Talk:Cubic function/Archive 4

2167: 275:

written algorithm is that in a program, names will be given to the expressions which appear several times, to avoid to compute them several times. For better readability, names have been given only to the quantities whose sign or equality to 0 have to be tested. I do not see any way to improve the algorithmic aspect of this page, while keeping a good readability, especially for beginners. Note that the first general formula is not useful from a logical point of view, and could thus be suppressed. But it has to appear, as it is the formula which is the most popular.

2092:: The geometric interpretation given in the caption of the figure at the top of the article should be recalled here. As given here, the geometric interpretation of Vieta's trigonometric formula appears to be simply an explanation of what a cosine and an arccosine are. Thus a figure is needed on which an equilateral triangle inscribed in a circle appears, whose edges project on the roots. A further explanation is needed, showing how the circle and the angle which has to be trisected may be graphically constructed. 2106:: As written, the object of this section is not clear, as it consider the roots as given. Thus I would have began it as: "With one real and two complex roots, the three roots can be represented as points in the complex plane, as well as the two roots of the derivative. There is an interesting relationship between all these roots ... " One may add somewhere the fact that the roots of the derivative are real or complex depending if the summit angle of the triangle is smaller or larger than 2257:

what is . Searching more, it appears that it is the version for printing of a page of MacTutor History of Mathematics and that is the second reference. MacTutor is usually reliable but it may be wrong on some points. Thus I need to read this second reference to verify. It seems not to be on the net, but it seems to be in a library in my town. Thus I'll consult it tomorrow. For the moment I'll supress the first reference and tagging this citation.

31: 6453:: I do not agree with the IP users on the change of name, but I agree on the expansion of the section: That is to introduce the successive derivatives and to explain that the root of the second derivative gives the inflexion point and that, up to a factor of 6, the 3d derivative is the leading coefficient, and that its sign says if the infinite branches are increasing or decreasing. 3219: 3498: 6429:: The remark of the IP user is worthwhile, even if it is not well expressed: All the displayed cubics have the same shape: two real roots for the derivative and a positive third derivative (leading coefficient). It would be useful to display all the four different possible shapes (in fact 6 if one counts the cases of a double root of the derivative). 4880: 2391: 2848:= 49 :1) but, even if so, the geometry is lost: a circle will be squeezed to an ellipse, a normal line will not seem perpendicular to tangency etc. This inconveniency can be avoided by means of the routine applied at "Trigonometric (and hyperbolic) method" where monic depressed cubic equation is, in fact, converted into its canonical form. 5165: 4881:

http://www.mathopenref.com/graphfunctions.html?fx=(3*b%5e2+1)*x%5e3+3*b*x%20&gx=a*(3*((c/d)%5e2+b)*(x-c/d)+872/7%5e3)+(1-a)*143/7%5e3&hx=a*6*((c/d)%5e2+b)*(x-c/d)+872/7%5e3&xh=2&yl=-1.5&ah=1&a=0&bh=1&bl=-1&b=-1&ch=99&c=16&dh=99&dl=1&d=14&cr=t&cx=1.14286

6223: 5753: 4118: 2134:. As for the other sections, unfortunately I have zero skills in drawing figures. I'll see if I can get someone to draw the relevant figures, and then I'll need instructions on how to bring them into Knowledge (XXG). Is it necessary to use a particular graphics package in order to get it into Knowledge (XXG)? 5914: 6268:

On the contrary, my intention was to show how this long article can be shortened rooming three images into the slightly expanded image from chapter “Derivative” the textual part of which would be completed with few additional lines as D. Lazard formulated below. You claim above that in graphs 3 and 4

2256:

JamesBWatson has just restored this controversial assertion, sourcing it by 3 references. The first one is a blog which reproduces textually the first one, without reference. It should thus be removed as non reliable, plagiarism and not useful. The third reference refers to without any indication of

3738: 192:

algorithm, because computing a numerical approximation of the roots of a cubic equation with a root finding algorithm is usually much faster than evaluating numerically the formula. In fact evaluating the formula implies to compute a square root, a cubic root and several other operations. Cubic root

2252:

As I have written before, such an assertion seems an anachronism. In fact, at these time, trigonometric values were not, as far as I know, computed numerically , but graphically (recall five centuries later, division was taught only in universities, in highest courses). It is thus highly improbable

211:

You may be right about performance. I had sort of assumed that since cubics can be solved in closed form, that it would be faster than an iterative approach. Still, the "algorithms" on the page aren't at all clear. I should be able to use the information on this page to find the roots of a cubic

274:

looks perfectly like an algorithm. It specifies the auxiliary functions (square and cubic roots). Its main difference with a well written algorithm is that the simple cases are at the end while a program is easier to read if the simple cases are at the beginning. Another difference with a well

2887: 6401:

The summary box of the graph file was incorrect -- I should have been more careful in writing it, and I have now corrected it. Thanks for pointing it out! However, I don't have the graphics skills to redraw the graph for the shifted depressed case, and anyway I think there's value in showing

3225: 4871: 4490: 2218:

This figure is quite misleading. It looks like the vertical line through the point of intersection of the circle and parabola also goes through the center of the circle. This makes the solution (the length of the red segment) in this case appear to be the radius of the circle.

4287: 3911: 5022: 4660: 107:

I'm writing software that needs to do a cubic solve. In Python, numpy.roots works great, but is slow for solving thousands of them in parallel, so I came here in search of the algorithm... but it isn't at all clear what the full algorithm is. Other web pages fail me too.

2328:...on Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. 2015:...on Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations. In that his solution is not a direct path to a numerical solution and in fact his solutions are not numbers but rather line segments. 1998:

He is the author of one of the most important treatises on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cubic equations by intersecting a hyperbola with a

301:

Some times ago I saw that sympy implement Cardano's formulas. However, I would have to consult the source again to find out if it is in any way stabilized. It is not to much effort to first implement the shift and then use the simple steps of the

5028: 6019: 5546: 3915: 1967:. OK, but this is not an indication that Omar Khayyám did use them. As I am unable to source my belief about graphical computation at that time, and I wish not to begin an edit war, I simply propose to remove the questionable assertion 2099:: Here also a figure is needed. Probably one should add a proof that this interpretation of the complex roots is correct, but this may be discussed. I have done an algebraic proof. A purely geometric proof would be much better if any. 5775: 2829:= 286) to be point of the inflection, of uneven Symmetry as well (see Knowledge (XXG) article on Inflection point where from the graph is overtaken and where negative sign before 400 at lower turning point is also missing). 3519: 3214:{\displaystyle {\frac {p_{3}(X)}{a_{3}}}={\frac {a_{3}X^{3}+a_{2}X^{2}+a_{1}X+a_{0}}{a_{3}}}=(X-s)^{3}+p(X-s)+q=0{\big |}*{\frac {4}{R^{3}}},s={\frac {-a_{2}}{3a_{3}}},R=-{\frac {2q}{|q|}}{\sqrt {\frac {|p|}{3}}}\Rightarrow } 3493:{\displaystyle 4\left({\frac {X-s}{R}}\right)^{3}+{\frac {3p}{|p|}}\left({\frac {X-s}{R}}\right)=4x^{3}+{\frac {3p}{|p|}}x={\frac {4q}{R^{3}}}\geq 0{\text{ and }}X=s+Rx=s-{\frac {2q}{|q|}}{\sqrt {\frac {|p|}{3}}}x.} 2747: 2365:. Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent (see 4664: 4291: 187:

Do you really need a cubic solver whose output is thousands formulas with cubic and square roots inside? This seems incredible. I guess that you want numerical output. For this it is clearly better to use a

5261: 5513: 4125: 3744: 6013: 4925: 4498: 1955:

is not sourced. On the other hand, the use of abaci to solve graphically numerical problem was very usual during the XVIe century and surely much before . Therefore I have edited this page, replacing

2085:

This section is a very good idea. However, as it is, without any figure, it is difficult to understand, even for an expert (I believe I am one). Even with figures, more explanation would be needed:

6359:

with three real roots, the roots form an equilateral triangle with vertices A, B, and C in the circle. The angle... It confuses me. The easiest way to avoid this incoherence is to rename abscise

5415: 336:

algorithm and it seems to work up to rounding error. For reference. This implementation solves many 4xn vectors at the same time (hence the lack of if/else and instead the use of masked arrays):

6248:

Everything looks fine to me as is, and I think putting in the extended information that you suggest would make an already very long article too long. Also, see my inserted comments above.

2813: 5160:{\displaystyle {\frac {143}{7^{3}}}=f\left(-0.1429\approx -{\frac {2}{14}}\right)=f\left(-0.7857\approx -{\frac {11}{14}}\right)=f\left(0.9286\approx {\frac {13}{14}}\right)} 6357: 2040:

By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.

2512: 153: 6309: 6218:{\displaystyle Re(x_{1;2})=x_{H}=-{\frac {x_{R}}{2}}=-{\frac {1}{28}}\approx 0.0357,x_{A}\approx -0.8325,Im(x_{1;2})=x_{A}-x_{H}={\frac {\sqrt {591}}{28}}\approx 0,8682.} 5748:{\displaystyle Re(x_{1;2})=x_{H}=-{\frac {x_{R}}{2}}=-{\frac {4}{7}}\approx -0.5714,x_{A}\approx -0.0922,Im(x_{1;2})=x_{A}-x_{H}={\frac {\sqrt {45}}{14}}\approx 0,4792.} 4113:{\displaystyle 4\left({\frac {X-1}{-2*7}}\right)^{3}-3\left({\frac {X-1}{-2*7}}\right)=4x^{3}-3x=f(x)=g(x)={\frac {143}{7^{3}}}\approx 0.4169<1{\text{ and }}p<0.} 6483:) where the infinite branches are increasing. Entering minus at the beginning of f-box we get remaining three shapes where the infinite branches are decreasing. Stap 5909:{\displaystyle h(x)={\frac {74}{7^{3}}}\approx 0.2157{\text{ intercepts }}f(x)=4x^{3}+3x{\text{ at R where }}x_{R}\approx 0.0714={\frac {1}{14}}={\frac {c}{d}}} 272: 252: 2586:

is a parameter of the depressed case, which none of the graphs show. However, all cubics are linear transformations of the depressed case, in which p=0 or p: -->

2300:

Khayyam took an indirect approach , ending up with a slightly different cubic, which he solved numerically via the intersection of two classic geometric curves.

6311:

along with its roots in next line. Moreover there is comment at File history authorized by Duoduoduo: ({{Information |Description = For the depressed cubic

3733:{\displaystyle s={\frac {-a_{2}}{3a_{3}}}=1,{\text{ }}p={\frac {p'_{3}(s)}{a_{3}}}=\mp 3*7^{2}{\text{ and }}q={\frac {p_{3}(s)}{a_{3}}}=2*143;-2*872;-2*74.} 2253:

that Khayyam preferred to report in (graphical) the length obtained by the intersection of a circle and an ellipse than to simply measure this length.

2587:

0 is a sufficient (but not necessary) condition for one real root. The next-to-last graph shows the case of a non-depressed cubic with one real root.

5541:) at three points R, A and B determining real (Re) and imaginary (Im) part of conjugate roots as presented at chapter 3.8.2.1 of the article. 4864:{\displaystyle 4\left({\frac {X-1}{2*7}}\right)^{3}+3\left({\frac {X-1}{2*7}}\right)=4x^{3}+3x=f(x)=h(x)={\frac {74}{7^{3}}}\approx 0,21574: --> 4484:{\displaystyle 4\left({\frac {X-1}{2*7}}\right)^{3}-3\left({\frac {X-1}{2*7}}\right)=4x^{3}-3x=f(x)=h(x)={\frac {872}{7^{3}}}\approx 1.7055: --> 2166: 2649: 6484: 6376: 6234: 2226: 1949:

Omar Khayyám (1048–1131) ... found a geometric solution which could be used to get a numerical answer by consulting trigonometric tables

4282:{\displaystyle X^{3}-3X^{2}-144X-1598=(X-1)^{3}-3*7^{2}(X-1)-2*872=0{\big |}*{\frac {4}{14^{3}}}{\text{ which can be bisected into:}}} 3906:{\displaystyle X^{3}-3X^{2}-144X+432=(X-1)^{3}-3*7^{2}(X-1)+2*143=0{\big |}*{\frac {-4}{14^{3}}}{\text{ which can be bisected into:}}} 5017:{\displaystyle g(x)={\frac {143}{7^{3}}}<1{\text{ intercepts }}f(x)=4x^{3}-3x{\text{ at the points determining 3 real roots:}}} 4655:{\displaystyle X^{3}-3X^{2}+152X-300=(X-1)^{3}+3*7^{2}(X-1)-2*74=0{\big |}*{\frac {4}{14^{3}}}{\text{ which can be bisected into:}}} 4920:) at only one. The abscises of these points are real roots of canonical cubic form for first and second example where p < 0: 5172: 5421: 5921: 2366: 2316:. However, it seems to label as false the passage that I quoted in an earlier section on this talk page, from the Wiki page 5268: 1951:. I do not believe that, at that time, it was usual to use numerical tables to solve equations. In any case the assertion 94: 86: 81: 69: 64: 59: 111:

Am I just blind, or are none of the algorithms listed completely? In the "General formula of roots" section, it says if

5416:

1{\text{ intercepts }}f(x)=4x^{3}-3x{\text{ at R where }}x_{R}\approx 1.14286={\frac {16}{14}}={\frac {c}{d}}}" /: -->

6402:

graphically that the statement in the text that a trig approach is possible remains true for the non-depressed case.

5269:

1{\text{ intercepts }}f(x)=4x^{3}-3x{\text{ at R where }}x_{R}\approx 1.14286={\frac {16}{14}}={\frac {c}{d}}}": -->

2170:

and its explanation. This suggest to include another subsection, including this image and explaining this solution.

38: 6507: 2358:

This article has been edited by a user who is known to have misused sources to unduly promote certain views. See

5410:

1{\text{ intercepts }}f(x)=4x^{3}-3x{\text{ at R where }}x_{R}\approx 1.14286={\frac {16}{14}}={\frac {c}{d}}}

2446:(inserted comment) The graphs are not of depressed cubics, and are not intended to be, so they are in terms of 6488: 6380: 6238: 2756: 2130:

Thanks for the suggestions. I've incorporated the wording changes and additions you suggested in the section

2230: 1928: 217: 177: 3502:

This conversion is applied below to three characteristic examples composed of the variables being abscise (

2851:

Graphical Function Explorer (GFE) is an easily handled but very suitable graphing tool where the variables

2222: 193:

extraction is not really faster than a general root finding algorithm applied to a general cubic equation.

2284:

posed and solved several problems involving cubic equations, though his solutions were always geometrical.

1920: 1986:

I'm not taking sides here because I don't know the history. But I would mention that the article on

6407: 6253: 2634: 2592: 2557: 2526: 2459: 2345: 2197: 2139: 2046: 47: 17: 2153: 114: 6458: 6434: 6314: 2403: 2262: 2175: 2117: 1976: 1924: 280: 213: 198: 173: 2472: 6272: 2376: 6230:

is already included as External link. Perhaps the author could update it as suggested above.

307: 2317: 1987: 6515: 6403: 6249: 2630: 2588: 2553: 2522: 2455: 2341: 2193: 2135: 2066: 2042: 257: 237: 6454: 6430: 2399: 2258: 2171: 2113: 1972: 276: 234:

I do not agree with you about the clarity of the "algorithms": The formula involving

194: 6519: 6492: 6462: 6438: 6411: 6384: 6257: 6242: 2638: 2596: 2561: 2530: 2463: 2407: 2380: 2349: 2266: 2234: 2201: 2179: 2143: 2121: 2070: 2050: 1980: 1932: 311: 284: 221: 202: 181: 2372: 2029: 1963:. This seems less anachronistic. This has been reverted with the explanation that 189: 6227: 2536:

In case of Depressed cubic the point of the inflection and of uneven Symmetry S(

1969:

which could be used to get a numerical answer by consulting trigonometric tables

1953:

which could be used to get a numerical answer by consulting trigonometric tables

46:

If you wish to start a new discussion or revive an old one, please do so on the

2340:

get numerical solutions from his geometric figures, contrary to that passage.

1919:

I may rephrase the description of this algorithm to make the description more

303: 2156:

there is some information. I had a look on the available images by searching

156:, then the solutions are ___. But then it gives caveats and the version with 6511: 2469:

Since we deal with polynomials of degree 3 it seems to me suitable to write

2062: 2552:(inserted comment) Those graphs are of the general, not depressed, case. 2248:

Using trigonometric tables for solving a cubic equation in 11th centuries

212:

function, but attempts to follow the steps failed, usually in ambiguity.

2548:) must be on y-axis which isn’t respected at fourth and fifth graph. 2742:{\displaystyle p''_{3}(x)=6ax+2b=0{\text{ for }}x=s={\frac {-b}{3a}}} 2023:

That's a pretty specific assertion: his solutions are not numbers.

4887:

an image enabling graphical resolving of all cubic equation types

2450:. But sometimes the text mentions the depressed case in terms of 2454:, since that is sometimes easier to explain. So that is fine. 172:

version complete? If so, it doesn't read like an algorithm...

25: 6375:)-axis at the inflection point and center of the circle. Stap 6269:

are not depressed cubic but in second line of joined text is

6506:— See multiple talk page abuse warnings to IP (aka STAP) at 5409:{\displaystyle h(x)={\frac {872}{7^{3}}}\approx 2.5423: --> 5256:{\displaystyle X_{k}=1-{\frac {14*q}{|q|}}*x_{k}=3;\pm 12.} 2162:

File:Omar Kayyám - Geometric solution to cubic equation.svg

1965:

Hipparchus compiled trigonometric tables in the 1st century

5508:{\displaystyle X_{R}=1-{\frac {14*q}{|q|}}*x_{R}=1+16=17.} 2274:

Unknown Quantity: A Real and Imaginary History of Algebra

1961:

which could be used to get graphically a numerical answer

6479:

successively to –1, 0, +1 GFE returns three shapes of f(

6008:{\displaystyle X_{R}=1-{\frac {14*q}{|q|}}*x_{R}=1+1=2.} 2568:

None of six graphs is presenting the cubic where either

2521:

is a commonly used function name--no need to change it.

2152:

I, also, have few skills in drawing figure. However, in

2362: 2359: 2057: 2007:

No mention of trig. In fact, later on the article says

328:

I must have been mistaken. I just re-tried coding the

6317: 6275: 6022: 5924: 5778: 5549: 5424: 5272: 5175: 5031: 4928: 4669: 4501: 4295: 4128: 3918: 3747: 3522: 3228: 2890: 2759: 2652: 2475: 260: 240: 118: 2602:The chapter-name Derivative could be set in plural 2418:

There are now six graphs plotted in the Cartesian (

6351: 6303: 6217: 6007: 5908: 5747: 5507: 5408: 5255: 5159: 5016: 4863: 4654: 4483: 4281: 4112: 3905: 3732: 3492: 3213: 2807: 2741: 2506: 266: 246: 146: 2609:(inserted comment) There is only one derivative: 2422:) plane, but in corresponding text we have often 5762:= +1 and replacing 872 by 74 in the boxes for f( 2882:(upper case) as unknown of cubic general form. 817:# Four cases: zero or not for Q and for b2m3ac: 2817:which, since it doesn’t change a sign for any 2308:This gets tantalizingly close, using the word 6228:http://www.mathopenref.com/cubicexplorer.html 4622: 4249: 3868: 3077: 8: 2840:aspect ratio should be at least mentioned ( 6322: 6316: 6280: 6274: 6191: 6182: 6169: 6147: 6116: 6093: 6076: 6070: 6058: 6036: 6021: 5981: 5966: 5958: 5944: 5929: 5923: 5896: 5883: 5868: 5859: 5844: 5817: 5803: 5794: 5777: 5721: 5712: 5699: 5677: 5646: 5620: 5603: 5597: 5585: 5563: 5548: 5481: 5466: 5458: 5444: 5429: 5423: 5396: 5383: 5368: 5359: 5344: 5317: 5297: 5288: 5271: 5232: 5217: 5209: 5195: 5180: 5174: 5142: 5110: 5072: 5041: 5032: 5030: 5009: 4994: 4967: 4953: 4944: 4927: 4847: 4821: 4812: 4764: 4724: 4708: 4678: 4668: 4647: 4639: 4630: 4621: 4620: 4581: 4562: 4522: 4506: 4500: 4467: 4447: 4438: 4390: 4350: 4334: 4304: 4294: 4274: 4266: 4257: 4248: 4247: 4208: 4189: 4149: 4133: 4127: 4096: 4076: 4067: 4019: 3976: 3960: 3927: 3917: 3898: 3890: 3876: 3867: 3866: 3827: 3808: 3768: 3752: 3746: 3680: 3660: 3653: 3642: 3636: 3612: 3589: 3582: 3572: 3554: 3539: 3529: 3521: 3472: 3464: 3460: 3452: 3444: 3433: 3401: 3387: 3373: 3359: 3351: 3340: 3331: 3299: 3287: 3279: 3268: 3259: 3237: 3227: 3196: 3188: 3184: 3176: 3168: 3157: 3136: 3121: 3111: 3094: 3085: 3076: 3075: 3036: 3009: 2998: 2982: 2969: 2959: 2946: 2936: 2929: 2918: 2898: 2891: 2889: 2769: 2764: 2758: 2719: 2702: 2657: 2651: 2486: 2474: 259: 239: 129: 117: 2808:{\displaystyle p_{3}^{(3)}(x)=6a\not =0} 5011:at the points determining 3 real roots: 4908:= 14) black horizontal straight line g( 2272:For what it's worth, Derbyshire, John, 44:Do not edit the contents of this page. 2878:is selected for variables as well as 2081:Geometric interpretation of the roots 164:, but that also has caveats. Is that 7: 2871:(lower case) are default. Therefore 2751:which is necessary condition and 147:{\displaystyle -27a^{2}\Delta : --> 2028:On the other hand, the article on 135: 24: 4916:) at three points but red one h( 2821:, is sufficient condition for S( 2389: 2165: 1947:In the history section, I read: 29: 4491:1{\text{ and }}p<0.}" /: --> 6159: 6140: 6048: 6029: 5967: 5959: 5831: 5825: 5788: 5782: 5689: 5670: 5575: 5556: 5467: 5459: 5331: 5325: 5282: 5276: 5218: 5210: 5169:giving after back-conversion: 4981: 4975: 4938: 4932: 4806: 4800: 4791: 4785: 4599: 4587: 4559: 4546: 4432: 4426: 4417: 4411: 4226: 4214: 4186: 4173: 4061: 4055: 4046: 4040: 3845: 3833: 3805: 3792: 3672: 3666: 3604: 3598: 3473: 3465: 3453: 3445: 3360: 3352: 3288: 3280: 3208: 3197: 3189: 3177: 3169: 3060: 3048: 3033: 3020: 2910: 2904: 2787: 2781: 2776: 2770: 2672: 2666: 2498: 2492: 1: 6427:Shape of the displayed cubics 5918:giving after back-conversion 5529:) at point H intercepting f( 5418:giving after back-conversion 4292:1{\text{ and }}p<0.}": --> 2276:, Plume, 2007, says on p. 54: 2180:01:48, 16 November 2011 (UTC) 2144:22:30, 15 November 2011 (UTC) 2122:21:39, 15 November 2011 (UTC) 6520:12:22, 21 January 2012 (UTC) 6493:18:29, 24 January 2012 (UTC) 6463:11:16, 21 January 2012 (UTC) 6439:11:16, 21 January 2012 (UTC) 6412:21:32, 24 January 2012 (UTC) 6385:18:29, 24 January 2012 (UTC) 6352:{\displaystyle t^{3}+pt+q=0} 6258:10:19, 21 January 2012 (UTC) 6243:05:42, 21 January 2012 (UTC) 3514:) of the inflection point: 2639:10:19, 21 January 2012 (UTC) 2597:10:19, 21 January 2012 (UTC) 2562:10:19, 21 January 2012 (UTC) 2531:10:19, 21 January 2012 (UTC) 2464:10:19, 21 January 2012 (UTC) 2408:16:45, 17 January 2012 (UTC) 2398:See note 8 in the article — 2381:15:43, 17 January 2012 (UTC) 2202:17:32, 2 December 2011 (UTC) 2071:11:15, 8 November 2011 (UTC) 2051:19:16, 6 November 2011 (UTC) 1981:18:54, 6 November 2011 (UTC) 1933:14:53, 17 October 2011 (UTC) 312:13:35, 13 October 2011 (UTC) 285:10:58, 14 October 2011 (UTC) 222:13:08, 13 October 2011 (UTC) 203:21:08, 12 October 2011 (UTC) 182:13:40, 11 October 2011 (UTC) 4649:which can be bisected into: 4276:which can be bisected into: 3900:which can be bisected into: 2507:{\displaystyle y=p_{3}(x).} 2350:21:20, 4 January 2012 (UTC) 2312:, but doesn't use the word 2267:18:35, 4 January 2012 (UTC) 6535: 6304:{\displaystyle t^{3}+pt+q} 2387: 2164:which contains this image 6508:User talk:188.127.120.154 2235:13:44, 16 July 2012 (UTC) 5525:) becomes tangency of f( 343: 6475:Setting unary operator 4485:1{\text{ and }}p<0.} 2292:and then says on p. 55: 1957:which could be used ... 6353: 6305: 6219: 6009: 5910: 5749: 5509: 5411: 5257: 5161: 5018: 4867: 4656: 4486: 4283: 4114: 3907: 3734: 3494: 3215: 2809: 2743: 2508: 2154:Knowledge (XXG):Images 2097:In the Cartesian plane 268: 248: 149: 6354: 6306: 6220: 6010: 5911: 5750: 5510: 5412: 5258: 5162: 5019: 4892:For initial setting ( 4872:0{\text{ and }}p: --> 4868: 4865:0{\text{ and }}p: --> 4665:0{\text{ and }}p: --> 4657: 4487: 4284: 4115: 3908: 3735: 3495: 3216: 2810: 2744: 2509: 2158:files: cubic equation 269: 249: 150: 103:Needs clear algorithm 42:of past discussions. 6315: 6273: 6020: 5922: 5776: 5547: 5422: 5270: 5173: 5029: 4926: 4667: 4499: 4293: 4126: 3916: 3745: 3520: 3226: 2888: 2757: 2650: 2473: 2367:WP:Jagged 85 cleanup 2132:In the complex plane 2104:In the complex plane 258: 238: 116: 5533:) at point R but h( 3597: 2780: 2665: 2582:(inserted comment) 2517:(inserted comment) 2438:even if unknown is 2160:, and I have found 18:Talk:Cubic function 6349: 6301: 6215: 6005: 5906: 5745: 5505: 5406: 5253: 5157: 5014: 4861: 4652: 4481: 4279: 4110: 3903: 3730: 3585: 3490: 3211: 2805: 2760: 2739: 2653: 2644:and extended by: 2504: 2442:(4 and 5 graph). 2336:Apparently he did 264: 244: 144: 6201: 6197: 6101: 6085: 5972: 5904: 5891: 5862: 5820: 5809: 5731: 5727: 5628: 5612: 5472: 5404: 5391: 5362: 5320: 5303: 5223: 5150: 5118: 5080: 5047: 5012: 4970: 4959: 4850: 4827: 4748: 4702: 4650: 4645: 4470: 4453: 4374: 4328: 4277: 4272: 4099: 4082: 4003: 3954: 3901: 3896: 3686: 3645: 3618: 3574: 3561: 3482: 3481: 3458: 3404: 3393: 3365: 3315: 3293: 3253: 3206: 3205: 3182: 3143: 3100: 3015: 2924: 2737: 2705: 2314:trigonometrically 2225:comment added by 267:{\displaystyle Q} 247:{\displaystyle C} 100: 99: 54: 53: 48:current talk page 6526: 6358: 6356: 6355: 6350: 6327: 6326: 6310: 6308: 6307: 6302: 6285: 6284: 6224: 6222: 6221: 6216: 6202: 6193: 6192: 6187: 6186: 6174: 6173: 6158: 6157: 6121: 6120: 6102: 6094: 6086: 6081: 6080: 6071: 6063: 6062: 6047: 6046: 6014: 6012: 6011: 6006: 5986: 5985: 5973: 5971: 5970: 5962: 5956: 5945: 5934: 5933: 5915: 5913: 5912: 5907: 5905: 5897: 5892: 5884: 5873: 5872: 5863: 5860: 5849: 5848: 5821: 5818: 5810: 5808: 5807: 5795: 5754: 5752: 5751: 5746: 5732: 5723: 5722: 5717: 5716: 5704: 5703: 5688: 5687: 5651: 5650: 5629: 5621: 5613: 5608: 5607: 5598: 5590: 5589: 5574: 5573: 5514: 5512: 5511: 5506: 5486: 5485: 5473: 5471: 5470: 5462: 5456: 5445: 5434: 5433: 5417: 5414: 5413: 5407: 5405: 5397: 5392: 5384: 5373: 5372: 5363: 5360: 5349: 5348: 5321: 5318: 5304: 5302: 5301: 5289: 5262: 5260: 5259: 5254: 5237: 5236: 5224: 5222: 5221: 5213: 5207: 5196: 5185: 5184: 5166: 5164: 5163: 5158: 5156: 5152: 5151: 5143: 5124: 5120: 5119: 5111: 5086: 5082: 5081: 5073: 5048: 5046: 5045: 5033: 5023: 5021: 5020: 5015: 5013: 5010: 4999: 4998: 4971: 4968: 4960: 4958: 4957: 4945: 4874: 4870: 4869: 4862: 4851: 4848: 4828: 4826: 4825: 4813: 4769: 4768: 4753: 4749: 4747: 4736: 4725: 4713: 4712: 4707: 4703: 4701: 4690: 4679: 4661: 4659: 4658: 4653: 4651: 4648: 4646: 4644: 4643: 4631: 4626: 4625: 4586: 4585: 4567: 4566: 4527: 4526: 4511: 4510: 4492: 4489: 4488: 4482: 4471: 4468: 4454: 4452: 4451: 4439: 4395: 4394: 4379: 4375: 4373: 4362: 4351: 4339: 4338: 4333: 4329: 4327: 4316: 4305: 4288: 4286: 4285: 4280: 4278: 4275: 4273: 4271: 4270: 4258: 4253: 4252: 4213: 4212: 4194: 4193: 4154: 4153: 4138: 4137: 4119: 4117: 4116: 4111: 4100: 4097: 4083: 4081: 4080: 4068: 4024: 4023: 4008: 4004: 4002: 3988: 3977: 3965: 3964: 3959: 3955: 3953: 3939: 3928: 3912: 3910: 3909: 3904: 3902: 3899: 3897: 3895: 3894: 3885: 3877: 3872: 3871: 3832: 3831: 3813: 3812: 3773: 3772: 3757: 3756: 3739: 3737: 3736: 3731: 3687: 3685: 3684: 3675: 3665: 3664: 3654: 3646: 3643: 3641: 3640: 3619: 3617: 3616: 3607: 3593: 3583: 3575: 3573: 3562: 3560: 3559: 3558: 3545: 3544: 3543: 3530: 3510:) and ordinate ( 3499: 3497: 3496: 3491: 3483: 3477: 3476: 3468: 3462: 3461: 3459: 3457: 3456: 3448: 3442: 3434: 3405: 3402: 3394: 3392: 3391: 3382: 3374: 3366: 3364: 3363: 3355: 3349: 3341: 3336: 3335: 3320: 3316: 3311: 3300: 3294: 3292: 3291: 3283: 3277: 3269: 3264: 3263: 3258: 3254: 3249: 3238: 3220: 3218: 3217: 3212: 3207: 3201: 3200: 3192: 3186: 3185: 3183: 3181: 3180: 3172: 3166: 3158: 3144: 3142: 3141: 3140: 3127: 3126: 3125: 3112: 3101: 3099: 3098: 3086: 3081: 3080: 3041: 3040: 3016: 3014: 3013: 3004: 3003: 3002: 2987: 2986: 2974: 2973: 2964: 2963: 2951: 2950: 2941: 2940: 2930: 2925: 2923: 2922: 2913: 2903: 2902: 2892: 2814: 2812: 2811: 2806: 2779: 2768: 2748: 2746: 2745: 2740: 2738: 2736: 2728: 2720: 2706: 2703: 2661: 2513: 2511: 2510: 2505: 2491: 2490: 2410: 2393: 2392: 2237: 2169: 2090:Three real roots 2061:that statement. 2060: 1910: 1907: 1904: 1901: 1898: 1895: 1892: 1889: 1886: 1883: 1880: 1877: 1874: 1871: 1868: 1865: 1862: 1859: 1856: 1853: 1850: 1847: 1844: 1841: 1838: 1835: 1832: 1829: 1826: 1823: 1820: 1817: 1814: 1811: 1808: 1805: 1802: 1799: 1796: 1793: 1790: 1787: 1784: 1781: 1778: 1775: 1772: 1769: 1766: 1763: 1760: 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1733: 1730: 1727: 1724: 1721: 1718: 1715: 1712: 1709: 1706: 1703: 1700: 1697: 1694: 1691: 1688: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1664: 1661: 1658: 1655: 1652: 1649: 1646: 1643: 1640: 1637: 1634: 1631: 1628: 1625: 1622: 1619: 1616: 1613: 1610: 1607: 1604: 1601: 1598: 1595: 1592: 1589: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1559: 1556: 1553: 1550: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1526: 1523: 1520: 1517: 1514: 1511: 1508: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1451: 1448: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1412: 1409: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1379: 1376: 1373: 1370: 1367: 1364: 1361: 1358: 1355: 1352: 1349: 1346: 1343: 1340: 1337: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1313: 1310: 1307: 1304: 1301: 1298: 1295: 1292: 1289: 1286: 1283: 1280: 1277: 1274: 1271: 1268: 1265: 1262: 1259: 1256: 1253: 1250: 1247: 1244: 1241: 1238: 1235: 1232: 1229: 1226: 1223: 1220: 1217: 1214: 1211: 1208: 1205: 1202: 1199: 1196: 1193: 1190: 1187: 1184: 1181: 1178: 1175: 1172: 1169: 1166: 1163: 1160: 1157: 1154: 1151: 1148: 1145: 1142: 1139: 1136: 1133: 1130: 1127: 1124: 1121: 1118: 1115: 1112: 1109: 1106: 1103: 1100: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 1049: 1046: 1043: 1040: 1037: 1034: 1031: 1028: 1025: 1022: 1019: 1016: 1013: 1010: 1007: 1004: 1001: 998: 995: 992: 989: 986: 983: 980: 977: 974: 971: 968: 965: 962: 959: 956: 953: 950: 947: 944: 941: 938: 935: 932: 929: 926: 923: 920: 917: 914: 911: 908: 905: 902: 899: 896: 893: 890: 887: 884: 881: 878: 875: 872: 869: 866: 863: 860: 857: 854: 851: 848: 845: 842: 839: 836: 833: 830: 827: 824: 821: 818: 815: 812: 809: 806: 803: 800: 797: 794: 791: 788: 785: 782: 779: 776: 773: 770: 767: 764: 761: 758: 755: 752: 749: 746: 743: 740: 737: 734: 731: 728: 725: 722: 719: 716: 713: 710: 707: 704: 701: 698: 695: 692: 689: 686: 683: 680: 677: 674: 671: 668: 665: 662: 659: 656: 653: 650: 647: 644: 641: 638: 635: 632: 629: 626: 623: 620: 617: 614: 611: 608: 605: 602: 599: 596: 593: 590: 587: 584: 581: 578: 575: 572: 569: 566: 563: 560: 557: 554: 551: 548: 545: 542: 539: 536: 533: 530: 527: 524: 521: 518: 515: 512: 509: 506: 503: 500: 497: 494: 491: 488: 485: 482: 479: 476: 473: 470: 467: 464: 461: 458: 455: 452: 449: 446: 443: 440: 437: 434: 431: 428: 425: 422: 419: 416: 413: 410: 407: 404: 401: 398: 395: 392: 389: 386: 383: 380: 377: 374: 371: 368: 365: 362: 359: 356: 353: 350: 347: 273: 271: 270: 265: 253: 251: 250: 245: 155: 152: 151: 145: 134: 133: 78: 56: 55: 33: 32: 26: 6534: 6533: 6529: 6528: 6527: 6525: 6524: 6523: 6318: 6313: 6312: 6276: 6271: 6270: 6178: 6165: 6143: 6112: 6072: 6054: 6032: 6018: 6017: 5977: 5957: 5946: 5925: 5920: 5919: 5864: 5840: 5799: 5774: 5773: 5708: 5695: 5673: 5642: 5599: 5581: 5559: 5545: 5544: 5537:) intercepts f( 5477: 5457: 5446: 5425: 5420: 5419: 5364: 5340: 5293: 5267: 5266: 5228: 5208: 5197: 5176: 5171: 5170: 5135: 5131: 5097: 5093: 5059: 5055: 5037: 5027: 5026: 4990: 4949: 4924: 4923: 4912:) intercepts f( 4817: 4760: 4737: 4726: 4720: 4691: 4680: 4674: 4673: 4663: 4662: 4635: 4577: 4558: 4518: 4502: 4497: 4496: 4443: 4386: 4363: 4352: 4346: 4317: 4306: 4300: 4299: 4290: 4289: 4262: 4204: 4185: 4145: 4129: 4124: 4123: 4072: 4015: 3989: 3978: 3972: 3940: 3929: 3923: 3922: 3914: 3913: 3886: 3878: 3823: 3804: 3764: 3748: 3743: 3742: 3676: 3656: 3655: 3632: 3608: 3584: 3550: 3546: 3535: 3531: 3518: 3517: 3463: 3443: 3435: 3383: 3375: 3350: 3342: 3327: 3301: 3295: 3278: 3270: 3239: 3233: 3232: 3224: 3223: 3187: 3167: 3159: 3132: 3128: 3117: 3113: 3090: 3032: 3005: 2994: 2978: 2965: 2955: 2942: 2932: 2931: 2914: 2894: 2893: 2886: 2885: 2877: 2832:The warning to 2755: 2754: 2729: 2721: 2648: 2647: 2482: 2471: 2470: 2416: 2411: 2397: 2395: 2390: 2250: 2220: 2083: 2056: 1945: 1925:—Ben FrantzDale 1912: 1911: 1908: 1905: 1902: 1899: 1896: 1893: 1890: 1887: 1884: 1881: 1878: 1875: 1872: 1869: 1866: 1863: 1860: 1857: 1854: 1851: 1848: 1845: 1842: 1839: 1836: 1833: 1830: 1827: 1824: 1821: 1818: 1815: 1812: 1809: 1806: 1803: 1800: 1797: 1794: 1791: 1788: 1785: 1782: 1779: 1776: 1773: 1770: 1767: 1764: 1761: 1758: 1755: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1731: 1728: 1725: 1722: 1719: 1716: 1713: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1680: 1677: 1674: 1671: 1668: 1665: 1662: 1659: 1656: 1653: 1650: 1647: 1644: 1641: 1638: 1635: 1632: 1629: 1626: 1623: 1620: 1617: 1614: 1611: 1608: 1605: 1602: 1599: 1596: 1593: 1590: 1587: 1584: 1581: 1578: 1575: 1572: 1569: 1566: 1563: 1560: 1557: 1554: 1551: 1548: 1545: 1542: 1539: 1536: 1533: 1530: 1527: 1524: 1521: 1518: 1515: 1512: 1509: 1506: 1503: 1500: 1497: 1494: 1491: 1488: 1485: 1482: 1479: 1476: 1473: 1470: 1467: 1464: 1461: 1458: 1455: 1452: 1449: 1446: 1443: 1440: 1437: 1434: 1431: 1428: 1425: 1422: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1398: 1395: 1392: 1389: 1386: 1383: 1380: 1377: 1374: 1371: 1368: 1365: 1362: 1359: 1356: 1353: 1350: 1347: 1344: 1341: 1338: 1335: 1332: 1329: 1326: 1323: 1320: 1317: 1314: 1311: 1308: 1305: 1302: 1299: 1296: 1293: 1290: 1287: 1284: 1281: 1278: 1275: 1272: 1269: 1266: 1263: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 1239: 1236: 1233: 1230: 1227: 1224: 1221: 1218: 1215: 1212: 1209: 1206: 1203: 1200: 1197: 1194: 1191: 1188: 1185: 1182: 1179: 1176: 1173: 1170: 1167: 1164: 1161: 1158: 1155: 1152: 1149: 1146: 1143: 1140: 1137: 1134: 1131: 1128: 1125: 1122: 1119: 1116: 1113: 1110: 1107: 1104: 1101: 1098: 1095: 1092: 1089: 1086: 1083: 1080: 1077: 1074: 1071: 1068: 1065: 1062: 1059: 1056: 1053: 1050: 1047: 1044: 1041: 1038: 1035: 1032: 1029: 1026: 1023: 1020: 1017: 1014: 1011: 1008: 1005: 1002: 999: 996: 993: 990: 987: 984: 981: 978: 975: 972: 969: 966: 963: 960: 957: 954: 951: 948: 945: 942: 939: 936: 933: 930: 927: 924: 921: 918: 915: 912: 909: 906: 903: 900: 897: 894: 891: 888: 885: 882: 879: 876: 873: 870: 867: 864: 861: 858: 855: 852: 849: 846: 843: 840: 837: 834: 831: 828: 825: 822: 819: 816: 813: 810: 807: 804: 801: 798: 795: 792: 789: 786: 783: 780: 777: 774: 771: 768: 765: 762: 759: 756: 753: 750: 747: 744: 741: 738: 735: 732: 729: 726: 723: 720: 717: 714: 711: 708: 705: 702: 699: 696: 693: 690: 687: 684: 681: 678: 675: 672: 669: 666: 663: 660: 657: 654: 651: 648: 645: 642: 639: 636: 633: 630: 627: 624: 621: 618: 615: 612: 609: 606: 603: 600: 597: 594: 591: 588: 585: 582: 579: 576: 573: 570: 567: 564: 561: 558: 555: 552: 549: 546: 543: 540: 537: 534: 531: 528: 525: 522: 519: 516: 513: 510: 507: 504: 501: 498: 495: 492: 489: 486: 483: 480: 477: 474: 471: 468: 465: 462: 459: 456: 453: 450: 447: 444: 441: 438: 435: 432: 429: 426: 423: 420: 417: 414: 411: 408: 405: 403:solveCubicRoots 402: 399: 396: 393: 390: 387: 384: 381: 378: 375: 372: 369: 366: 363: 360: 357: 354: 351: 349:solveCubicRoots 348: 345: 256: 255: 236: 235: 214:—Ben FrantzDale 174:—Ben FrantzDale 125: 113: 112: 105: 74: 30: 22: 21: 20: 12: 11: 5: 6532: 6530: 6502: 6500: 6499: 6498: 6497: 6496: 6495: 6468: 6467: 6466: 6465: 6444: 6443: 6442: 6441: 6421: 6420: 6419: 6418: 6417: 6416: 6415: 6414: 6392: 6391: 6390: 6389: 6388: 6387: 6348: 6345: 6342: 6339: 6336: 6333: 6330: 6325: 6321: 6300: 6297: 6294: 6291: 6288: 6283: 6279: 6261: 6260: 6214: 6211: 6208: 6205: 6200: 6196: 6190: 6185: 6181: 6177: 6172: 6168: 6164: 6161: 6156: 6153: 6150: 6146: 6142: 6139: 6136: 6133: 6130: 6127: 6124: 6119: 6115: 6111: 6108: 6105: 6100: 6097: 6092: 6089: 6084: 6079: 6075: 6069: 6066: 6061: 6057: 6053: 6050: 6045: 6042: 6039: 6035: 6031: 6028: 6025: 6004: 6001: 5998: 5995: 5992: 5989: 5984: 5980: 5976: 5969: 5965: 5961: 5955: 5952: 5949: 5943: 5940: 5937: 5932: 5928: 5903: 5900: 5895: 5890: 5887: 5882: 5879: 5876: 5871: 5867: 5858: 5855: 5852: 5847: 5843: 5839: 5836: 5833: 5830: 5827: 5824: 5816: 5813: 5806: 5802: 5798: 5793: 5790: 5787: 5784: 5781: 5770:) GFE returns 5757: 5744: 5741: 5738: 5735: 5730: 5726: 5720: 5715: 5711: 5707: 5702: 5698: 5694: 5691: 5686: 5683: 5680: 5676: 5672: 5669: 5666: 5663: 5660: 5657: 5654: 5649: 5645: 5641: 5638: 5635: 5632: 5627: 5624: 5619: 5616: 5611: 5606: 5602: 5596: 5593: 5588: 5584: 5580: 5577: 5572: 5569: 5566: 5562: 5558: 5555: 5552: 5504: 5501: 5498: 5495: 5492: 5489: 5484: 5480: 5476: 5469: 5465: 5461: 5455: 5452: 5449: 5443: 5440: 5437: 5432: 5428: 5403: 5400: 5395: 5390: 5387: 5382: 5379: 5376: 5371: 5367: 5358: 5355: 5352: 5347: 5343: 5339: 5336: 5333: 5330: 5327: 5324: 5316: 5313: 5310: 5307: 5300: 5296: 5292: 5287: 5284: 5281: 5278: 5275: 5265: 5252: 5249: 5246: 5243: 5240: 5235: 5231: 5227: 5220: 5216: 5212: 5206: 5203: 5200: 5194: 5191: 5188: 5183: 5179: 5155: 5149: 5146: 5141: 5138: 5134: 5130: 5127: 5123: 5117: 5114: 5109: 5106: 5103: 5100: 5096: 5092: 5089: 5085: 5079: 5076: 5071: 5068: 5065: 5062: 5058: 5054: 5051: 5044: 5040: 5036: 5008: 5005: 5002: 4997: 4993: 4989: 4986: 4983: 4980: 4977: 4974: 4966: 4963: 4956: 4952: 4948: 4943: 4940: 4937: 4934: 4931: 4860: 4857: 4854: 4846: 4843: 4840: 4837: 4834: 4831: 4824: 4820: 4816: 4811: 4808: 4805: 4802: 4799: 4796: 4793: 4790: 4787: 4784: 4781: 4778: 4775: 4772: 4767: 4763: 4759: 4756: 4752: 4746: 4743: 4740: 4735: 4732: 4729: 4723: 4719: 4716: 4711: 4706: 4700: 4697: 4694: 4689: 4686: 4683: 4677: 4672: 4642: 4638: 4634: 4629: 4624: 4619: 4616: 4613: 4610: 4607: 4604: 4601: 4598: 4595: 4592: 4589: 4584: 4580: 4576: 4573: 4570: 4565: 4561: 4557: 4554: 4551: 4548: 4545: 4542: 4539: 4536: 4533: 4530: 4525: 4521: 4517: 4514: 4509: 4505: 4495: 4480: 4477: 4474: 4466: 4463: 4460: 4457: 4450: 4446: 4442: 4437: 4434: 4431: 4428: 4425: 4422: 4419: 4416: 4413: 4410: 4407: 4404: 4401: 4398: 4393: 4389: 4385: 4382: 4378: 4372: 4369: 4366: 4361: 4358: 4355: 4349: 4345: 4342: 4337: 4332: 4326: 4323: 4320: 4315: 4312: 4309: 4303: 4298: 4269: 4265: 4261: 4256: 4251: 4246: 4243: 4240: 4237: 4234: 4231: 4228: 4225: 4222: 4219: 4216: 4211: 4207: 4203: 4200: 4197: 4192: 4188: 4184: 4181: 4178: 4175: 4172: 4169: 4166: 4163: 4160: 4157: 4152: 4148: 4144: 4141: 4136: 4132: 4122: 4109: 4106: 4103: 4095: 4092: 4089: 4086: 4079: 4075: 4071: 4066: 4063: 4060: 4057: 4054: 4051: 4048: 4045: 4042: 4039: 4036: 4033: 4030: 4027: 4022: 4018: 4014: 4011: 4007: 4001: 3998: 3995: 3992: 3987: 3984: 3981: 3975: 3971: 3968: 3963: 3958: 3952: 3949: 3946: 3943: 3938: 3935: 3932: 3926: 3921: 3893: 3889: 3884: 3881: 3875: 3870: 3865: 3862: 3859: 3856: 3853: 3850: 3847: 3844: 3841: 3838: 3835: 3830: 3826: 3822: 3819: 3816: 3811: 3807: 3803: 3800: 3797: 3794: 3791: 3788: 3785: 3782: 3779: 3776: 3771: 3767: 3763: 3760: 3755: 3751: 3729: 3726: 3723: 3720: 3717: 3714: 3711: 3708: 3705: 3702: 3699: 3696: 3693: 3690: 3683: 3679: 3674: 3671: 3668: 3663: 3659: 3652: 3649: 3639: 3635: 3631: 3628: 3625: 3622: 3615: 3611: 3606: 3603: 3600: 3596: 3592: 3588: 3581: 3578: 3571: 3568: 3565: 3557: 3553: 3549: 3542: 3538: 3534: 3528: 3525: 3489: 3486: 3480: 3475: 3471: 3467: 3455: 3451: 3447: 3441: 3438: 3432: 3429: 3426: 3423: 3420: 3417: 3414: 3411: 3408: 3400: 3397: 3390: 3386: 3381: 3378: 3372: 3369: 3362: 3358: 3354: 3348: 3345: 3339: 3334: 3330: 3326: 3323: 3319: 3314: 3310: 3307: 3304: 3298: 3290: 3286: 3282: 3276: 3273: 3267: 3262: 3257: 3252: 3248: 3245: 3242: 3236: 3231: 3210: 3204: 3199: 3195: 3191: 3179: 3175: 3171: 3165: 3162: 3156: 3153: 3150: 3147: 3139: 3135: 3131: 3124: 3120: 3116: 3110: 3107: 3104: 3097: 3093: 3089: 3084: 3079: 3074: 3071: 3068: 3065: 3062: 3059: 3056: 3053: 3050: 3047: 3044: 3039: 3035: 3031: 3028: 3025: 3022: 3019: 3012: 3008: 3001: 2997: 2993: 2990: 2985: 2981: 2977: 2972: 2968: 2962: 2958: 2954: 2949: 2945: 2939: 2935: 2928: 2921: 2917: 2912: 2909: 2906: 2901: 2897: 2875: 2804: 2801: 2798: 2795: 2792: 2789: 2786: 2783: 2778: 2775: 2772: 2767: 2763: 2735: 2732: 2727: 2724: 2718: 2715: 2712: 2709: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2680: 2677: 2674: 2671: 2668: 2664: 2660: 2656: 2642: 2641: 2600: 2599: 2565: 2564: 2534: 2533: 2503: 2500: 2497: 2494: 2489: 2485: 2481: 2478: 2467: 2466: 2415: 2412: 2388: 2386: 2385: 2384: 2383: 2370: 2353: 2352: 2333: 2332: 2331: 2330: 2322: 2321: 2305: 2304: 2303: 2302: 2294: 2293: 2289: 2288: 2287: 2286: 2278: 2277: 2249: 2246: 2245: 2244: 2243: 2242: 2241: 2240: 2239: 2238: 2209: 2208: 2207: 2206: 2205: 2204: 2185: 2184: 2183: 2182: 2147: 2146: 2126: 2082: 2079: 2078: 2077: 2076: 2075: 2074: 2073: 2034: 2033: 2025: 2024: 2020: 2019: 2018: 2017: 2009: 2008: 2004: 2003: 2002: 2001: 1992: 1991: 1944: 1941: 1940: 1939: 1938: 1937: 1936: 1935: 344: 342: 341: 340: 339: 338: 337: 321: 320: 319: 318: 317: 316: 315: 314: 292: 291: 290: 289: 288: 287: 263: 243: 227: 226: 225: 224: 206: 205: 143: 140: 137: 132: 128: 124: 121: 104: 101: 98: 97: 92: 89: 84: 79: 72: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 6531: 6522: 6521: 6517: 6513: 6509: 6505: 6494: 6490: 6486: 6485:217.197.142.0 6482: 6478: 6474: 6473: 6472: 6471: 6470: 6469: 6464: 6460: 6456: 6452: 6448: 6447: 6446: 6445: 6440: 6436: 6432: 6428: 6425: 6424: 6423: 6422: 6413: 6409: 6405: 6400: 6399: 6398: 6397: 6396: 6395: 6394: 6393: 6386: 6382: 6378: 6377:217.197.142.0 6374: 6370: 6366: 6362: 6346: 6343: 6340: 6337: 6334: 6331: 6328: 6323: 6319: 6298: 6295: 6292: 6289: 6286: 6281: 6277: 6267: 6266: 6265: 6264: 6263: 6262: 6259: 6255: 6251: 6247: 6246: 6245: 6244: 6240: 6236: 6235:217.197.142.0 6231: 6229: 6225: 6212: 6209: 6206: 6203: 6198: 6194: 6188: 6183: 6179: 6175: 6170: 6166: 6162: 6154: 6151: 6148: 6144: 6137: 6134: 6131: 6128: 6125: 6122: 6117: 6113: 6109: 6106: 6103: 6098: 6095: 6090: 6087: 6082: 6077: 6073: 6067: 6064: 6059: 6055: 6051: 6043: 6040: 6037: 6033: 6026: 6023: 6015: 6002: 5999: 5996: 5993: 5990: 5987: 5982: 5978: 5974: 5963: 5953: 5950: 5947: 5941: 5938: 5935: 5930: 5926: 5916: 5901: 5898: 5893: 5888: 5885: 5880: 5877: 5874: 5869: 5865: 5856: 5853: 5850: 5845: 5841: 5837: 5834: 5828: 5822: 5814: 5811: 5804: 5800: 5796: 5791: 5785: 5779: 5771: 5769: 5765: 5761: 5755: 5742: 5739: 5736: 5733: 5728: 5724: 5718: 5713: 5709: 5705: 5700: 5696: 5692: 5684: 5681: 5678: 5674: 5667: 5664: 5661: 5658: 5655: 5652: 5647: 5643: 5639: 5636: 5633: 5630: 5625: 5622: 5617: 5614: 5609: 5604: 5600: 5594: 5591: 5586: 5582: 5578: 5570: 5567: 5564: 5560: 5553: 5550: 5542: 5540: 5536: 5532: 5528: 5524: 5520: 5515: 5502: 5499: 5496: 5493: 5490: 5487: 5482: 5478: 5474: 5463: 5453: 5450: 5447: 5441: 5438: 5435: 5430: 5426: 5401: 5398: 5393: 5388: 5385: 5380: 5377: 5374: 5369: 5365: 5356: 5353: 5350: 5345: 5341: 5337: 5334: 5328: 5322: 5314: 5311: 5308: 5305: 5298: 5294: 5290: 5285: 5279: 5273: 5263: 5250: 5247: 5244: 5241: 5238: 5233: 5229: 5225: 5214: 5204: 5201: 5198: 5192: 5189: 5186: 5181: 5177: 5167: 5153: 5147: 5144: 5139: 5136: 5132: 5128: 5125: 5121: 5115: 5112: 5107: 5104: 5101: 5098: 5094: 5090: 5087: 5083: 5077: 5074: 5069: 5066: 5063: 5060: 5056: 5052: 5049: 5042: 5038: 5034: 5024: 5006: 5003: 5000: 4995: 4991: 4987: 4984: 4978: 4972: 4964: 4961: 4954: 4950: 4946: 4941: 4935: 4929: 4921: 4919: 4915: 4911: 4907: 4903: 4899: 4895: 4890: 4888: 4883: 4882: 4878: 4875: 4858: 4855: 4852: 4844: 4841: 4838: 4835: 4832: 4829: 4822: 4818: 4814: 4809: 4803: 4797: 4794: 4788: 4782: 4779: 4776: 4773: 4770: 4765: 4761: 4757: 4754: 4750: 4744: 4741: 4738: 4733: 4730: 4727: 4721: 4717: 4714: 4709: 4704: 4698: 4695: 4692: 4687: 4684: 4681: 4675: 4670: 4640: 4636: 4632: 4627: 4617: 4614: 4611: 4608: 4605: 4602: 4596: 4593: 4590: 4582: 4578: 4574: 4571: 4568: 4563: 4555: 4552: 4549: 4543: 4540: 4537: 4534: 4531: 4528: 4523: 4519: 4515: 4512: 4507: 4503: 4493: 4478: 4475: 4472: 4464: 4461: 4458: 4455: 4448: 4444: 4440: 4435: 4429: 4423: 4420: 4414: 4408: 4405: 4402: 4399: 4396: 4391: 4387: 4383: 4380: 4376: 4370: 4367: 4364: 4359: 4356: 4353: 4347: 4343: 4340: 4335: 4330: 4324: 4321: 4318: 4313: 4310: 4307: 4301: 4296: 4267: 4263: 4259: 4254: 4244: 4241: 4238: 4235: 4232: 4229: 4223: 4220: 4217: 4209: 4205: 4201: 4198: 4195: 4190: 4182: 4179: 4176: 4170: 4167: 4164: 4161: 4158: 4155: 4150: 4146: 4142: 4139: 4134: 4130: 4120: 4107: 4104: 4101: 4093: 4090: 4087: 4084: 4077: 4073: 4069: 4064: 4058: 4052: 4049: 4043: 4037: 4034: 4031: 4028: 4025: 4020: 4016: 4012: 4009: 4005: 3999: 3996: 3993: 3990: 3985: 3982: 3979: 3973: 3969: 3966: 3961: 3956: 3950: 3947: 3944: 3941: 3936: 3933: 3930: 3924: 3919: 3891: 3887: 3882: 3879: 3873: 3863: 3860: 3857: 3854: 3851: 3848: 3842: 3839: 3836: 3828: 3824: 3820: 3817: 3814: 3809: 3801: 3798: 3795: 3789: 3786: 3783: 3780: 3777: 3774: 3769: 3765: 3761: 3758: 3753: 3749: 3740: 3727: 3724: 3721: 3718: 3715: 3712: 3709: 3706: 3703: 3700: 3697: 3694: 3691: 3688: 3681: 3677: 3669: 3661: 3657: 3650: 3647: 3637: 3633: 3629: 3626: 3623: 3620: 3613: 3609: 3601: 3594: 3590: 3586: 3579: 3576: 3569: 3566: 3563: 3555: 3551: 3547: 3540: 3536: 3532: 3526: 3523: 3515: 3513: 3509: 3505: 3500: 3487: 3484: 3478: 3469: 3449: 3439: 3436: 3430: 3427: 3424: 3421: 3418: 3415: 3412: 3409: 3406: 3398: 3395: 3388: 3384: 3379: 3376: 3370: 3367: 3356: 3346: 3343: 3337: 3332: 3328: 3324: 3321: 3317: 3312: 3308: 3305: 3302: 3296: 3284: 3274: 3271: 3265: 3260: 3255: 3250: 3246: 3243: 3240: 3234: 3229: 3221: 3202: 3193: 3173: 3163: 3160: 3154: 3151: 3148: 3145: 3137: 3133: 3129: 3122: 3118: 3114: 3108: 3105: 3102: 3095: 3091: 3087: 3082: 3072: 3069: 3066: 3063: 3057: 3054: 3051: 3045: 3042: 3037: 3029: 3026: 3023: 3017: 3010: 3006: 2999: 2995: 2991: 2988: 2983: 2979: 2975: 2970: 2966: 2960: 2956: 2952: 2947: 2943: 2937: 2933: 2926: 2919: 2915: 2907: 2899: 2895: 2883: 2881: 2874: 2870: 2866: 2862: 2858: 2854: 2849: 2847: 2843: 2839: 2835: 2830: 2828: 2824: 2820: 2815: 2802: 2799: 2796: 2793: 2790: 2784: 2773: 2765: 2761: 2752: 2749: 2733: 2730: 2725: 2722: 2716: 2713: 2710: 2707: 2699: 2696: 2693: 2690: 2687: 2684: 2681: 2678: 2675: 2669: 2662: 2658: 2654: 2645: 2640: 2636: 2632: 2628: 2624: 2620: 2616: 2612: 2608: 2607: 2606: 2605: 2598: 2594: 2590: 2585: 2581: 2580: 2579: 2578: 2576: 2571: 2563: 2559: 2555: 2551: 2550: 2549: 2547: 2543: 2539: 2532: 2528: 2524: 2520: 2516: 2515: 2514: 2501: 2495: 2487: 2483: 2479: 2476: 2465: 2461: 2457: 2453: 2449: 2445: 2444: 2443: 2441: 2437: 2433: 2429: 2425: 2421: 2413: 2409: 2405: 2401: 2382: 2378: 2374: 2371: 2368: 2364: 2361: 2357: 2356: 2355: 2354: 2351: 2347: 2343: 2339: 2335: 2334: 2329: 2326: 2325: 2324: 2323: 2319: 2315: 2311: 2307: 2306: 2301: 2298: 2297: 2296: 2295: 2291: 2290: 2285: 2282: 2281: 2280: 2279: 2275: 2271: 2270: 2269: 2268: 2264: 2260: 2254: 2247: 2236: 2232: 2228: 2227:66.188.89.180 2224: 2217: 2216: 2215: 2214: 2213: 2212: 2211: 2210: 2203: 2199: 2195: 2191: 2190: 2189: 2188: 2187: 2186: 2181: 2177: 2173: 2168: 2163: 2159: 2155: 2151: 2150: 2149: 2148: 2145: 2141: 2137: 2133: 2129: 2128: 2127: 2124: 2123: 2119: 2115: 2111: 2109: 2105: 2100: 2098: 2093: 2091: 2086: 2080: 2072: 2068: 2064: 2059: 2054: 2053: 2052: 2048: 2044: 2041: 2038: 2037: 2036: 2035: 2031: 2027: 2026: 2022: 2021: 2016: 2013: 2012: 2011: 2010: 2006: 2005: 2000: 1996: 1995: 1994: 1993: 1989: 1985: 1984: 1983: 1982: 1978: 1974: 1970: 1966: 1962: 1958: 1954: 1950: 1942: 1934: 1930: 1926: 1922: 1918: 1917: 1916: 1915: 1914: 1913: 335: 331: 327: 326: 325: 324: 323: 322: 313: 309: 305: 302:derivation.-- 300: 299: 298: 297: 296: 295: 294: 293: 286: 282: 278: 261: 241: 233: 232: 231: 230: 229: 228: 223: 219: 215: 210: 209: 208: 207: 204: 200: 196: 191: 186: 185: 184: 183: 179: 175: 171: 167: 163: 159: 141: 138: 130: 126: 122: 119: 109: 102: 96: 93: 90: 88: 85: 83: 80: 77: 73: 71: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 6503: 6501: 6480: 6476: 6450: 6426: 6372: 6368: 6364: 6360: 6232: 6226: 6016: 5917: 5772: 5767: 5763: 5759: 5756: 5543: 5538: 5534: 5530: 5526: 5522: 5518: 5516: 5264: 5168: 5025: 4922: 4917: 4913: 4909: 4905: 4901: 4897: 4893: 4891: 4886: 4884: 4879: 4876: 4494: 4121: 3741: 3516: 3511: 3507: 3503: 3501: 3222: 2884: 2879: 2872: 2868: 2867:and unknown 2864: 2860: 2856: 2852: 2850: 2845: 2841: 2837: 2833: 2831: 2826: 2822: 2818: 2816: 2753: 2750: 2646: 2643: 2626: 2622: 2618: 2614: 2610: 2603: 2601: 2583: 2574: 2569: 2567: 2566: 2545: 2541: 2537: 2535: 2518: 2468: 2451: 2447: 2439: 2435: 2431: 2427: 2423: 2419: 2417: 2337: 2327: 2318:Omar Khayyám 2313: 2309: 2299: 2283: 2273: 2255: 2251: 2221:— Preceding 2161: 2157: 2131: 2125: 2112: 2107: 2103: 2101: 2096: 2094: 2089: 2087: 2084: 2039: 2030:trigonometry 2014: 1997: 1988:Omar Khayyám 1968: 1964: 1960: 1956: 1952: 1948: 1946: 1943:Anachronism? 333: 329: 190:root finding 169: 165: 161: 157: 110: 106: 75: 43: 37: 5861:at R where 5819:intercepts 5361:at R where 5319:intercepts 4969:intercepts 4873:0.}" /: --> 2604:Derivatives 2434:and always 2414:Suggestions 2310:numerically 1990:simply says 1309:logical_and 934:logical_and 898:logical_and 862:logical_and 826:logical_and 36:This is an 6451:Derivative 3506:), slope ( 2876:0; 1; 2; 3 2430:) instead 1921:imperative 154:0}" /: --> 6404:Duoduoduo 6250:Duoduoduo 4904:= 16 and 4877:Opening 4666:0.}": --> 2631:Duoduoduo 2589:Duoduoduo 2554:Duoduoduo 2523:Duoduoduo 2456:Duoduoduo 2342:Duoduoduo 2194:Duoduoduo 2136:Duoduoduo 2058:cn-tagged 2043:Duoduoduo 95:Archive 8 87:Archive 6 82:Archive 5 76:Archive 4 70:Archive 3 65:Archive 2 60:Archive 1 6455:D.Lazard 6449:Section 6431:D.Lazard 6367:setting 5766:) and h( 5758:Setting 2400:D.Lazard 2394:Resolved 2259:D.Lazard 2223:unsigned 2172:D.Lazard 2114:D.Lazard 2102:Section 2095:Section 2088:Section 1973:D.Lazard 277:D.Lazard 195:D.Lazard 115:0}": --> 5378:1.14286 4885:we get 2420:x&y 2373:Tobby72 2338:somehow 2055:I have 1999:circle. 1399:complex 811:complex 367:asarray 39:archive 6129:0.8325 6107:0.0357 5878:0.0714 5815:0.2157 5659:0.0922 5637:0.5714 5521:= 1 g( 5309:2.5423 5137:0.9286 5102:0.7857 5064:0.1429 4900:= –1, 4459:1.7055 4088:0.4169 2192:Done. 1909:result 1906:return 1876:result 1675:result 1501:result 1417:result 1315:b2m3ac 1126:result 1042:result 991:result 952:b2m3ac 916:b2m3ac 880:b2m3ac 844:b2m3ac 781:result 676:return 460:b2m3ac 400:return 6213:8682. 5743:4792. 5312:: --> 4896:= 0, 4856:: --> 4842:: --> 4839:21574 4462:: --> 2617:) = 3 2573:0 or 2572:: --> 2363:edits 805:dtype 799:shape 787:zeros 304:LutzL 139:: --> 16:< 6516:talk 6512:DVdm 6510:. - 6504:Note 6489:talk 6459:talk 6435:talk 6408:talk 6381:talk 6254:talk 6239:talk 6233:Stap 5517:For 4962:< 4849:and 4476:< 4469:and 4168:1598 4105:< 4098:and 4091:< 3644:and 3403:and 2836:vs. 2825:= 1, 2704:for 2635:talk 2593:talk 2577:= 0! 2558:talk 2527:talk 2460:talk 2404:talk 2377:talk 2360:this 2346:talk 2263:talk 2231:talk 2198:talk 2176:talk 2140:talk 2118:talk 2110:/3. 2067:talk 2063:DVdm 2047:talk 2032:says 1977:talk 1929:talk 1777:sqrt 1729:sqrt 1603:sqrt 1555:sqrt 1360:mask 1345:CofQ 1303:mask 1297:copy 1285:mask 1267:CofQ 631:CofQ 499:sqrt 421:cast 388:ndim 308:talk 281:talk 254:and 218:talk 199:talk 178:talk 168:and 160:and 6363:to 6195:591 5503:17. 5291:872 5251:12. 5035:143 4947:143 4889:. 4866:0.} 4541:300 4532:152 4441:872 4239:872 4159:144 4070:143 3858:143 3787:432 3778:144 3728:74. 3713:872 3698:143 2621:+ 2 2540:=0, 1959:by 1291:QNZ 1279:QNZ 1240:QNZ 682:0.5 628:def 346:def 6518:) 6491:) 6461:) 6437:) 6410:) 6383:) 6256:) 6241:) 6204:≈ 6199:28 6176:− 6126:− 6123:≈ 6104:≈ 6099:28 6091:− 6068:− 6003:2. 5975:∗ 5951:∗ 5948:14 5942:− 5889:14 5875:≈ 5812:≈ 5797:74 5734:≈ 5729:14 5725:45 5706:− 5656:− 5653:≈ 5634:− 5631:≈ 5618:− 5595:− 5497:16 5475:∗ 5451:∗ 5448:14 5442:− 5389:14 5386:16 5375:≈ 5351:− 5306:≈ 5248:± 5226:∗ 5202:∗ 5199:14 5193:− 5148:14 5145:13 5140:≈ 5116:14 5113:11 5108:− 5105:≈ 5099:− 5078:14 5070:− 5067:≈ 5061:− 5001:− 4859:0. 4830:≈ 4815:74 4742:∗ 4731:− 4696:∗ 4685:− 4637:14 4628:∗ 4612:74 4609:∗ 4603:− 4594:− 4575:∗ 4553:− 4538:− 4513:− 4479:0. 4456:≈ 4397:− 4368:∗ 4357:− 4341:− 4322:∗ 4311:− 4264:14 4255:∗ 4236:∗ 4230:− 4221:− 4202:∗ 4196:− 4180:− 4165:− 4156:− 4140:− 4108:0. 4085:≈ 4026:− 3997:∗ 3991:− 3983:− 3967:− 3948:∗ 3942:− 3934:− 3888:14 3880:− 3874:∗ 3855:∗ 3840:− 3821:∗ 3815:− 3799:− 3775:− 3759:− 3725:∗ 3719:− 3710:∗ 3704:− 3695:∗ 3630:∗ 3624:∓ 3533:− 3431:− 3396:≥ 3306:− 3244:− 3209:⇒ 3155:− 3115:− 3083:∗ 3055:− 3027:− 2863:, 2859:, 2855:, 2844:: 2723:− 2637:) 2629:. 2625:+ 2623:bx 2619:ax 2613:′( 2595:) 2560:) 2529:) 2462:) 2406:) 2396:– 2379:) 2369:). 2348:) 2265:) 2233:) 2200:) 2178:) 2142:) 2120:) 2069:) 2049:) 1979:) 1971:. 1931:) 1923:. 1798:** 1786:)) 1738:)) 1624:** 1612:)) 1564:)) 1447:** 1330:== 1318:== 1300:() 1252:!= 1237:)) 1231:** 1186:** 1144:** 1123:)) 1117:** 955:!= 943:!= 928:NN 919:== 907:!= 892:NZ 883:!= 871:== 856:ZN 847:== 835:== 820:ZZ 775:3. 769:1. 763:** 760:)) 748:** 739:27 706:** 646:): 619:** 592:** 571:** 556:** 547:27 514:** 502:(( 469:** 391:== 379:if 358:): 332:, 310:) 283:) 220:) 201:) 180:) 148:0} 136:Δ 123:27 120:− 91:→ 6514:( 6487:( 6481:x 6477:b 6457:( 6433:( 6406:( 6379:( 6373:t 6371:( 6369:f 6365:t 6361:x 6347:0 6344:= 6341:q 6338:+ 6335:t 6332:p 6329:+ 6324:3 6320:t 6299:q 6296:+ 6293:t 6290:p 6287:+ 6282:3 6278:t 6252:( 6237:( 6210:, 6207:0 6189:= 6184:H 6180:x 6171:A 6167:x 6163:= 6160:) 6155:2 6152:; 6149:1 6145:x 6141:( 6138:m 6135:I 6132:, 6118:A 6114:x 6110:, 6096:1 6088:= 6083:2 6078:R 6074:x 6065:= 6060:H 6056:x 6052:= 6049:) 6044:2 6041:; 6038:1 6034:x 6030:( 6027:e 6024:R 6000:= 5997:1 5994:+ 5991:1 5988:= 5983:R 5979:x 5968:| 5964:q 5960:| 5954:q 5939:1 5936:= 5931:R 5927:X 5902:d 5899:c 5894:= 5886:1 5881:= 5870:R 5866:x 5857:x 5854:3 5851:+ 5846:3 5842:x 5838:4 5835:= 5832:) 5829:x 5826:( 5823:f 5805:3 5801:7 5792:= 5789:) 5786:x 5783:( 5780:h 5768:x 5764:x 5760:b 5740:, 5737:0 5719:= 5714:H 5710:x 5701:A 5697:x 5693:= 5690:) 5685:2 5682:; 5679:1 5675:x 5671:( 5668:m 5665:I 5662:, 5648:A 5644:x 5640:, 5626:7 5623:4 5615:= 5610:2 5605:R 5601:x 5592:= 5587:H 5583:x 5579:= 5576:) 5571:2 5568:; 5565:1 5561:x 5557:( 5554:e 5551:R 5539:x 5535:x 5531:x 5527:x 5523:x 5519:a 5500:= 5494:+ 5491:1 5488:= 5483:R 5479:x 5468:| 5464:q 5460:| 5454:q 5439:1 5436:= 5431:R 5427:X 5402:d 5399:c 5394:= 5381:= 5370:R 5366:x 5357:x 5354:3 5346:3 5342:x 5338:4 5335:= 5332:) 5329:x 5326:( 5323:f 5315:1 5299:3 5295:7 5286:= 5283:) 5280:x 5277:( 5274:h 5245:; 5242:3 5239:= 5234:k 5230:x 5219:| 5215:q 5211:| 5205:q 5190:1 5187:= 5182:k 5178:X 5154:) 5133:( 5129:f 5126:= 5122:) 5095:( 5091:f 5088:= 5084:) 5075:2 5057:( 5053:f 5050:= 5043:3 5039:7 5007:x 5004:3 4996:3 4992:x 4988:4 4985:= 4982:) 4979:x 4976:( 4973:f 4965:1 4955:3 4951:7 4942:= 4939:) 4936:x 4933:( 4930:g 4918:x 4914:x 4910:x 4906:d 4902:c 4898:b 4894:a 4853:p 4845:0 4836:, 4833:0 4823:3 4819:7 4810:= 4807:) 4804:x 4801:( 4798:h 4795:= 4792:) 4789:x 4786:( 4783:f 4780:= 4777:x 4774:3 4771:+ 4766:3 4762:x 4758:4 4755:= 4751:) 4745:7 4739:2 4734:1 4728:X 4722:( 4718:3 4715:+ 4710:3 4705:) 4699:7 4693:2 4688:1 4682:X 4676:( 4671:4 4641:3 4633:4 4623:| 4618:0 4615:= 4606:2 4600:) 4597:1 4591:X 4588:( 4583:2 4579:7 4572:3 4569:+ 4564:3 4560:) 4556:1 4550:X 4547:( 4544:= 4535:X 4529:+ 4524:2 4520:X 4516:3 4508:3 4504:X 4473:p 4465:1 4449:3 4445:7 4436:= 4433:) 4430:x 4427:( 4424:h 4421:= 4418:) 4415:x 4412:( 4409:f 4406:= 4403:x 4400:3 4392:3 4388:x 4384:4 4381:= 4377:) 4371:7 4365:2 4360:1 4354:X 4348:( 4344:3 4336:3 4331:) 4325:7 4319:2 4314:1 4308:X 4302:( 4297:4 4268:3 4260:4 4250:| 4245:0 4242:= 4233:2 4227:) 4224:1 4218:X 4215:( 4210:2 4206:7 4199:3 4191:3 4187:) 4183:1 4177:X 4174:( 4171:= 4162:X 4151:2 4147:X 4143:3 4135:3 4131:X 4102:p 4094:1 4078:3 4074:7 4065:= 4062:) 4059:x 4056:( 4053:g 4050:= 4047:) 4044:x 4041:( 4038:f 4035:= 4032:x 4029:3 4021:3 4017:x 4013:4 4010:= 4006:) 4000:7 3994:2 3986:1 3980:X 3974:( 3970:3 3962:3 3957:) 3951:7 3945:2 3937:1 3931:X 3925:( 3920:4 3892:3 3883:4 3869:| 3864:0 3861:= 3852:2 3849:+ 3846:) 3843:1 3837:X 3834:( 3829:2 3825:7 3818:3 3810:3 3806:) 3802:1 3796:X 3793:( 3790:= 3784:+ 3781:X 3770:2 3766:X 3762:3 3754:3 3750:X 3722:2 3716:; 3707:2 3701:; 3692:2 3689:= 3682:3 3678:a 3673:) 3670:s 3667:( 3662:3 3658:p 3651:= 3648:q 3638:2 3634:7 3627:3 3621:= 3614:3 3610:a 3605:) 3602:s 3599:( 3595:′ 3591:3 3587:p 3580:= 3577:p 3570:, 3567:1 3564:= 3556:3 3552:a 3548:3 3541:2 3537:a 3527:= 3524:s 3512:q 3508:p 3504:s 3488:. 3485:x 3479:3 3474:| 3470:p 3466:| 3454:| 3450:q 3446:| 3440:q 3437:2 3428:s 3425:= 3422:x 3419:R 3416:+ 3413:s 3410:= 3407:X 3399:0 3389:3 3385:R 3380:q 3377:4 3371:= 3368:x 3361:| 3357:p 3353:| 3347:p 3344:3 3338:+ 3333:3 3329:x 3325:4 3322:= 3318:) 3313:R 3309:s 3303:X 3297:( 3289:| 3285:p 3281:| 3275:p 3272:3 3266:+ 3261:3 3256:) 3251:R 3247:s 3241:X 3235:( 3230:4 3203:3 3198:| 3194:p 3190:| 3178:| 3174:q 3170:| 3164:q 3161:2 3152:= 3149:R 3146:, 3138:3 3134:a 3130:3 3123:2 3119:a 3109:= 3106:s 3103:, 3096:3 3092:R 3088:4 3078:| 3073:0 3070:= 3067:q 3064:+ 3061:) 3058:s 3052:X 3049:( 3046:p 3043:+ 3038:3 3034:) 3030:s 3024:X 3021:( 3018:= 3011:3 3007:a 3000:0 2996:a 2992:+ 2989:X 2984:1 2980:a 2976:+ 2971:2 2967:X 2961:2 2957:a 2953:+ 2948:3 2944:X 2938:3 2934:a 2927:= 2920:3 2916:a 2911:) 2908:X 2905:( 2900:3 2896:p 2880:X 2873:a 2869:x 2865:d 2861:c 2857:b 2853:a 2846:x 2842:y 2838:x 2834:y 2827:q 2823:s 2819:x 2803:0 2800:≠ 2797:a 2794:6 2791:= 2788:) 2785:x 2782:( 2777:) 2774:3 2771:( 2766:3 2762:p 2734:a 2731:3 2726:b 2717:= 2714:s 2711:= 2708:x 2700:0 2697:= 2694:b 2691:2 2688:+ 2685:x 2682:a 2679:6 2676:= 2673:) 2670:x 2667:( 2663:″ 2659:3 2655:p 2633:( 2627:c 2615:x 2611:f 2591:( 2584:p 2575:p 2570:p 2556:( 2546:q 2544:= 2542:y 2538:t 2525:( 2519:f 2502:. 2499:) 2496:x 2493:( 2488:3 2484:p 2480:= 2477:y 2458:( 2452:t 2448:x 2440:t 2436:x 2432:y 2428:x 2426:( 2424:f 2402:( 2375:( 2344:( 2320:: 2261:( 2229:( 2196:( 2174:( 2138:( 2116:( 2108:π 2065:( 2045:( 1975:( 1927:( 1903:) 1900:a 1897:* 1894:3 1891:( 1888:/ 1885:b 1882:- 1879:= 1873:p 1870:= 1867:d 1864:, 1861:c 1858:, 1855:b 1852:, 1849:a 1846:) 1843:C 1840:* 1837:a 1834:* 1831:6 1828:( 1825:/ 1822:) 1819:c 1816:* 1813:a 1810:* 1807:3 1804:- 1801:2 1795:b 1792:( 1789:* 1783:3 1780:( 1774:* 1771:i 1768:+ 1765:1 1762:( 1759:+ 1756:) 1753:a 1750:* 1747:6 1744:( 1741:/ 1735:3 1732:( 1726:* 1723:i 1720:- 1717:1 1714:( 1711:* 1708:C 1705:+ 1702:) 1699:a 1696:* 1693:3 1690:( 1687:/ 1684:b 1681:- 1678:= 1672:) 1669:C 1666:* 1663:a 1660:* 1657:6 1654:( 1651:/ 1648:) 1645:c 1642:* 1639:a 1636:* 1633:3 1630:- 1627:2 1621:b 1618:( 1615:* 1609:3 1606:( 1600:* 1597:i 1594:- 1591:1 1588:( 1585:+ 1582:) 1579:a 1576:* 1573:6 1570:( 1567:/ 1561:3 1558:( 1552:* 1549:i 1546:+ 1543:1 1540:( 1537:* 1534:C 1531:+ 1528:) 1525:a 1522:* 1519:3 1516:( 1513:/ 1510:b 1507:- 1504:= 1498:) 1495:a 1492:* 1489:3 1486:( 1483:/ 1480:) 1477:C 1474:/ 1471:) 1468:c 1465:* 1462:a 1459:* 1456:3 1453:- 1450:2 1444:b 1441:( 1438:- 1435:C 1432:- 1429:b 1426:- 1423:( 1420:= 1414:) 1411:1 1408:, 1405:0 1402:( 1396:= 1393:i 1390:p 1387:= 1384:d 1381:, 1378:c 1375:, 1372:b 1369:, 1366:a 1363:) 1357:, 1354:Q 1351:- 1348:( 1342:= 1339:C 1336:) 1333:0 1327:C 1324:, 1321:0 1312:( 1306:= 1294:. 1288:= 1282:) 1276:, 1273:Q 1270:( 1264:= 1261:C 1258:) 1255:0 1249:Q 1246:( 1243:= 1234:2 1228:b 1225:- 1222:c 1219:* 1216:a 1213:* 1210:3 1207:( 1204:* 1201:a 1198:( 1195:/ 1192:) 1189:3 1183:b 1180:+ 1177:c 1174:* 1171:b 1168:* 1165:a 1162:* 1159:4 1156:- 1153:d 1150:* 1147:2 1141:a 1138:* 1135:9 1132:( 1129:= 1120:2 1114:b 1111:- 1108:c 1105:* 1102:a 1099:* 1096:3 1093:( 1090:* 1087:2 1084:( 1081:/ 1078:) 1075:d 1072:* 1069:a 1066:* 1063:9 1060:- 1057:c 1054:* 1051:b 1048:( 1045:= 1039:p 1036:= 1033:d 1030:, 1027:c 1024:, 1021:b 1018:, 1015:a 1012:a 1009:* 1006:3 1003:/ 1000:b 997:- 994:= 988:p 985:= 982:d 979:, 976:c 973:, 970:b 967:, 964:a 961:) 958:0 949:, 946:0 940:Q 937:( 931:= 925:) 922:0 913:, 910:0 904:Q 901:( 895:= 889:) 886:0 877:, 874:0 868:Q 865:( 859:= 853:) 850:0 841:, 838:0 832:Q 829:( 823:= 814:) 808:= 802:, 796:. 793:p 790:( 784:= 778:) 772:/ 766:( 757:d 754:* 751:2 745:a 742:* 736:+ 733:c 730:* 727:b 724:* 721:a 718:* 715:9 712:- 709:3 703:b 700:* 697:2 694:+ 691:Q 688:( 685:* 679:( 673:p 670:= 667:d 664:, 661:c 658:, 655:b 652:, 649:a 643:m 640:, 637:Q 634:( 625:) 622:3 616:) 613:c 610:* 607:a 604:* 601:3 598:- 595:2 589:b 586:( 583:* 580:4 577:- 574:2 568:) 565:d 562:* 559:2 553:a 550:* 544:+ 541:c 538:* 535:b 532:* 529:a 526:* 523:9 520:- 517:3 511:b 508:* 505:2 496:= 493:Q 490:c 487:* 484:a 481:* 478:3 475:- 472:2 466:b 463:= 457:p 454:= 451:d 448:, 445:c 442:, 439:b 436:, 433:a 430:) 427:p 424:( 418:= 415:p 412:) 409:p 406:( 397:: 394:1 385:. 382:p 376:) 373:p 370:( 364:= 361:p 355:p 352:( 334:C 330:Q 306:( 279:( 262:Q 242:C 216:( 197:( 176:( 170:C 166:Q 162:C 158:Q 142:0 131:2 127:a 50:.

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

Index