2615:, with symmetry, would help illustrate the structure of common asymptotic bounds that the Master Theorem spits out. I feel such tables add an intuitive understanding of things. They are commonly seen in many computer science articles, in which I also feel they (there) add significantly to a quick, intuitive understanding of things. The coloring is automatically calculated such that the same exponents are the same color, and the column that states divergence is an important part of highlighting what happens intuitively when b=1. The table sizes were chosen to fit comfortably on most resolutions (though if someone has specialty in wikipedia mobile, they should feel free to chime in). The table was reverted as being "gaudy", but I feel color is acceptable when used for things like plots and graphs if it adds value (in this case, equal things have the same color, and the scale is a log scale).
731:
any content is to be added, please make sure it is added below the existing content. The original content is very explanatory and the examples are very thourough and excellently explained. Some of the final answers where in a font rather small and hard to see on a 1024x768 resolution (please verify if this is a problem on other browsers) but otherwise, this is the most concise article on a mathmatical theorem i have read on wikipedia. It's greatness is in it's simplicity. Please do not dilute it, and keep up the good work!
2841:
blocks of text is harder than reading paragraphs. I do not think the table is an improvement. (By contrast, the smaller table at the bottom of the article shows a place where a tabular presentation can allow for easy comparison by eye.) The redundant words-ful formulas are also terrible, they are helpful only if you already know what they mean -- the cleaner formula with a sentence explaining how to interpret its terms is much better. I have not made any attempt to evaluate any smaller-scale edits. --
2217:
towards others in the past, in this very article, which can be noted in the history). I have spent the better part of a day hoping to contribute to the pedagogy of this article and to, after significant amounts of consideration and revision, distill the salient and most important parts of the article into a more concise, elegant, and understandable form, especially for students new to the art. I have also spent the better part of a day trying to improve the article by creating a custom-made table.
320:
243:
222:
191:
2840:
I think that tables can be a good way of conveying some kinds of information, but in the present case every single entry in the table is itself a block of text -- there is no advantage created, you still have to read every single block of text to understand what it's saying, and reading small cramped
1117:
It is not very helpful that both versions are used inconsistently throughout the article. I'd be fine if everything was "=Ī" but it's not. I prefer the element version as it's more accurate semantically, though I agree that more people may use the other one for simplicity. Just be consistent. In any
730:
I have read many wikipedia pages that describe mathematical theorems. and many are very obtuse, and usually digress in many directions preventing the main idea from being conveyed. This article is of utmost excellence. And did a better job describing the theorem than my college book or professor. If
2231:
User David
Eppstein has brashly reverted parts he did not even object to or discuss (likely by accident at first), and continues to do so, and has not edited, while I attempted to edit things to address his specific concerns. Below I compare the two additions to the article, and people unaffiliated
1100:
It is not for us here to promote nonstandard notation. "=Ī" is the standard notation for "has the same order of magnitude as", a relation between two functions rather than a membership relation between a function and a set. It is perfectly valid mathematically as long as one regards "=Ī" as being a
1963:
The name "Master theorem" for this article is silly. It is the "master theorem" for solving a certain class of recurrences that are found in algorithmic analysis. But there are other master theorems in other fields. Just because a lot of writers of wikipedia have a computer science background, and
2216:
I am taking the initiative to start a discussion here, because user David
Eppstein has provably started an edit war in bad faith (as evidenced by his revision comments, "pedantic notation unused by anyone except pedants, not accidental", as well as his history of other vitriolic revision comments
2325:
I felt the three cases were not sufficiently stated as a spectrum vis-a-vis each other. The tabular layout, specifically the
Description column, clarifies their relation. It also clarifies the log terms, and uses similar language to highlight symmetry in the regimes (such as how each regime is
2729:
where you didn't also include the 50000 bytes of colorization and commented-out Python code for colorization (?) that added nothing of value to the article. Look, you have been on
Knowledge for over 10 years; how have you managed not to pick up even the basic points of the manual of style?
2055:, only 4 are about this theorem. And of those 4, 3 explicitly call it "a" "master theorem for divide-and-conquer recurrences". That is, not only do they qualify it by its field of application, they don't call it "the" master theorem, but just "a" master theorem.
2661:
Sesquipedlian writing-around-the-point instead of simple direct language ("Whenever we have a recurrence of this form, we can consider it as occurring in one of three regimes, based on how the work to split/recombine the problem relates to the special
341:
2821:
Also, the new text "Whenever we have a recurrence of this form, we can consider it as occurring in one of three regimes" is objectively false. There are plenty of recurrences of this form that do not fall under any case of the theorem.
153:
2047:
It seems clear that this is the "Master theorem" for recurrences found in the analysis of algorithms, but that this name is not used outside that subfield. It doesn't even seem to be used in the mathematical study of recurrences in
2310:
The examples were unavoidably verbose, and while important, were intermingling too much with the definition to the point that one would have trouble comparing and contrasting the three regimes without wading through lots of text.
1942:
If f=0, then f is obviously O-big of anything. It is reasonable to expect that Theta=const=0 is also possible as a solution, but the theorem states that it can't be the case. Maybe it is missing somewhere the requirement that f:
692:
I believe there should be "The" in front of "Master theorem". In either case, can someone at least redirect it to here? As it currently stand, if you search "The Master
Theorem", this page will actually NOT show up as a result!
1603:
836:
This theorem is all well and good, but it just sits in space providing no reason why anyone would want to use it. It would be much better if an example or description of its relevance to the world, or mathematics was added.
2103:
is also terrible, though -- unguessable, unsearchable, and a horrible mouthful. "Master theorem (algorithms)" or "Master theorem (analysis of algorithms)" would be better (though the latter is also a pretty bad mouthful).
2784:
to be a design element, not available to
Gutenberg and his first successors, and still somewhat rare within books not intended for infantile readership, the good use of which greatly enhances intelligibility of complex
1682:
757:
I completely agree with the above sentiments. I am learning this theorem from the very book that this article cites, yet this article does a MUCH BETTER JOB. Whoever wrote this is amazing and they did a great job.
2968:
2100:
2065:
1211:
1868:
1426:
1964:
therefore are likely to be more familiar with that usage, doesn't mean that this is a good title. I would suggest something like Master theorem (recurrences) or Master theorem (analysis of algorithms). --
1076:
2762:
In a similar effort, those who feel themselves comfortable in their elaborate use of the
English language as it is correctly spoken/written in WP, should be able to save the half step in sesquipedaling.
1004:
1322:
1263:
1766:
147:
1490:
2550:
is a term derived from work splitting and recombining. I assume here that mathematicians and computer scientists might have some innate biases here, but we should feel free to discuss.
2807:
I can't even get to what has changed in the function because the form is unreadable. And that includes being written in unreadable
English, not just minor details of capitalization. ā
2869:
I have tried to work in some of your feedback about the verbosity of the regime table, and the textual equation, and some other concerns of David
Eppstein's in the following edit.
1921:
2714:
44:
1101:
single compound symbol rather than as two separate = and Ī symbols. The other notation is much less standard and should be avoided for that reason, logical though it may be. ā
2972:
1799:
2831:
2816:
2451:
2188:, because that is what makes sense in context. I think that would actually have been a better name for the article, but that didn't seem to appeal to other editors.... --
2964:
2907:
2613:
2374:
365:
2144:
It is unguessable, I agree. Except that the "search
Knowledge" box has typeahead, so you should see the suggestion as soon as you type "Master t", just as you do today.
2130:
Agreed that "Master theorem (divide and conquer recurrences)" is clumsy, but that is what is called in the literature, which is what we're supposed to use to guide us.
1110:
681:
2864:
2850:
1090:
771:
661:
583:
293:
2548:
2403:
719:
715:
2739:
2082:
2038:
2881:
1949:
1127:
699:
505:
2559:
2519:
2499:
2476:
2462:
2428:
2414:
2335:
2014:
1992:
2163:
2113:
2320:
2903:
2376:
should be noted appears mainly as an exponent, and the associated polynomial should be stressed as an important quantity: the regime is determined by how the
1334:
1123:
905:
2629:
I hope that any objectors can make their full objections clear here rather than reverting, and that together we can work to improve the article. Thank you.
2197:
870:
422:
360:
1338:
2241:
890:
867:
848:
2755:
certain violations of capitalizing, inappropriate direct addressing of readers, projecting anthropomorphism to theorems (is it allowed to have one such
1496:
2991:
2943:
2915:
2226:
1924:
696:
650:
The Case 2 showed here is not canon; it was supposed either to follow the CLRS' Master theorem, or should state it is instead a more general theorem.
638:
283:
851:
1932:
168:
1330:
1094:
135:
2947:
1953:
738:
665:
620:
2996:
587:
908:
779:
2939:
2801:
2263:
687:
634:
2896:
How is binary tree traversal complexity C(n) = 2 C(n/2) + O(1)? Is the tree assumed to be almost complete to have both sides of size n/2?
2638:
2624:
2574:
1608:
623:
1973:
2304:
893:
642:
259:
2285:
1081:
Though the first writing is commonly used, the second is mathematically correct and the first is not, because a set of functions(N-: -->
750:
2986:
1144:
704:
I believe this is not true. I looked in the book quoted by the article and the word "The" is not capitalized in any references to it.
467:
129:
79:
927:
2133:"A general method and a master theorem for divide-and-conquer recurrences with applications", RM Verma - Journal of Algorithms, 1994
2720:) and this notation is objectively wrong (doesn't work for situations where the O-notation is used for only part of an expression).
1805:
125:
2751:
Slightly violating the WP:MOS, even after editing for decades on WP, should not disqualify any edits, but attract some gnomes to
2185:
2029:
nomenclature, which a -to me- good encyclopedia should avoid, even when the sages of a particular field request their dominance.
1355:
734:
24:
2658:
Anthropomorphic fallacy ("The Master Theorem says" ... no, it doesn't, the action of speaking is not something a theorem can do)
918:
I assume it needs to be rewritten. Some portion of this page is to be removed and some more descriptive portion is to be added.
1133:
1015:
441:
306:
250:
227:
2246:
2061:
a) can we find any uses of the term "master theorem" outside the analysis of algorithms that refer to this particular theorem?
2911:
530:
175:
85:
943:
1086:
857:
775:
657:
579:
2952:
2139:"A master theorem for discrete divide and conquer recurrences", M Drmota, W Szpankowski - Journal of the ACM (JACM), 2013
711:
413:
2855:
P.S. Whatever way this gets resolved, the "See also" section is in a very weird place and will need to be relocated. --
2565:
Yes, there is still room for improvement after these improvements, but I believe they're a step in the right direction.
1281:
1222:
672:
The Case 2 showed here is exactly the one from the Goodrich-Tamassia book. I have added that reference to the article. ā
2273:
This is a snapshot of the Generic Form section, before my proposed changes, to which the article is being reverted to:
2211:
1945:
30:
2290:
1689:
394:
2268:
2136:"An improved master theorem for divide-and-conquer recurrences", S Roura - Automata, Languages and Programming, 1997
2921:
258:
related articles on Knowledge. If you would like to participate, please visit the project page, where you can join
141:
2745:
598:
Looking into other sources about the issue, I find no reference for the exact stating of the "Case 2" given here.
486:
99:
1432:
1343:
831:
746:
104:
20:
2891:
1009:
Theta represents a set of functions and T(n) is only a special function of these it should rather be written
74:
451:
332:
202:
2274:
2252:
1998:
461:
375:
65:
1875:
2797:
2647:
Ignoring the manual of style for capitalization ("Generic form" vs "Generic Form"; "The Master Theorem"
2034:
1937:
932:
631:
Agreed... it was very confusing and I have never seen that either. I just made the change to the page.
614:
496:
2052:
604:
Having looked into Cormen's book, I see the same there: no 'k' mentioned in the second case's context.
2668:
523:
881:
The discussion of a/b should be n/b, at least as given in Cormen, Leiserson and Rivest 1st Ed. p62.
2827:
2812:
2735:
1988:
1106:
677:
593:
2665:
Ridiculous set-theoretic notation that does not work to describe how O-notation is actually used (
2860:
2846:
2109:
1958:
109:
1772:
1268:
However, the explanation is inconsistent with the description of Case 3 of the master theorem.
602:
Could it be that there is a mistake here? or is it that all other places are less general forms?
2726:
803:
563:
2436:
2793:
2650:
Ignoring the manual of style re use of first person ("The Master Theorem sometimes gives us")
2582:
2343:
2030:
1928:
904:
I think that Special Cases should be discussed as well, as there are gaps between the cases.
899:
813:
725:
432:
208:
2870:
2147:
I don't think it's unsearchable. Both Google search and Knowledge search already return the
572:
for the theorem, but the article by Bentley et al. does not use the term "master theorem".
2899:
2877:
2634:
2620:
2570:
2555:
2472:
2458:
2424:
2410:
2331:
2316:
2300:
2281:
2259:
2237:
2222:
913:
763:
707:
653:
575:
8:
2957:
This phrase does not make sense: "T(n) = Theta(1) when n is less than some bound kappa".
2823:
2808:
2731:
2524:
2379:
1997:
It is actually ambiguous. There are things called "master theorems" in combinatorics and
1984:
1102:
767:
673:
190:
55:
2856:
2842:
2193:
2159:
2105:
2078:
2010:
1980:
1969:
876:
742:
610:
70:
2504:
2484:
2453:
because it is a simpler definition (but as with everything here, open to discussion).
2275:
https://en.wikipedia.org/search/?title=Master_theorem&oldid=784218527#Generic_form
2253:
https://en.wikipedia.org/search/?title=Master_theorem&oldid=785156913#Generic_Form
1271:
I suspect this was just a simple inversion typo and that the explanation should read:
342:
Requested articles/Applied arts and sciences/Computer science, computing, and Internet
2717:
2579:
Furthermore, I added a table. I felt a lookup table for very common values of a,b in
923:
882:
799:
351:
51:
1598:{\displaystyle T(n)/2^{2n}=T\left({\frac {n}{2}}\right)/2^{n}+({\frac {n}{4}})^{n}}
403:
255:
161:
2873:
2630:
2616:
2566:
2551:
2468:
2454:
2420:
2406:
2327:
2312:
2296:
2277:
2255:
2233:
2218:
2184:
OK, I have moved the article and updated links to it. Most of them now look like
2405:
term asymptotically relates to it, which the article did not yet properly show.
2005:
have alternative names doesn't make this one the definitive "master theorem". --
794:
page links here; but I suspect for the wrong reason. Ramanujan had some sort of
569:
2765:
Personally, I would like to avoid writing about some notation others use to be
2148:
1119:
477:
2960:
If n is bounded then asymptotic stuff about a function of n don't make sense.
2769:, at least as long as the intention is reconstructible. Considering that even
2481:
Added a more verbose formula which instantly clarifies the difference between
889:
Not sure which part you meant? Probably the error has already been corrected.
2980:
2189:
2155:
2074:
2006:
1965:
1348:
Inadmissible equations says "... cannot be solved using the master theorem".
847:
Disagreed, the first sentence of the article gives exactly what you ask for.
838:
825:
807:
2653:
Incorrect use of mathematics formatting (hint: in your coding <math: -->
2251:
This is a snapshot of the Generic Form section, after my proposed changes:
919:
600:
Instead it appears to be described everywhere as the special case of (k=0).
1983:ā disambiguation should be used only when a title is actually ambiguous. ā
2871:
https://en.wikipedia.org/search/?title=Master_theorem&oldid=785547309
862:
Also, does it really make sense to put this under 'Computer Languages'Ā ?
319:
2295:
I made the changes to help address some shortcomings with this article:
1677:{\displaystyle S(n)=S\left({\frac {n}{2}}\right)+({\frac {n}{4}})^{n}}
937:
This writing is probably not correct (not only in this special case):
2467:
Added symmetry of examples, that was stressed by the tabular layout.
791:
384:
2759:
a proposition?), and other nitpicking like non-italicizing op-names.
2655:, you are juxtaposing four individual variable names w, o, r, and k)
615:
http://www.columbia.edu/~cs2035/courses/csor4231.F03/recurrences.pdf
2232:
with either of us should feel free to comment to reach a consensus:
1206:{\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+{\frac {1}{n}}.}
1118:
case, please don't call people pedantic when reverting changes. ā
2773:
notation is in heavy use, I suggest to check if editing out the
242:
221:
1769:
which is true almost everywhere for arbitrary choosen constant
460:
Find pictures for the biographies of computer scientists (see
1863:{\displaystyle S(n)=\Theta \left(({\frac {n}{4}})^{n}\right)}
2788:
Are there any non-formal objections to the proposed content?
1421:{\displaystyle T(n)=2^{n}T\left({\frac {n}{2}}\right)+n^{n}}
785:
2099:
I agree with the criticisms of the name "master theorem."
2716:). We should not be pushing idiosyncratic notations here (
1071:{\displaystyle T(n)\in \Theta \left(n^{\log _{b}a}\right)}
568:
Where did the term "master theorem" originate? CLRS cites
2963:
Just say "T(n) is bounded" instead of "T(n) = Theta(1)".
866:
It does not, but it seems to have been resolved already.
843:
I feel the same way, studying it right now for a class!
999:{\displaystyle T(n)=\Theta \left(n^{\log _{b}a}\right)}
160:
2671:
2585:
2527:
2507:
2487:
2439:
2382:
2346:
1878:
1808:
1775:
1692:
1611:
1499:
1435:
1358:
1284:
1225:
1147:
1018:
946:
2051:
For example, in the first 20 results for the search
1685:
and, with Master Theorem case 3 under the condition
611:
http://www.cs.concordia.ca/~chvatal/notes/master.pdf
254:, a collaborative effort to improve the coverage of
15:
2326:dominated by various parts of the recursion tree).
1317:{\displaystyle 2{\frac {n}{2}}\leq c{\frac {1}{n}}}
1258:{\displaystyle 2{\frac {2}{n}}\leq c{\frac {1}{n}}}
2708:
2644:Some specific things I object to in your rewrite:
2607:
2542:
2513:
2493:
2445:
2397:
2368:
1915:
1862:
1793:
1760:
1676:
1597:
1484:
1420:
1316:
1257:
1205:
1070:
998:
366:Computer science articles needing expert attention
2186:master theorem for divide-and-conquer recurrences
2978:
1761:{\displaystyle (n/8)^{n/2}\leq c\cdot (n/4)^{n}}
33:for general discussion of the article's subject.
2151:article in position #1. with (e.g.) the search
2101:Master theorem (divide and conquer recurrences)
2066:Master theorem (divide and conquer recurrences)
506:WikiProject Computer science/Unreferenced BLPs
2792:Look, how could one rank form over function?
802:, as I recall. This one looks like it's from
174:
1138:One example in section Inadmissible states:
1485:{\displaystyle S(n)={\frac {T(n)}{2^{2n}}}}
423:Computer science articles without infoboxes
361:Computer science articles needing attention
1082:R) can not equal a single function(N-: -->
327:Here are some tasks awaiting attention:
2992:High-importance Computer science articles
2926:In the pseudocode we have the statement:
806:. The MacMahon one is a big thing from
188:
25:Master theorem (analysis of algorithms)
2979:
2934:in case examples under the section of
268:Knowledge:WikiProject Computer science
2997:WikiProject Computer science articles
2965:2601:985:4380:C00:980B:EA88:5DB3:A164
1923:by using the Master Theorem I think.
688:Page Title Change: The Master Theorem
271:Template:WikiProject Computer science
1351:IsnĀ“t it the case that you can solve
824:definitely needs a disambig. page.
248:This article is within the scope of
184:
2938:when glancing through the article.
1916:{\displaystyle T(n)=\Theta (n^{n})}
207:It is of interest to the following
23:for discussing improvements to the
13:
2053:"master theorem" on Google Scholar
1894:
1872:This leads to the asymptotic term
1824:
1034:
962:
442:Timeline of computing 2020āpresent
14:
3008:
2987:C-Class Computer science articles
468:Computing articles needing images
2953:Minor glitch in use of Big Theta
2930:. This can be confused with the
2709:{\displaystyle f(x)\in O(n^{c})}
318:
241:
220:
189:
45:Click here to start a new topic.
2419:Added numerous clarifications.
2064:b) any objections to the title
570:Bentley, Haken, and Saxe (1980)
288:This article has been rated as
2916:09:22, 17 September 2017 (UTC)
2703:
2690:
2681:
2675:
2537:
2531:
2392:
2386:
1910:
1897:
1888:
1882:
1846:
1832:
1818:
1812:
1749:
1734:
1708:
1693:
1665:
1651:
1621:
1615:
1586:
1572:
1509:
1503:
1463:
1457:
1445:
1439:
1368:
1362:
1157:
1151:
1134:Error in inadmissible example?
1028:
1022:
956:
950:
909:19:13, 21 September 2006 (UTC)
1:
1933:00:33, 18 February 2011 (UTC)
928:20:21, 30 December 2007 (UTC)
780:16:07, 6 September 2020 (UTC)
751:05:46, 20 February 2007 (UTC)
682:00:42, 25 February 2009 (UTC)
666:00:12, 25 February 2009 (UTC)
522:Tag all relevant articles in
262:and see a list of open tasks.
42:Put new text under old text.
2973:15:08, 2 November 2022 (UTC)
1339:23:35, 3 November 2009 (UTC)
858:Category: Computer Languages
720:23:52, 16 October 2013 (UTC)
700:17:33, 5 December 2006 (UTC)
624:23:32, 1 December 2006 (UTC)
531:WikiProject Computer science
307:WikiProject Computer science
251:WikiProject Computer science
7:
2198:21:30, 7 October 2017 (UTC)
2025:without any qualifier as a
1794:{\displaystyle 0<c<1}
462:List of computer scientists
50:New to Knowledge? Welcome!
10:
3013:
2212:Proposed changes June 11th
2015:22:29, 28 April 2017 (UTC)
1993:15:42, 27 April 2017 (UTC)
1974:13:07, 27 April 2017 (UTC)
643:09:55, 17 March 2008 (UTC)
294:project's importance scale
2928:if n < some constant k
2882:04:46, 14 June 2017 (UTC)
2865:15:02, 13 June 2017 (UTC)
2851:14:57, 13 June 2017 (UTC)
2832:06:18, 13 June 2017 (UTC)
2817:14:38, 12 June 2017 (UTC)
2802:10:09, 12 June 2017 (UTC)
2740:06:17, 12 June 2017 (UTC)
2727:your latest cut-down diff
2639:06:11, 12 June 2017 (UTC)
2625:06:11, 12 June 2017 (UTC)
2575:06:11, 12 June 2017 (UTC)
2560:06:11, 12 June 2017 (UTC)
2477:06:11, 12 June 2017 (UTC)
2463:06:11, 12 June 2017 (UTC)
2446:{\displaystyle \epsilon }
2429:06:11, 12 June 2017 (UTC)
2415:06:11, 12 June 2017 (UTC)
2336:06:11, 12 June 2017 (UTC)
2321:06:11, 12 June 2017 (UTC)
2305:06:11, 12 June 2017 (UTC)
2286:06:11, 12 June 2017 (UTC)
2264:06:11, 12 June 2017 (UTC)
2242:06:11, 12 June 2017 (UTC)
2227:06:11, 12 June 2017 (UTC)
2164:17:58, 14 June 2017 (UTC)
2114:15:02, 13 June 2017 (UTC)
2083:09:26, 13 June 2017 (UTC)
2039:09:15, 12 June 2017 (UTC)
1954:15:25, 27 July 2011 (UTC)
1128:14:32, 18 July 2017 (UTC)
894:08:34, 21 June 2007 (UTC)
885:09:25, 26 Oct 2004 (UTC)
871:08:34, 21 June 2007 (UTC)
852:08:34, 21 June 2007 (UTC)
816:15:50, 16 Apr 2004 (UTC)
524:Category:Computer science
300:
287:
274:Computer science articles
236:
215:
80:Be welcoming to newcomers
2922:Small note on usage of k
2608:{\displaystyle log_{b}a}
2369:{\displaystyle log_{b}a}
1274:"There does not exist a
1111:23:13, 4 July 2009 (UTC)
1095:22:56, 4 July 2009 (UTC)
820:You're right, Charles.
588:17:02, 12 May 2011 (UTC)
526:and sub-categories with
2948:15:04, 1 May 2021 (UTC)
2058:So, just a final check:
1215:There does not exist a
2710:
2609:
2544:
2515:
2495:
2447:
2399:
2370:
2021:I agree in estimating
1917:
1864:
1795:
1762:
1678:
1599:
1486:
1422:
1344:Inadmissible equation?
1318:
1259:
1207:
1072:
1000:
832:Relevance and Examples
804:analysis of algorithms
487:Computer science stubs
197:This article is rated
75:avoid personal attacks
2892:Binary tree traversal
2777:improves the article.
2711:
2654:work(n)</math: -->
2610:
2545:
2516:
2496:
2448:
2400:
2371:
1918:
1865:
1796:
1763:
1679:
1600:
1487:
1423:
1319:
1260:
1208:
1073:
1001:
737:comment was added by
100:Neutral point of view
2669:
2583:
2543:{\displaystyle f(x)}
2525:
2505:
2485:
2437:
2398:{\displaystyle f(x)}
2380:
2344:
2001:. Just because they
1876:
1806:
1773:
1690:
1609:
1497:
1433:
1429:by the substitution
1356:
1282:
1223:
1145:
1016:
944:
305:Things you can help
105:No original research
2725:And that's just in
2521:on sight, and that
1327:What do you think?
2706:
2605:
2540:
2511:
2491:
2443:
2395:
2366:
1938:Case 1 formal note
1913:
1860:
1791:
1758:
1674:
1595:
1482:
1418:
1314:
1255:
1203:
1068:
996:
933:Formal correctness
800:Laplace transforms
798:, but it involved
203:content assessment
86:dispute resolution
47:
2918:
2902:comment added by
2514:{\displaystyle b}
2494:{\displaystyle a}
1843:
1662:
1642:
1583:
1548:
1480:
1399:
1312:
1296:
1253:
1237:
1198:
1181:
782:
766:comment added by
754:
710:comment added by
656:comment added by
578:comment added by
561:
560:
557:
556:
553:
552:
549:
548:
545:
544:
183:
182:
66:Assume good faith
43:
3004:
2897:
2715:
2713:
2712:
2707:
2702:
2701:
2614:
2612:
2611:
2606:
2601:
2600:
2549:
2547:
2546:
2541:
2520:
2518:
2517:
2512:
2500:
2498:
2497:
2492:
2452:
2450:
2449:
2444:
2404:
2402:
2401:
2396:
2375:
2373:
2372:
2367:
2362:
2361:
2247:Proposed changes
1922:
1920:
1919:
1914:
1909:
1908:
1869:
1867:
1866:
1861:
1859:
1855:
1854:
1853:
1844:
1836:
1800:
1798:
1797:
1792:
1767:
1765:
1764:
1759:
1757:
1756:
1744:
1724:
1723:
1719:
1703:
1683:
1681:
1680:
1675:
1673:
1672:
1663:
1655:
1647:
1643:
1635:
1604:
1602:
1601:
1596:
1594:
1593:
1584:
1576:
1568:
1567:
1558:
1553:
1549:
1541:
1529:
1528:
1516:
1491:
1489:
1488:
1483:
1481:
1479:
1478:
1466:
1452:
1427:
1425:
1424:
1419:
1417:
1416:
1404:
1400:
1392:
1383:
1382:
1323:
1321:
1320:
1315:
1313:
1305:
1297:
1289:
1264:
1262:
1261:
1256:
1254:
1246:
1238:
1230:
1212:
1210:
1209:
1204:
1199:
1191:
1186:
1182:
1174:
1077:
1075:
1074:
1069:
1067:
1063:
1062:
1055:
1054:
1005:
1003:
1002:
997:
995:
991:
990:
983:
982:
814:Charles Matthews
761:
732:
722:
668:
618:and many others.
594:Case2 skepticism
590:
535:
529:
404:Computer science
333:Article requests
322:
315:
314:
302:
301:
276:
275:
272:
269:
266:
265:Computer science
256:Computer science
245:
238:
237:
232:
228:Computer science
224:
217:
216:
200:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
3012:
3011:
3007:
3006:
3005:
3003:
3002:
3001:
2977:
2976:
2955:
2924:
2904:Quentin Fortier
2894:
2748:
2697:
2693:
2670:
2667:
2666:
2596:
2592:
2584:
2581:
2580:
2526:
2523:
2522:
2506:
2503:
2502:
2486:
2483:
2482:
2438:
2435:
2434:
2381:
2378:
2377:
2357:
2353:
2345:
2342:
2341:
2293:
2271:
2249:
2214:
1961:
1959:Name of article
1940:
1904:
1900:
1877:
1874:
1873:
1849:
1845:
1835:
1831:
1827:
1807:
1804:
1803:
1774:
1771:
1770:
1752:
1748:
1740:
1715:
1711:
1707:
1699:
1691:
1688:
1687:
1668:
1664:
1654:
1634:
1630:
1610:
1607:
1606:
1589:
1585:
1575:
1563:
1559:
1554:
1540:
1536:
1521:
1517:
1512:
1498:
1495:
1494:
1471:
1467:
1453:
1451:
1434:
1431:
1430:
1412:
1408:
1391:
1387:
1378:
1374:
1357:
1354:
1353:
1346:
1304:
1288:
1283:
1280:
1279:
1245:
1229:
1224:
1221:
1220:
1190:
1173:
1169:
1146:
1143:
1142:
1136:
1050:
1046:
1045:
1041:
1037:
1017:
1014:
1013:
978:
974:
973:
969:
965:
945:
942:
941:
935:
916:
902:
879:
860:
844:
834:
788:
733:āThe preceding
728:
705:
690:
651:
596:
573:
566:
541:
538:
533:
527:
515:Project-related
510:
491:
472:
446:
427:
408:
389:
370:
346:
290:High-importance
273:
270:
267:
264:
263:
231:Highāimportance
230:
201:on Knowledge's
198:
121:
116:
115:
114:
91:
61:
12:
11:
5:
3010:
3000:
2999:
2994:
2989:
2954:
2951:
2923:
2920:
2893:
2890:
2889:
2888:
2887:
2886:
2885:
2884:
2853:
2835:
2834:
2824:David Eppstein
2819:
2809:David Eppstein
2790:
2789:
2786:
2778:
2763:
2760:
2747:
2744:
2743:
2742:
2732:David Eppstein
2723:
2722:
2721:
2705:
2700:
2696:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2663:
2659:
2656:
2651:
2648:
2604:
2599:
2595:
2591:
2588:
2563:
2562:
2539:
2536:
2533:
2530:
2510:
2490:
2479:
2465:
2442:
2431:
2417:
2394:
2391:
2388:
2385:
2365:
2360:
2356:
2352:
2349:
2338:
2323:
2292:
2289:
2270:
2267:
2248:
2245:
2213:
2210:
2209:
2208:
2207:
2206:
2205:
2204:
2203:
2202:
2201:
2200:
2173:
2172:
2171:
2170:
2169:
2168:
2167:
2166:
2152:
2149:Master theorem
2145:
2142:
2141:
2140:
2137:
2134:
2121:
2120:
2119:
2118:
2117:
2116:
2092:
2091:
2090:
2089:
2088:
2087:
2086:
2085:
2062:
2059:
2056:
2049:
2042:
2041:
2023:Master theorem
2019:
2018:
2017:
1985:David Eppstein
1960:
1957:
1939:
1936:
1912:
1907:
1903:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1871:
1858:
1852:
1848:
1842:
1839:
1834:
1830:
1826:
1823:
1820:
1817:
1814:
1811:
1802:
1790:
1787:
1784:
1781:
1778:
1768:
1755:
1751:
1747:
1743:
1739:
1736:
1733:
1730:
1727:
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1695:
1686:
1684:
1671:
1667:
1661:
1658:
1653:
1650:
1646:
1641:
1638:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1592:
1588:
1582:
1579:
1574:
1571:
1566:
1562:
1557:
1552:
1547:
1544:
1539:
1535:
1532:
1527:
1524:
1520:
1515:
1511:
1508:
1505:
1502:
1493:
1477:
1474:
1470:
1465:
1462:
1459:
1456:
1450:
1447:
1444:
1441:
1438:
1428:
1415:
1411:
1407:
1403:
1398:
1395:
1390:
1386:
1381:
1377:
1373:
1370:
1367:
1364:
1361:
1352:
1345:
1342:
1311:
1308:
1303:
1300:
1295:
1292:
1287:
1252:
1249:
1244:
1241:
1236:
1233:
1228:
1202:
1197:
1194:
1189:
1185:
1180:
1177:
1172:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1135:
1132:
1131:
1130:
1114:
1113:
1103:David Eppstein
1087:95.223.140.220
1079:
1078:
1066:
1061:
1058:
1053:
1049:
1044:
1040:
1036:
1033:
1030:
1027:
1024:
1021:
1007:
1006:
994:
989:
986:
981:
977:
972:
968:
964:
961:
958:
955:
952:
949:
934:
931:
915:
912:
906:66.130.236.213
901:
898:
897:
896:
878:
875:
874:
873:
859:
856:
855:
854:
842:
833:
830:
829:
828:
822:Master theorem
796:master theorem
787:
784:
768:Joaquin.Ravelo
727:
724:
689:
686:
685:
684:
674:David Eppstein
658:201.92.145.179
619:
617:
613:
609:
605:
603:
601:
599:
595:
592:
580:128.151.39.160
565:
564:Origin of term
562:
559:
558:
555:
554:
551:
550:
547:
546:
543:
542:
540:
539:
537:
536:
519:
511:
509:
508:
502:
492:
490:
489:
483:
473:
471:
470:
465:
457:
447:
445:
444:
438:
428:
426:
425:
419:
409:
407:
406:
400:
390:
388:
387:
381:
371:
369:
368:
363:
357:
347:
345:
344:
338:
326:
324:
323:
311:
310:
298:
297:
286:
280:
279:
277:
260:the discussion
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:
9:
6:
4:
3:
2:
3009:
2998:
2995:
2993:
2990:
2988:
2985:
2984:
2982:
2975:
2974:
2970:
2966:
2961:
2958:
2950:
2949:
2945:
2941:
2937:
2933:
2929:
2919:
2917:
2913:
2909:
2905:
2901:
2883:
2879:
2875:
2872:
2868:
2867:
2866:
2862:
2858:
2854:
2852:
2848:
2844:
2839:
2838:
2837:
2836:
2833:
2829:
2825:
2820:
2818:
2814:
2810:
2806:
2805:
2804:
2803:
2799:
2795:
2787:
2783:
2779:
2776:
2772:
2768:
2764:
2761:
2758:
2754:
2750:
2749:
2741:
2737:
2733:
2728:
2724:
2719:
2698:
2694:
2687:
2684:
2678:
2672:
2664:
2660:
2657:
2652:
2649:
2646:
2645:
2643:
2642:
2641:
2640:
2636:
2632:
2627:
2626:
2622:
2618:
2602:
2597:
2593:
2589:
2586:
2577:
2576:
2572:
2568:
2561:
2557:
2553:
2534:
2528:
2508:
2488:
2480:
2478:
2474:
2470:
2466:
2464:
2460:
2456:
2440:
2432:
2430:
2426:
2422:
2418:
2416:
2412:
2408:
2389:
2383:
2363:
2358:
2354:
2350:
2347:
2340:The quantity
2339:
2337:
2333:
2329:
2324:
2322:
2318:
2314:
2309:
2308:
2307:
2306:
2302:
2298:
2288:
2287:
2283:
2279:
2276:
2266:
2265:
2261:
2257:
2254:
2244:
2243:
2239:
2235:
2229:
2228:
2224:
2220:
2199:
2195:
2191:
2187:
2183:
2182:
2181:
2180:
2179:
2178:
2177:
2176:
2175:
2174:
2165:
2161:
2157:
2153:
2150:
2146:
2143:
2138:
2135:
2132:
2131:
2129:
2128:
2127:
2126:
2125:
2124:
2123:
2122:
2115:
2111:
2107:
2102:
2098:
2097:
2096:
2095:
2094:
2093:
2084:
2080:
2076:
2072:
2071:
2070:
2069:
2067:
2063:
2060:
2057:
2054:
2050:
2046:
2045:
2044:
2043:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1995:
1994:
1990:
1986:
1982:
1978:
1977:
1976:
1975:
1971:
1967:
1956:
1955:
1951:
1947:
1935:
1934:
1930:
1926:
1905:
1901:
1891:
1885:
1879:
1856:
1850:
1840:
1837:
1828:
1821:
1815:
1809:
1788:
1785:
1782:
1779:
1776:
1753:
1745:
1741:
1737:
1731:
1728:
1725:
1720:
1716:
1712:
1704:
1700:
1696:
1669:
1659:
1656:
1648:
1644:
1639:
1636:
1631:
1627:
1624:
1618:
1612:
1590:
1580:
1577:
1569:
1564:
1560:
1555:
1550:
1545:
1542:
1537:
1533:
1530:
1525:
1522:
1518:
1513:
1506:
1500:
1475:
1472:
1468:
1460:
1454:
1448:
1442:
1436:
1413:
1409:
1405:
1401:
1396:
1393:
1388:
1384:
1379:
1375:
1371:
1365:
1359:
1349:
1341:
1340:
1336:
1332:
1328:
1325:
1309:
1306:
1301:
1298:
1293:
1290:
1285:
1277:
1272:
1269:
1266:
1250:
1247:
1242:
1239:
1234:
1231:
1226:
1218:
1213:
1200:
1195:
1192:
1187:
1183:
1178:
1175:
1170:
1166:
1163:
1160:
1154:
1148:
1139:
1129:
1125:
1121:
1116:
1115:
1112:
1108:
1104:
1099:
1098:
1097:
1096:
1092:
1088:
1084:
1064:
1059:
1056:
1051:
1047:
1042:
1038:
1031:
1025:
1019:
1012:
1011:
1010:
992:
987:
984:
979:
975:
970:
966:
959:
953:
947:
940:
939:
938:
930:
929:
925:
921:
911:
910:
907:
900:Special Cases
895:
892:
888:
887:
886:
884:
872:
869:
865:
864:
863:
853:
850:
846:
845:
841:
840:
827:
823:
819:
818:
817:
815:
811:
809:
808:combinatorics
805:
801:
797:
793:
783:
781:
777:
773:
769:
765:
759:
755:
752:
748:
744:
740:
736:
726:Excellent job
723:
721:
717:
713:
712:184.58.117.18
709:
702:
701:
698:
694:
683:
679:
675:
671:
670:
669:
667:
663:
659:
655:
648:
645:
644:
640:
636:
632:
629:
626:
625:
622:
616:
612:
606:
591:
589:
585:
581:
577:
571:
532:
525:
521:
520:
518:
516:
512:
507:
504:
503:
501:
499:
498:
493:
488:
485:
484:
482:
480:
479:
474:
469:
466:
463:
459:
458:
456:
454:
453:
448:
443:
440:
439:
437:
435:
434:
429:
424:
421:
420:
418:
416:
415:
410:
405:
402:
401:
399:
397:
396:
391:
386:
383:
382:
380:
378:
377:
372:
367:
364:
362:
359:
358:
356:
354:
353:
348:
343:
340:
339:
337:
335:
334:
329:
328:
325:
321:
317:
316:
313:
312:
308:
304:
303:
299:
295:
291:
285:
282:
281:
278:
261:
257:
253:
252:
247:
244:
240:
239:
235:
229:
226:
223:
219:
218:
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:
2962:
2959:
2956:
2936:Generic form
2935:
2931:
2927:
2925:
2898:āĀ Preceding
2895:
2791:
2781:
2774:
2770:
2766:
2756:
2752:
2628:
2578:
2564:
2294:
2291:Improvements
2272:
2250:
2230:
2215:
2026:
2022:
2002:
1962:
1946:80.237.26.14
1941:
1350:
1347:
1329:
1326:
1275:
1273:
1270:
1267:
1216:
1214:
1140:
1137:
1085:
1080:
1008:
936:
917:
914:Meet quality
903:
891:A.N. Yzelman
883:Terrycojones
880:
868:A.N. Yzelman
861:
849:A.N. Yzelman
835:
821:
812:
795:
789:
762:āĀ Preceding
760:
756:
729:
706:ā Preceding
703:
695:
691:
649:
646:
633:
630:
627:
607:
597:
567:
514:
513:
497:Unreferenced
495:
494:
476:
475:
450:
449:
431:
430:
412:
411:
393:
392:
374:
373:
350:
349:
331:
330:
289:
249:
209:WikiProjects
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
2780:I consider
2269:Prior state
1981:WP:PARENDIS
1925:Joerg Bader
1605:or equally
697:24.222.8.32
652:āPreceding
574:āPreceding
148:free images
31:not a forum
2981:Categories
2874:Ninjagecko
2767:ridiculous
2746:Discussion
2718:WP:SOAPBOX
2662:quantity")
2631:Ninjagecko
2617:Ninjagecko
2567:Ninjagecko
2552:Ninjagecko
2469:Ninjagecko
2455:Ninjagecko
2421:Ninjagecko
2407:Ninjagecko
2328:Ninjagecko
2313:Ninjagecko
2297:Ninjagecko
2278:Ninjagecko
2256:Ninjagecko
2234:Ninjagecko
2219:Ninjagecko
1801:, you get
1278:such that
1219:such that
877:a/b vs n/b
608:Reference:
1120:Xiphoseer
792:Ramanujan
786:Ramanujan
385:Computing
88:if needed
71:Be polite
21:talk page
2912:contribs
2900:unsigned
2785:content.
2775:ridicule
2757:to state
2190:Macrakis
2156:Macrakis
2075:Macrakis
2048:general.
2007:Macrakis
1999:analysis
1966:Macrakis
1276:c < 1
1217:c < 1
839:Centroyd
826:Giftlite
776:contribs
764:unsigned
747:contribs
739:Jediborg
735:unsigned
708:unsigned
654:unsigned
621:Seftembr
576:unsigned
433:Maintain
376:Copyedit
56:get help
29:This is
27:article.
2771:abusive
2753:rectify
1492:to get
920:Nafsadh
414:Infobox
352:Cleanup
292:on the
199:C-class
154:WPĀ refs
142:scholar
2940:Dizqar
2433:Added
635:Mpercy
395:Expand
205:scale.
126:Google
2794:Purgy
2782:color
2031:Purgy
2027:silly
478:Stubs
452:Photo
309:with:
169:JSTOR
130:books
84:Seek
2969:talk
2944:talk
2908:talk
2878:talk
2861:talk
2847:talk
2828:talk
2813:talk
2798:talk
2736:talk
2635:talk
2621:talk
2571:talk
2556:talk
2501:and
2473:talk
2459:talk
2425:talk
2411:talk
2332:talk
2317:talk
2301:talk
2282:talk
2260:talk
2238:talk
2223:talk
2194:talk
2160:talk
2110:talk
2079:talk
2035:talk
2011:talk
2003:also
1989:talk
1979:See
1970:talk
1950:talk
1929:talk
1786:<
1780:<
1335:talk
1124:talk
1107:talk
1091:talk
924:talk
790:The
772:talk
743:talk
716:talk
678:talk
662:talk
647:---
639:talk
628:---
584:talk
284:High
162:FENS
136:news
73:and
2857:JBL
2843:JBL
2106:JBL
1944:0?
1943:-->
1331:smt
1324:."
1265:."
1083:R)
1048:log
976:log
176:TWL
2983::
2971:)
2946:)
2914:)
2910:ā¢
2880:)
2863:)
2849:)
2830:)
2815:)
2800:)
2738:)
2685:ā
2637:)
2623:)
2573:)
2558:)
2475:)
2461:)
2441:Ļµ
2427:)
2413:)
2334:)
2319:)
2303:)
2284:)
2262:)
2240:)
2225:)
2196:)
2162:)
2154:--
2112:)
2104:--
2081:)
2073:--
2068:?
2037:)
2013:)
1991:)
1972:)
1952:)
1931:)
1895:Ī
1825:Ī
1732:ā
1726:ā¤
1337:)
1299:ā¤
1240:ā¤
1126:)
1109:)
1093:)
1057:ā”
1035:Ī
1032:ā
985:ā”
963:Ī
926:)
810:.
778:)
774:ā¢
749:)
745:ā¢
718:)
680:)
664:)
641:)
586:)
534:}}
528:{{
156:)
54:;
2967:(
2942:(
2932:k
2906:(
2876:(
2859:(
2845:(
2826:(
2822:ā
2811:(
2796:(
2734:(
2730:ā
2704:)
2699:c
2695:n
2691:(
2688:O
2682:)
2679:x
2676:(
2673:f
2633:(
2619:(
2603:a
2598:b
2594:g
2590:o
2587:l
2569:(
2554:(
2538:)
2535:x
2532:(
2529:f
2509:b
2489:a
2471:(
2457:(
2423:(
2409:(
2393:)
2390:x
2387:(
2384:f
2364:a
2359:b
2355:g
2351:o
2348:l
2330:(
2315:(
2299:(
2280:(
2258:(
2236:(
2221:(
2192:(
2158:(
2108:(
2077:(
2033:(
2009:(
1987:(
1968:(
1948:(
1927:(
1911:)
1906:n
1902:n
1898:(
1892:=
1889:)
1886:n
1883:(
1880:T
1870:.
1857:)
1851:n
1847:)
1841:4
1838:n
1833:(
1829:(
1822:=
1819:)
1816:n
1813:(
1810:S
1789:1
1783:c
1777:0
1754:n
1750:)
1746:4
1742:/
1738:n
1735:(
1729:c
1721:2
1717:/
1713:n
1709:)
1705:8
1701:/
1697:n
1694:(
1670:n
1666:)
1660:4
1657:n
1652:(
1649:+
1645:)
1640:2
1637:n
1632:(
1628:S
1625:=
1622:)
1619:n
1616:(
1613:S
1591:n
1587:)
1581:4
1578:n
1573:(
1570:+
1565:n
1561:2
1556:/
1551:)
1546:2
1543:n
1538:(
1534:T
1531:=
1526:n
1523:2
1519:2
1514:/
1510:)
1507:n
1504:(
1501:T
1476:n
1473:2
1469:2
1464:)
1461:n
1458:(
1455:T
1449:=
1446:)
1443:n
1440:(
1437:S
1414:n
1410:n
1406:+
1402:)
1397:2
1394:n
1389:(
1385:T
1380:n
1376:2
1372:=
1369:)
1366:n
1363:(
1360:T
1333:(
1310:n
1307:1
1302:c
1294:2
1291:n
1286:2
1251:n
1248:1
1243:c
1235:n
1232:2
1227:2
1201:.
1196:n
1193:1
1188:+
1184:)
1179:2
1176:n
1171:(
1167:T
1164:2
1161:=
1158:)
1155:n
1152:(
1149:T
1141:"
1122:(
1105:(
1089:(
1065:)
1060:a
1052:b
1043:n
1039:(
1029:)
1026:n
1023:(
1020:T
993:)
988:a
980:b
971:n
967:(
960:=
957:)
954:n
951:(
948:T
922:(
770:(
753:.
741:(
714:(
676:(
660:(
637:(
582:(
517::
500::
481::
464:)
455::
436::
417::
398::
379::
355::
336::
296:.
211::
172:Ā·
166:Ā·
158:Ā·
151:Ā·
145:Ā·
139:Ā·
133:Ā·
128:(
58:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
