: 1835: 1832: 1828: 1823: 1822: 1819: 1815: 1808: 1800: 1797: 1793: 1792: 1791: 1790: 1789: 1788: 1787: 1786: 1783: 1761: 1758: 1750: 1744: 1738: 1735: 1732: 1717: 1712: 1708: 1704: 1698: 1692: 1685: 1684: 1682: 1663: 1660: 1657: 1639: 1635: 1630: 1627: 1608: 1605: 1602: 1596: 1588: 1569: 1566: 1563: 1545: 1544: 1543: 1529: 1526: 1522: 1519: 1515: 1503: 1499: 1496: 1492: 1488: 1487: 1486: 1485: 1482: 1471: 1470: 1467: 1466:24.25.211.241 1459:More Examples 1458: 1456: 1454: 1450: 1446: 1442: 1432: 1429: 1407: 1401: 1397: 1379: 1375: 1372: 1353: 1346: 1340: 1332: 1328: 1324: 1321: 1317: 1312: 1306: 1274: 1268: 1264: 1246: 1242: 1239: 1220: 1213: 1207: 1204: 1198: 1170: 1166: 1145: 1101: 1097: 1093: 1088: 1085: 1077: 1052: 1012: 1009: 1006: 1003: 1000: 992: 978: 977: 976: 973: 972: 969: 960: 958: 957: 954: 949: 943: 941: 938: 935: 933: 929: 923: 922: 914: 908: 907: 904: 901: 897: 893: 892: 891: 888: 887: 884: 871: 868: 863: 862: 857: 854: 850: 846: 845: 844: 843: 840: 837: 833: 829: 825: 824: 818: 814: 810: 806: 799: 795: 791: 787: 783: 779: 775: 771: 767: 763: 759: 755: 751: 748:compare with 747: 746: 745: 744: 741: 738: 734: 730: 729:Abelian Group 726: 722: 721: 717: 713: 712: 711: 705: 703: 702: 698: 694: 684: 682: 668: 665: 662: 654: 648: 642: 639: 636: 630: 627: 624: 618: 612: 605: 604: 601: 593: 591: 577: 574: 571: 563: 557: 551: 548: 545: 539: 536: 533: 527: 521: 514: 513: 510: 502: 500: 486: 480: 477: 474: 468: 465: 459: 456: 453: 447: 444: 437: 436: 433: 425: 423: 409: 403: 400: 397: 391: 388: 382: 379: 376: 370: 365: 361: 353: 352: 349: 346: 345: 341: 337: 329: 327: 311: 307: 301: 298: 297: 294: 277: 273: 269: 268: 260: 254: 249: 247: 244: 240: 239: 235: 229: 226: 223: 219: 214: 210: 204: 196: 192: 187: 186: 177: 173: 170: 167: 163: 159: 155: 152: 149: 146: 143: 140: 137: 134: 131: 127: 124: 123:Find sources: 120: 119: 111: 110:Verifiability 108: 106: 103: 101: 98: 97: 96: 87: 83: 81: 78: 76: 72: 69: 67: 64: 63: 57: 53: 52:Learn to edit 49: 46: 41: 40: 37: 36: 32: 26: 22: 18: 17: 5728: 5709: 5705: 5640: 5638: 5618:— Preceding 5610: 5602: 5582: 5578: 5574: 5569: 5546: 5543: 5518:source check 5497: 5491: 5488: 5461: 5458: 5435: 5410:source check 5389: 5383: 5368: 5364: 5360: 5358: 5319: 5316: 5293:— Preceding 5290: 5287: 5283: 5255:— Preceding 5251: 5248: 5245: 4972:To instead: 4556: 4315: 4311: 4293:— Preceding 4041: 3752: 3749: 3717:172.56.9.253 3714: 3695:. Retrieved 3682: 3674: 3662:TChapProctor 3618: 3549: 3523:— Preceding 3496: 3476:— Preceding 3463: 3435:— Preceding 3432: 3429: 3426: 3423: 3400:— Preceding 3397: 3378: 3375: 3372: 3368: 3360: 3345: 3342:Second Order 3328:129.2.175.79 3210: 3148: 3035: 3032: 3029: 3009: 3006: 2929: 2923:G(x,s)-: --> 2915:G(x,s)-: --> 2765:G(x,s)-: --> 2756: 2752: 2748: 2666: 2663: 2655: 2618: 2548: 2421: 2354: 2275: 2272: 2268: 2231: 2227: 2223: 2221: 2185: 2171: 2160:HowiAuckland 2156: 2139: 2119:TMHutchcroft 2099: 2087: 2001: 1984: 1961: 1947: 1869: 1865: 1849: 1837: 1824: 1816: 1812: 1779: 1680: 1637: 1633: 1625: 1586: 1507: 1477: 1462: 1435: 1428:HowiAuckland 974: 964: 950: 947: 939: 936: 931: 927: 924: 918: 889: 879: 827: 803:— Preceding 709: 690: 599: 508: 431: 347: 333: 325: 306:Mid-priority 305: 265: 231:Mid‑priority 209:WikiProjects 171: 165: 157: 150: 144: 138: 132: 122: 94: 19:This is the 5702:What is X11 5376:Sourcecheck 3322:—Preceding 2660:Convolution 2133:Merge from 2027:—Preceding 1439:—Preceding 883:Jonas Olson 770:Snell's law 281:Mathematics 272:mathematics 228:Mathematics 148:free images 31:not a forum 5747:Categories 5555:Report bug 3732:Johnkeevil 3675:References 3499:Enyokoyama 3441:Frankdj100 1988:Mel Etitis 1858:MathMartin 1831:MathMartin 5689:Tertionix 5538:this tool 5531:this tool 5430:this tool 5423:this tool 2916:G(x-s)\,} 2251:Thenub314 2236:Jjauregui 2204:mike40033 2103:GaborPete 1950:Nickvence 1504:Intuition 968:Dan Gluck 706:Page name 88:if needed 71:Be polite 21:talk page 5620:unsigned 5544:Cheers.— 5436:Cheers.— 5332:cbignore 5295:unsigned 5269:contribs 5261:Ephecina 5257:unsigned 5228:Holmansf 4544:Holmansf 4295:unsigned 3537:contribs 3529:PassPort 3525:unsigned 3514:PassPort 3490:contribs 3482:PassPort 3478:unsigned 3467:PassPort 3449:contribs 3437:unsigned 3402:unsigned 3324:unsigned 2029:unsigned 1796:Stevvers 1495:Stevvers 1481:Stevvers 1441:unsigned 944:bad form 805:unsigned 735:, etc.-- 693:Holmansf 336:Holmansf 56:get help 29:This is 27:article. 5614:SomeGuy 5468:my edit 5446::Online 5361:checked 5326:my edit 3697:15 July 3381:Aklamas 2619:solves 2182:example 900:Pjacobi 832:Zeidler 330:Adjoint 308:on the 199:B-class 154:WP refs 142:scholar 5667:, and 5595:MaxEnt 5585:MaxEnt 5369:failed 5340:nobots 2864:=: --> 2804:=: --> 2549:Hence 2276:Write 2270:term. 1809:kernel 1118:where 1025:where 953:Wolfch 205:scale. 126:Google 5706:: --> 3692:(PDF) 2888:: --> 2265:Eq. 3 2174:Ulner 2150:Lunch 2093:Ulner 2071:Thchr 2068:: --> 2015:Lunch 1981:Title 716:3mta3 169:JSTOR 130:books 84:Seek 5735:talk 5716:talk 5693:talk 5628:talk 5365:true 5303:talk 5265:talk 5232:talk 4548:talk 4303:talk 3736:talk 3721:talk 3699:2013 3666:talk 3533:talk 3518:talk 3503:talk 3486:talk 3471:talk 3445:talk 3410:talk 3385:talk 3353:talk 3332:talk 3231:and 3017:talk 3013:Mike 2255:talk 2240:talk 2226:and 2208:talk 2193:talk 2148:). 2123:talk 2107:talk 2075:talk 2057:talk 2037:talk 1993:Talk 1972:talk 1954:talk 1818:Atmd 1449:talk 1293:and 813:talk 697:talk 340:talk 162:FENS 136:news 73:and 5512:RfC 5482:to 5404:RfC 5381:). 5367:or 5352:to 5226:pi. 4614:to 4485:or 3427:-- 3320:. 3179:to 3071:to 2232:x-s 2144:or 1840:Sim 853:Tex 830:by 685:(3) 594:(3) 503:(2) 432:or 426:(1) 300:Mid 176:TWL 5749:: 5737:) 5718:) 5695:) 5647:, 5643:, 5630:) 5525:. 5520:}} 5516:{{ 5417:. 5412:}} 5408:{{ 5379:}} 5373:{{ 5338:{{ 5334:}} 5330:{{ 5305:) 5271:) 5267:• 5234:) 5203:^ 5200:σ 5190:⋅ 5168:ϕ 5161:∇ 5134:− 5104:∇ 5086:ϕ 5072:∫ 5031:ρ 4999:∫ 4995:− 4980:ϕ 4953:^ 4950:σ 4940:⋅ 4918:ϕ 4911:∇ 4884:− 4854:∇ 4836:ϕ 4822:∫ 4781:ρ 4749:∫ 4733:ϕ 4704:ρ 4701:− 4686:ϕ 4677:∇ 4647:ρ 4632:ϕ 4623:∇ 4593:ρ 4590:− 4575:ϕ 4566:∇ 4550:) 4521:ρ 4518:− 4503:ϕ 4494:∇ 4464:ρ 4449:ϕ 4440:∇ 4406:ρ 4403:− 4388:ϕ 4379:∇ 4349:ρ 4334:ϕ 4325:∇ 4305:) 4272:^ 4269:σ 4259:⋅ 4237:ϕ 4230:∇ 4173:∇ 4155:ϕ 4152:− 4138:∫ 4101:− 4090:π 4017:^ 4014:σ 4004:⋅ 3982:ϕ 3975:∇ 3948:− 3918:∇ 3900:ϕ 3886:∫ 3845:ρ 3813:∫ 3797:ϕ 3761:ϕ 3738:) 3723:) 3668:) 3648:ρ 3645:⁡ 3642:ln 3636:π 3593:∂ 3575:∂ 3559:Δ 3539:) 3535:• 3505:) 3492:) 3488:• 3465:-- 3451:) 3447:• 3412:) 3387:) 3355:) 3334:) 3285:^ 3239:σ 3019:) 3011:-- 2968:− 2956:∫ 2900:− 2885:− 2855:− 2846:δ 2816:− 2795:− 2693:∫ 2566:∫ 2496:∫ 2448:∫ 2399:− 2390:δ 2314:− 2305:δ 2302:∫ 2257:) 2242:) 2210:) 2195:) 2125:) 2109:) 2101:-- 2077:) 2059:) 2039:) 2005:TJ 1996:) 1986:-- 1974:) 1956:) 1897:∫ 1709:∫ 1606:− 1597:δ 1516:− 1451:) 1412:′ 1384:′ 1376:− 1358:′ 1347:ρ 1341:∫ 1329:ϵ 1325:π 1301:Φ 1279:′ 1251:′ 1243:− 1225:′ 1214:ρ 1208:∫ 1193:Φ 1167:ϵ 1146:ρ 1126:Φ 1098:ϵ 1094:ρ 1089:− 1083:Φ 1074:∇ 1053:ρ 1033:Φ 1013:ρ 1010:π 1004:− 998:Φ 989:∇ 867:TJ 851:-- 815:) 788:, 784:, 780:, 776:, 772:, 768:, 764:, 760:, 756:, 752:, 731:, 727:, 699:) 628:∫ 537:∫ 478:− 469:δ 401:− 392:δ 366:∗ 342:) 156:) 54:; 5733:( 5714:( 5691:( 5675:L 5655:L 5645:L 5641:L 5626:( 5557:) 5553:( 5540:. 5533:. 5432:. 5425:. 5301:( 5263:( 5230:( 5209:′ 5193:d 5186:] 5182:) 5178:′ 5175:x 5171:( 5164:′ 5157:) 5153:′ 5150:x 5146:, 5143:x 5140:( 5137:G 5131:) 5127:′ 5124:x 5120:, 5117:x 5114:( 5111:G 5107:′ 5100:) 5096:′ 5093:x 5089:( 5082:[ 5076:S 5068:+ 5064:′ 5061:x 5055:3 5051:d 5045:) 5041:′ 5038:x 5034:( 5028:) 5024:′ 5021:x 5017:, 5014:x 5011:( 5008:G 5003:V 4992:= 4989:) 4986:x 4983:( 4959:′ 4943:d 4936:] 4932:) 4928:′ 4925:x 4921:( 4914:′ 4907:) 4903:′ 4900:x 4896:, 4893:x 4890:( 4887:G 4881:) 4877:′ 4874:x 4870:, 4867:x 4864:( 4861:G 4857:′ 4850:) 4846:′ 4843:x 4839:( 4832:[ 4826:S 4818:+ 4814:′ 4811:x 4805:3 4801:d 4795:) 4791:′ 4788:x 4784:( 4778:) 4774:′ 4771:x 4767:, 4764:x 4761:( 4758:G 4753:V 4745:= 4742:) 4739:x 4736:( 4713:) 4710:x 4707:( 4698:= 4695:) 4692:x 4689:( 4681:2 4656:) 4653:x 4650:( 4644:= 4641:) 4638:x 4635:( 4627:2 4602:) 4599:x 4596:( 4587:= 4584:) 4581:x 4578:( 4570:2 4546:( 4530:) 4527:x 4524:( 4515:= 4512:) 4509:x 4506:( 4498:2 4473:) 4470:x 4467:( 4461:= 4458:) 4455:x 4452:( 4444:2 4415:) 4412:x 4409:( 4400:= 4397:) 4394:x 4391:( 4383:2 4358:) 4355:x 4352:( 4346:= 4343:) 4340:x 4337:( 4329:2 4301:( 4278:′ 4262:d 4255:] 4251:) 4247:′ 4244:x 4240:( 4233:′ 4226:) 4222:′ 4219:x 4215:, 4212:x 4209:( 4206:G 4203:+ 4200:) 4196:′ 4193:x 4189:, 4186:x 4183:( 4180:G 4176:′ 4169:) 4165:′ 4162:x 4158:( 4148:[ 4142:S 4117:) 4113:| 4108:′ 4105:x 4098:x 4094:| 4087:4 4084:( 4080:/ 4076:1 4073:= 4070:) 4066:′ 4063:x 4059:, 4056:x 4053:( 4050:G 4027:. 4023:′ 4007:d 4000:] 3996:) 3992:′ 3989:x 3985:( 3978:′ 3971:) 3967:′ 3964:x 3960:, 3957:x 3954:( 3951:G 3945:) 3941:′ 3938:x 3934:, 3931:x 3928:( 3925:G 3921:′ 3914:) 3910:′ 3907:x 3903:( 3896:[ 3890:S 3882:+ 3878:′ 3875:x 3869:3 3865:d 3859:) 3855:′ 3852:x 3848:( 3842:) 3838:′ 3835:x 3831:, 3828:x 3825:( 3822:G 3817:V 3809:= 3806:) 3803:x 3800:( 3770:) 3767:x 3764:( 3734:( 3719:( 3701:. 3664:( 3633:2 3629:1 3602:2 3597:y 3589:+ 3584:2 3579:x 3571:= 3566:D 3563:2 3531:( 3516:( 3501:( 3484:( 3469:( 3443:( 3408:( 3383:( 3351:( 3330:( 3308:A 3282:A 3259:A 3219:A 3194:n 3189:R 3165:m 3160:R 3134:) 3131:s 3128:, 3125:x 3122:( 3119:G 3099:) 3095:′ 3092:x 3088:, 3085:x 3082:( 3079:G 3059:) 3056:s 3053:, 3050:x 3047:( 3044:G 3015:( 2992:s 2989:d 2986:) 2983:s 2980:( 2977:f 2974:) 2971:s 2965:x 2962:( 2959:G 2953:= 2950:) 2947:x 2944:( 2941:u 2926:. 2906:) 2903:s 2897:x 2894:( 2891:G 2882:) 2879:s 2876:, 2873:x 2870:( 2867:G 2861:) 2858:s 2852:x 2849:( 2843:= 2840:) 2837:s 2834:, 2831:x 2828:( 2825:G 2822:) 2819:s 2813:x 2810:( 2807:L 2801:) 2798:s 2792:x 2789:( 2786:L 2783:= 2780:) 2777:x 2774:( 2771:L 2745:. 2732:s 2729:d 2724:) 2721:s 2718:, 2715:x 2712:( 2709:G 2706:) 2703:s 2700:( 2697:f 2690:= 2687:) 2684:x 2681:( 2678:u 2639:f 2636:= 2633:u 2630:L 2602:s 2599:d 2596:) 2593:s 2590:( 2587:f 2584:) 2581:s 2578:, 2575:x 2572:( 2569:G 2563:= 2560:u 2532:s 2529:d 2526:) 2523:s 2520:( 2517:f 2514:) 2511:s 2508:, 2505:x 2502:( 2499:G 2493:L 2490:= 2487:s 2484:d 2481:) 2478:s 2475:( 2472:f 2469:) 2466:s 2463:, 2460:x 2457:( 2454:G 2451:L 2445:= 2442:) 2439:x 2436:( 2433:f 2405:) 2402:s 2396:x 2393:( 2387:= 2384:) 2381:s 2378:, 2375:x 2372:( 2369:G 2366:L 2338:s 2335:d 2332:) 2329:s 2326:( 2323:f 2320:) 2317:s 2311:x 2308:( 2299:= 2296:) 2293:x 2290:( 2287:f 2253:( 2238:( 2228:s 2224:x 2206:( 2191:( 2121:( 2105:( 2073:( 2055:( 2035:( 1990:( 1970:( 1952:( 1933:y 1930:d 1927:) 1924:y 1921:( 1918:f 1915:) 1912:y 1909:, 1906:x 1903:( 1900:G 1894:= 1891:) 1888:x 1885:( 1882:u 1850:L 1774:. 1762:y 1759:d 1754:) 1751:y 1748:( 1745:f 1742:) 1739:y 1736:, 1733:x 1730:( 1725:G 1718:b 1713:a 1705:= 1702:) 1699:x 1696:( 1693:u 1681:V 1667:) 1664:y 1661:, 1658:x 1655:( 1650:G 1638:x 1634:y 1626:f 1612:) 1609:y 1603:x 1600:( 1587:u 1573:) 1570:y 1567:, 1564:x 1561:( 1556:G 1530:f 1527:= 1523:″ 1520:u 1493:. 1447:( 1408:x 1402:3 1398:d 1390:| 1380:x 1373:x 1369:| 1363:) 1354:x 1350:( 1333:0 1322:4 1318:1 1313:= 1310:) 1307:x 1304:( 1275:x 1269:3 1265:d 1257:| 1247:x 1240:x 1236:| 1230:) 1221:x 1217:( 1205:= 1202:) 1199:x 1196:( 1171:0 1102:0 1086:= 1078:2 1007:4 1001:= 993:2 811:( 695:( 669:. 666:s 663:d 658:) 655:s 652:( 649:f 646:) 643:x 640:, 637:s 634:( 631:G 625:= 622:) 619:x 616:( 613:u 578:. 575:s 572:d 567:) 564:s 561:( 558:f 555:) 552:s 549:, 546:x 543:( 540:G 534:= 531:) 528:x 525:( 522:u 487:. 484:) 481:s 475:x 472:( 466:= 463:) 460:s 457:, 454:x 451:( 448:G 445:L 410:, 407:) 404:s 398:x 395:( 389:= 386:) 383:s 380:, 377:x 374:( 371:G 362:L 338:( 312:. 211:: 172:· 166:· 158:· 151:· 145:· 139:· 133:· 128:( 58:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index