95:
85:
64:
31:
1251:
other sources in the section which are of much better quality (Rashed, Brummelen, Sidoli are all promminent historians of science and mathematics). Also, Gibbs himself does not claim that the coefficients or the triangle were known to ancient Indian mathematicians, rather, he discusses the matter and states that, given the current state of knowledge, al-Tusi must be recorded as the earliest reference to the triangle. Thus, i see no reason to re-add the removed content.
247:
194:
169:
361:
imports, try/catch) was helping either. Anyway I'm sure that the floodgates are open now, and we're going to have 200 versions of this program in different languages... which is OK by me, as long as everybody is striving towards comprehension of Pascal's triangle. I claim the existing program is an improvement over the previous one. I don't claim it's the best possible. Happy editing,
22:
313:
1250:
Although "historia mathematica" seems to be in general a reliable source for this kind of topic, I would like to draw your attention to the fact that the author of the article on the other hand is not an expert source on the history of mathematics, so this source cannot be used to counterbalance the
1634:
A recent edit added, with citation: "As the proportion of black numbers tends to zero with increasing n, a corollary is that the proportion of odd binomial coefficients tends to zero as n tends to infinity." In trying to clean this up, I realized that I didn't know what "proportion" meant exactly.
382:
More about the code. I see the C code has been restored. I don't think this is an improvement. Why don't we just cut out the code altogther (in any language). Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python
1429:
Actually i am very much interested to know truth even if halyudha discover it his name should definitely come to in history of pascal triangle. Also halyuddha guy give his commentary on pingala work. On indian mathematician wiki pages it was clearly mention pingala discover pascal triangle and
360:
Hello, Iprogram with a much shorter, and, I hope, more comprehensible program. The goal here is to help people understand Pascal's triangle. I don't think the gyrations needed to get the numbers to line up were aiding that understanding. I also don't think the standard Java cruft (class name,
1192:. It was used to identify the poetic metre and to combine the short and long syllables to produce the required rythem. Although Biggs did not conclude if this triangle originated by the Hindu scholars, he definitely mentions about further research on the origin of the Arthmatic triangle.'
1212:
We have several high quality sources in the article that attribute the binomial coefficients and the triangle to medieval Muslim mathematicians and so far, i've never seen similar high quality sources supporting this kind of edits. Besides, some weeks ago, i already
1609:
Also, is there a neat way to split the article into one shorter and more accessible article , and another picking up on more periferal or advanced topics? The latter would by nature be very fragmented (or it would be a ridiculous number of very short articles).
402:
I agree with Wile that the code does not add any information on Pascal's triangle. The algorithm, based on Pascal's identity is already explained in
English in the lead section, and it is straightforward to translate it in a specific programming language. --
1320:
Well, i knew about Biggs' Wiki article, that's where i found that he is a mathematician, and while he has written some papers about the history of maths, he is not a prominent historian of maths simply because that field is not his main field, unlike
1928:
1974:
Jia Xian is already mentioned in the history section of this article. According to what is written there, his name is not usually attached to it, even in China; do you have any sources that support the contrary view? (The only source in
1551:(which I rewrote some time ago, as far as I recall making it quite a bit longer, but more readable), also seem to be of marginal interest only, making this article overly long and much less accessible than it could be. Thoughts on that?
651:
460:
Am I wrong? This assumes that the row numbering starts at one (which is consistent with other parts of the text). In particular, only the odd numbered rows have a "middle element", and odd numbered rows must be of the form
1344:
most of the article written in science direct is not reliable and many authors claimed to be an expert.I saw a article on cow urine therapy treatment for black fungus in science direct and itself a pseudoscience article.
793:
1566:
Having not surveyed the literature myself, but having only read this article, I agree with you. In your proposed sentence, we should remove the clause "and it is rarely of any use", which verges on speculation and
1000:
This truth is demonstrated by the example in this section, showing that squares of the terms of the 5th row (where row numbering starts from 1) add up to 70. However, the math formula demonstrating this sums over
931:
373:, I really don't mind you reverting to the Python code; in fact, I probably shouldn't have converted it in the first place, since it's against my policy of leaving real-language code alone. Sorry about that.
1329:. I have no problem with trying to find the source you mentioned, but honestly, i don't see how the addition of what a single source says about the topic of this article would be a notable improvement of it.
1377:
When I remvoed the fairly recent paragraph on
Pingala, I was not aware that other material on pre-islamic insights into binomial coefficients (arranged in a triangle, or not) har been removed in this edit:
1543:
I suggest replacing the whole section by this statement, perhaps with the same source cited at the last comma, and possibly with a a link to a separate article containing the present section. Thoughts?
952:
Jitse, thanks for that example, which helps clarify some things for me. However, it also points out the other inconsistencies of this article. The crux of the problem is that the text uses the letter
1415:
Even if it were, it doesn't support the changes you want to make. It explains that Pascal's
Triangle doesn't appear in what survives of Pingala's work, but in Halayudha's more than 1,000 years later.
1748:
1956:
Jia Xian described the Pascal's triangle (Jia Xian triangle) around the middle of the 11th century, about six centuries before Pascal. Jia used it as a tool for extracting square and cubic roots
1306:
1265) to be the earliest reference. I suggest to try to find the source (Ahmedev and
Rosenfeld 1963) given by Biggs, and then add an appropriate remark, quite different from the deleted one. -
1181:
The following material was added by an IP user, and subsequently edited several times; e.g., the spelling "rythem" was corrected to "rythm" (along with other changes), but that was reverted.
151:
418:
I've cut the section with the computer code. As I said in the edit summary, "source code doesn't shed any light on Pascal's triangle, and it's a language war magnet". For what it's worth,
2023:
1471:
1443:
1081:
1539:
Pascal's triangle may be extended upwards, above the 1 at the apex, preserving the additive property, but there is more than one way to do so, and it is rarely of any use.
1791:
1535:, extending the triangle upwards to the left and right, relies on a single source, and I think it has undue weight, as it is of very marginal interest. Briefly stated,
2013:
1149:
995:
829:
383:
because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars.
1802:
484:
Some further explanations prompted by the edits of Chad.nezar: The article assumes that the row with just one 1 is row number one. It then follows that row number
2028:
493:
35:
1456:
The Indian version of
Knowledge is a separate project with vastly different policies and standards. What happens there has no bearing on the English wiki.
1606:"sometinh someone cooked up one afternoon after class", but still I'm not sure its presense makes the article better. Thoughts on that, or other sections?
2038:
1197:
I have for now removed it; it may be sound (I do not have access to the source), but perhaps we can find an acceptable version here before re-adding it.
141:
2008:
2018:
117:
2033:
670:
1436:
Fowler, David (1996), "Binomial
Coefficient Function", The American Mathematical Monthly, 103 (1): 1–17, doi:10.2307/2975209, JSTOR 2975209
1402:
200:
174:
1186:
Earliest evidence about this triangle was found in the year 200 B.C. Pingalas
Sanskrit Text Chandah-Sutra . The triangle was called
836:
268:
264:
260:
108:
69:
1266:
Biggs' article, so I can't yet comment on your last sentences. However, I wonder how you decide who is an expert and who isn't. -
370:
1475:
1447:
2003:
1103:
row", but something more appropriate. Either the text should be changed to indicate that we consider the first row to be the
1009:
is equal to 5 (as the text would indicate), that sum has a total of 6 terms, not 5. Clearly it is written assuming that the
1439:
I request you Pls visit indian mathematocs wiki page and see pingala section and revise pascal triangle history accordingly.
1664:
44:
298:
1364:
1359:
A judgment about general reliablilty of a journal can't be based on an (unsourced rumor about a) single article. -
1311:
1289:
1271:
1241:
1099:", with all the math formulas based on n starting at zero, and the text should be updated to not use the phrase "
277:
1406:
1979:
is not available online as far as I can see, so I am not able to immediately check what is written there.) --
1588:
and I think basically everyone agrees that section is excessive but no one has done anything about it :). --
419:
384:
362:
338:
1360:
1350:
1307:
1285:
1267:
1237:
961:
row, where the row numbering starts at 1. However, clearly the math formulas and notation require that
50:
94:
1346:
1152:
1570:
Similarly, I think that the recently added section "To arbitrary bases" is overly long and detailed.
1303:
1229:
205:
179:
21:
1984:
1959:
This appears to be a kinda important historical fact and probably should be included here as well.
1593:
346:
1047:
1020:
In the first section, there is also some confusion in that the page states "for positive integers
116:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1548:
1532:
1326:
283:
100:
1960:
84:
63:
1923:{\displaystyle \lim _{n\to \infty }{\frac {\sum _{i=1}^{n}O(i)}{\sum _{i=1}^{n}O(i)+E(i)}}=0?}
1757:
1461:
1420:
1167:
938:
404:
345:
Please help fix the broken anchors. You can remove this template after fixing the problems. |
449:
But it appears to me that the claim should be "the sum of the squares of the elements of the
1937:
1575:
325:
279:
246:
1964:
1399:
1281:
1122:
968:
802:
646:{\displaystyle {n-1 \choose 0},{n-1 \choose 1},{n-1 \choose 2},\ldots ,{n-1 \choose n-1}.}
1615:
1556:
1387:
1202:
1980:
1589:
1299:
1997:
1322:
474:
392:
374:
1602:
Done. How about the section on calculating a row or column by itself? I think it is
1166:
I think your (Chad.netzer's) edits make the article clearer, so thanks for that. --
1083:
is irrelevant). I therefore think the wording should be "for non-negative integers
1457:
1416:
1933:
1571:
391:
So algorithms (though tied to a specific language) are not encyclopedia worthy?
113:
193:
168:
1988:
1968:
1941:
1619:
1611:
1597:
1579:
1560:
1552:
1479:
1465:
1451:
1424:
1410:
1391:
1383:
1368:
1354:
1335:
1315:
1293:
1275:
1257:
1245:
1233:
1223:
1206:
1198:
90:
281:
1470:
Actually it was english wikipedia only thats why i am saying pls check it
1976:
1952:
1433:
the reference of this content was given by some Fowler and david person.
1951:
We probably should mention the Jia Xian triangle here as well. The Page
788:{\displaystyle \sum _{k=0}^{n-1}{n-1 \choose k}^{2}={2n-2 \choose n-1}.}
337:] The anchor (#To_the_integers) is no longer available because it was
333:
This article links to one or more target anchors that no longer exist.
1230:
https://www.sciencedirect.com/science/article/pii/0315086079900740
926:{\displaystyle \sum _{k=0}^{m}{m \choose k}^{2}={2m \choose m},}
965:
start at 0. In the example you gave above, you have to define
657:
Hence, the text "the sum of the squares of the elements of the
1527:
Extension to negative rows and columns, and other fringe stuff
307:
284:
240:
15:
1111:) to indicate the row number and give it's relationship to
1549:
Pascal's triangle#Calculating a row or diagonal by itself
1400:
http://5010.mathed.usu.edu/Fall2022/EHumes/history.html
1382:. I wonder if some of that material should be rescued.
1379:
1214:
1743:{\displaystyle \lim _{n\to \infty }O(n)/(O(n)+E(n))=0}
473:
I believe you're right — this is an off-by-one error.
1805:
1760:
1667:
1125:
1050:
971:
839:
805:
673:
496:
1646:) be the number of odd binomial coefficients in row
112:, a collaborative effort to improve the coverage of
1922:
1785:
1742:
1143:
1075:
989:
925:
823:
787:
645:
1236:. So I intend to re-add it and fix the errors. -
1067:
1054:
914:
896:
878:
865:
776:
744:
726:
705:
634:
605:
587:
566:
554:
533:
521:
500:
1807:
1669:
1585:
1298:Now I read p.130-131 of Biggs, and I agree with
203:, a project which is currently considered to be
2024:Knowledge level-5 vital articles in Mathematics
1232:, and in fact confirms the material deleted by
438:"the sum of the squares of the elements of the
1509:Biggs, N.L. (1979). "Roots of Combinatorics".
292:This page has archives. Sections older than
8:
1228:The cited article is publicly available at
937:which is the formula given in the text. --
1280:Biggs apparently has a wikipedia article:
163:
58:
1875:
1864:
1840:
1829:
1822:
1810:
1804:
1771:
1759:
1696:
1672:
1666:
1396:Does this source is acceptable on pingala
1124:
1066:
1053:
1051:
1049:
970:
913:
895:
893:
884:
877:
864:
862:
855:
844:
838:
804:
775:
743:
741:
732:
725:
704:
702:
689:
678:
672:
633:
604:
602:
586:
565:
563:
553:
532:
530:
520:
499:
497:
495:
2014:Knowledge vital articles in Mathematics
1498:
661:row equals the middle element of the (2
165:
60:
19:
1044:start at zero (otherwise, the case of
302:when more than 5 sections are present.
2029:C-Class vital articles in Mathematics
1517:(1979): 130–131 – via Science Direct.
453:row equals the middle element of the
442:row equals the middle element of the
7:
1504:
1502:
435:The page lists the following claim:
199:This article is within the scope of
106:This article is within the scope of
1955:says about this relationship: : -->
1430:halyuddha do commentary on his work
1107:row, or use another variable (like
1040:to start at 0, and implicitly that
215:Knowledge:WikiProject Introductions
49:It is of interest to the following
1817:
1750:, because all coefficients in row
1679:
1058:
900:
869:
748:
709:
609:
570:
537:
504:
218:Template:WikiProject Introductions
14:
2039:Mid-priority mathematics articles
1533:Pascal's triangle#To the integers
371:Knowledge: Wikicode/Specification
296:may be automatically archived by
126:Knowledge:WikiProject Mathematics
2009:Knowledge level-5 vital articles
1472:2409:4050:2D05:5C64:0:0:A89:770F
1444:2409:4050:2D05:5C64:0:0:A89:770F
1036:", however the examples require
1005:which starts at zero. Thus, if
431:Question about a listed property
311:
245:
192:
167:
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
146:This article has been rated as
2019:C-Class level-5 vital articles
1942:12:06, 12 September 2023 (UTC)
1905:
1899:
1890:
1884:
1855:
1849:
1814:
1796:So is the claim instead that
1731:
1728:
1722:
1713:
1707:
1701:
1693:
1687:
1676:
1658:) be the number of even ones.
1630:Proportion of odd coefficients
1151:row, where needed. Comments?
1138:
1126:
1013:row numbering starts with the
488:has the binomial coefficients
1:
1076:{\displaystyle {n \choose n}}
120:and see a list of open tasks.
2034:C-Class mathematics articles
1586:#Extending Pascal's triangle
1584:This was discussed above at
1480:04:30, 2 February 2024 (UTC)
1466:03:26, 2 February 2024 (UTC)
1452:03:12, 2 February 2024 (UTC)
1425:02:10, 2 February 2024 (UTC)
1411:02:06, 2 February 2024 (UTC)
2055:
1989:23:27, 10 March 2024 (UTC)
1969:01:22, 10 March 2024 (UTC)
1947:Mention Jia Xian triangle?
1547:Other sections, including
1369:10:33, 23 March 2023 (UTC)
1355:21:39, 22 March 2023 (UTC)
1336:22:04, 23 March 2023 (UTC)
1316:11:43, 23 March 2023 (UTC)
1294:10:40, 23 March 2023 (UTC)
1276:10:30, 23 March 2023 (UTC)
1258:18:13, 22 March 2023 (UTC)
1246:10:12, 22 March 2023 (UTC)
1224:20:34, 21 March 2023 (UTC)
1207:09:59, 21 March 2023 (UTC)
1786:{\displaystyle n=2^{k}-1}
1620:12:37, 29 June 2023 (UTC)
1598:17:35, 27 June 2023 (UTC)
1580:12:15, 27 June 2023 (UTC)
1561:11:48, 27 June 2023 (UTC)
1392:16:14, 6 April 2023 (UTC)
201:WikiProject Introductions
187:
145:
78:
57:
1170:11:51, 11 Apr 2005 (UTC)
957:to indicate rows as the
477:08:37, 24 Mar 2005 (UTC)
387:06:47, 6 Mar 2005 (UTC)
377:17:57, 10 Oct 2004 (UTC)
365:05:57, 8 Jun 2004 (UTC)
152:project's priority scale
1155:22:07, 8 Apr 2005 (UTC)
941:10:57, 6 Apr 2005 (UTC)
422:15:14, 7 Mar 2005 (UTC)
407:15:23, 6 Mar 2005 (UTC)
395:08:36, 6 Mar 2005 (UTC)
109:WikiProject Mathematics
2004:C-Class vital articles
1924:
1880:
1845:
1787:
1744:
1145:
1077:
997:for just this purpose.
991:
927:
860:
825:
789:
700:
647:
299:Lowercase sigmabot III
221:Introductions articles
1925:
1860:
1825:
1788:
1745:
1146:
1144:{\displaystyle (n+1)}
1078:
992:
990:{\displaystyle m=n-1}
928:
840:
826:
824:{\displaystyle m=n-1}
790:
674:
648:
36:level-5 vital article
1803:
1758:
1665:
1661:It is not true that
1531:The lengthy section
1511:Historia Mathematica
1262:I still didn't read
1217:some similar claims.
1123:
1119:row in the text, to
1048:
969:
837:
803:
671:
494:
132:mathematics articles
1302:that Biggs claims (
1977:Jia Xian#Biography
1953:Jia_Xian#Biography
1920:
1821:
1783:
1740:
1683:
1327:Glen Van Brummelen
1141:
1115:. Or just change
1073:
987:
923:
821:
785:
643:
420:Wile E. Heresiarch
385:Wile E. Heresiarch
369:As I mentioned on
363:Wile E. Heresiarch
101:Mathematics portal
45:content assessment
1909:
1806:
1668:
1065:
912:
876:
774:
724:
632:
585:
552:
519:
353:
352:
339:deleted by a user
328:in most browsers.
306:
305:
237:
236:
233:
232:
229:
228:
162:
161:
158:
157:
2046:
1929:
1927:
1926:
1921:
1910:
1908:
1879:
1874:
1858:
1844:
1839:
1823:
1820:
1792:
1790:
1789:
1784:
1776:
1775:
1749:
1747:
1746:
1741:
1700:
1682:
1519:
1518:
1506:
1361:Jochen Burghardt
1333:
1308:Jochen Burghardt
1286:Jochen Burghardt
1268:Jochen Burghardt
1255:
1238:Jochen Burghardt
1221:
1150:
1148:
1147:
1142:
1082:
1080:
1079:
1074:
1072:
1071:
1070:
1057:
996:
994:
993:
988:
932:
930:
929:
924:
919:
918:
917:
908:
899:
889:
888:
883:
882:
881:
868:
859:
854:
830:
828:
827:
822:
794:
792:
791:
786:
781:
780:
779:
773:
762:
747:
737:
736:
731:
730:
729:
720:
708:
699:
688:
652:
650:
649:
644:
639:
638:
637:
631:
620:
608:
592:
591:
590:
581:
569:
559:
558:
557:
548:
536:
526:
525:
524:
515:
503:
347:Reporting errors
315:
314:
308:
301:
285:
249:
241:
223:
222:
219:
216:
213:
196:
189:
188:
183:
171:
164:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
2054:
2053:
2049:
2048:
2047:
2045:
2044:
2043:
1994:
1993:
1949:
1859:
1824:
1801:
1800:
1767:
1756:
1755:
1663:
1662:
1632:
1529:
1524:
1523:
1522:
1508:
1507:
1500:
1331:
1282:Norman L. Biggs
1253:
1219:
1179:
1121:
1120:
1052:
1046:
1045:
967:
966:
901:
894:
863:
861:
835:
834:
801:
800:
763:
749:
742:
710:
703:
701:
669:
668:
621:
610:
603:
571:
564:
538:
531:
505:
498:
492:
491:
433:
375:Derrick Coetzee
358:
349:
331:
330:
329:
312:
297:
286:
280:
254:
220:
217:
214:
211:
210:
177:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2052:
2050:
2042:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
1996:
1995:
1992:
1991:
1948:
1945:
1931:
1930:
1919:
1916:
1913:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1878:
1873:
1870:
1867:
1863:
1857:
1854:
1851:
1848:
1843:
1838:
1835:
1832:
1828:
1819:
1816:
1813:
1809:
1782:
1779:
1774:
1770:
1766:
1763:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1699:
1695:
1692:
1689:
1686:
1681:
1678:
1675:
1671:
1631:
1628:
1627:
1626:
1625:
1624:
1623:
1622:
1607:
1568:
1541:
1540:
1528:
1525:
1521:
1520:
1497:
1496:
1492:
1491:
1490:
1489:
1488:
1487:
1486:
1485:
1484:
1483:
1482:
1440:
1437:
1434:
1431:
1403:122.161.48.206
1397:
1394:
1375:
1374:
1373:
1372:
1371:
1342:
1341:
1340:
1339:
1338:
1296:
1278:
1226:
1195:
1194:
1178:
1175:
1174:
1173:
1172:
1171:
1161:
1160:
1159:
1158:
1157:
1156:
1140:
1137:
1134:
1131:
1128:
1069:
1064:
1061:
1056:
1018:
998:
986:
983:
980:
977:
974:
945:
944:
943:
942:
935:
934:
933:
922:
916:
911:
907:
904:
898:
892:
887:
880:
875:
872:
867:
858:
853:
850:
847:
843:
820:
817:
814:
811:
808:
797:
796:
795:
784:
778:
772:
769:
766:
761:
758:
755:
752:
746:
740:
735:
728:
723:
719:
716:
713:
707:
698:
695:
692:
687:
684:
681:
677:
655:
654:
653:
642:
636:
630:
627:
624:
619:
616:
613:
607:
601:
598:
595:
589:
584:
580:
577:
574:
568:
562:
556:
551:
547:
544:
541:
535:
529:
523:
518:
514:
511:
508:
502:
479:
478:
432:
429:
428:
427:
426:
425:
424:
423:
411:
410:
409:
408:
397:
396:
381:
379:
378:
357:
354:
351:
350:
344:
343:
342:
326:case-sensitive
320:
319:
318:
316:
304:
303:
291:
288:
287:
282:
278:
276:
273:
272:
256:
255:
250:
244:
235:
234:
231:
230:
227:
226:
224:
197:
185:
184:
172:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2051:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2001:
1999:
1990:
1986:
1982:
1978:
1973:
1972:
1971:
1970:
1966:
1962:
1957:
1954:
1946:
1944:
1943:
1939:
1935:
1917:
1914:
1911:
1902:
1896:
1893:
1887:
1881:
1876:
1871:
1868:
1865:
1861:
1852:
1846:
1841:
1836:
1833:
1830:
1826:
1811:
1799:
1798:
1797:
1794:
1780:
1777:
1772:
1768:
1764:
1761:
1754:are odd when
1753:
1737:
1734:
1725:
1719:
1716:
1710:
1704:
1697:
1690:
1684:
1673:
1659:
1657:
1653:
1649:
1645:
1641:
1636:
1629:
1621:
1617:
1613:
1608:
1605:
1601:
1600:
1599:
1595:
1591:
1587:
1583:
1582:
1581:
1577:
1573:
1569:
1565:
1564:
1563:
1562:
1558:
1554:
1550:
1545:
1538:
1537:
1536:
1534:
1526:
1516:
1512:
1505:
1503:
1499:
1495:
1481:
1477:
1473:
1469:
1468:
1467:
1463:
1459:
1455:
1454:
1453:
1449:
1445:
1441:
1438:
1435:
1432:
1428:
1427:
1426:
1422:
1418:
1414:
1413:
1412:
1408:
1404:
1401:
1398:
1395:
1393:
1389:
1385:
1381:
1376:
1370:
1366:
1362:
1358:
1357:
1356:
1352:
1348:
1343:
1337:
1334:
1332:---Wikaviani
1328:
1324:
1323:Roshdi Rashed
1319:
1318:
1317:
1313:
1309:
1305:
1301:
1297:
1295:
1291:
1287:
1283:
1279:
1277:
1273:
1269:
1265:
1261:
1260:
1259:
1256:
1254:---Wikaviani
1249:
1248:
1247:
1243:
1239:
1235:
1231:
1227:
1225:
1222:
1220:---Wikaviani
1216:
1211:
1210:
1209:
1208:
1204:
1200:
1193:
1190:
1189:Meru Prastara
1187:
1184:
1183:
1182:
1176:
1169:
1165:
1164:
1163:
1162:
1154:
1135:
1132:
1129:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1062:
1059:
1043:
1039:
1035:
1031:
1027:
1023:
1019:
1016:
1012:
1008:
1004:
999:
984:
981:
978:
975:
972:
964:
960:
956:
951:
950:
949:
948:
947:
946:
940:
936:
920:
909:
905:
902:
890:
885:
873:
870:
856:
851:
848:
845:
841:
833:
832:
818:
815:
812:
809:
806:
799:Substituting
798:
782:
770:
767:
764:
759:
756:
753:
750:
738:
733:
721:
717:
714:
711:
696:
693:
690:
685:
682:
679:
675:
667:
666:
664:
660:
656:
640:
628:
625:
622:
617:
614:
611:
599:
596:
593:
582:
578:
575:
572:
560:
549:
545:
542:
539:
527:
516:
512:
509:
506:
490:
489:
487:
483:
482:
481:
480:
476:
472:
471:
470:
468:
464:
458:
456:
452:
447:
445:
441:
436:
430:
421:
417:
416:
415:
414:
413:
412:
406:
401:
400:
399:
398:
394:
390:
389:
388:
386:
376:
372:
368:
367:
366:
364:
355:
348:
340:
336:
335:
334:
327:
323:
317:
310:
309:
300:
295:
290:
289:
275:
274:
271:
270:
266:
262:
258:
257:
253:
248:
243:
242:
239:
225:
212:Introductions
208:
207:
202:
198:
195:
191:
190:
186:
181:
176:
175:Introductions
173:
170:
166:
153:
149:
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
1958:
1950:
1932:
1795:
1751:
1660:
1655:
1651:
1647:
1643:
1639:
1637:
1633:
1603:
1546:
1542:
1530:
1514:
1510:
1493:
1330:
1263:
1252:
1218:
1196:
1191:
1188:
1185:
1180:
1168:Jitse Niesen
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1041:
1037:
1033:
1029:
1025:
1021:
1014:
1010:
1006:
1002:
962:
958:
954:
939:Jitse Niesen
665:- 1)" means
662:
658:
485:
466:
462:
459:
454:
450:
448:
443:
439:
437:
434:
405:Jitse Niesen
380:
359:
332:
324:Anchors are
321:
293:
259:
251:
238:
204:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
34:
1347:Ppppphgtygd
1153:Chad.netzer
123:Mathematics
114:mathematics
70:Mathematics
1998:Categories
1650:, and let
1567:editorial.
1494:References
1300:Wikaviani
39:is rated
393:Cburnett
294:365 days
252:Archives
1458:MrOllie
1442:Thanne
1417:MrOllie
1304:al-Tusi
1215:removed
1177:Pingala
831:yields
341:before.
206:defunct
180:defunct
150:on the
41:C-class
1934:Mgnbar
1572:Mgnbar
1264:Gibbs'
1091:where
1028:where
465:, not
47:scale.
1961:Agowa
457:-1."
28:This
1985:talk
1965:talk
1938:talk
1638:Let
1616:talk
1604:less
1594:talk
1576:talk
1557:talk
1476:talk
1462:talk
1448:talk
1421:talk
1407:talk
1388:talk
1380:diff
1365:talk
1351:talk
1312:talk
1290:talk
1284:. -
1272:talk
1242:talk
1203:talk
1087:and
1024:and
1017:row.
475:Deco
463:2n-1
356:Code
322:Tip:
1981:JBL
1808:lim
1670:lim
1590:JBL
1325:or
446:."
142:Mid
2000::
1987:)
1967:)
1940:)
1862:∑
1827:∑
1818:∞
1815:→
1793:.
1778:−
1680:∞
1677:→
1618:)
1612:Nø
1596:)
1578:)
1559:)
1553:Nø
1513:.
1501:^
1478:)
1464:)
1450:)
1423:)
1409:)
1390:)
1384:Nø
1367:)
1353:)
1314:)
1292:)
1274:)
1244:)
1234:Nø
1205:)
1199:Nø
1095:≥
1032:≥
982:−
842:∑
816:−
768:−
757:−
715:−
694:−
676:∑
626:−
615:−
597:…
576:−
543:−
510:−
469:.
467:2n
455:2n
444:2n
267:,
263:,
1983:(
1963:(
1936:(
1918:?
1915:0
1912:=
1906:)
1903:i
1900:(
1897:E
1894:+
1891:)
1888:i
1885:(
1882:O
1877:n
1872:1
1869:=
1866:i
1856:)
1853:i
1850:(
1847:O
1842:n
1837:1
1834:=
1831:i
1812:n
1781:1
1773:k
1769:2
1765:=
1762:n
1752:n
1738:0
1735:=
1732:)
1729:)
1726:n
1723:(
1720:E
1717:+
1714:)
1711:n
1708:(
1705:O
1702:(
1698:/
1694:)
1691:n
1688:(
1685:O
1674:n
1656:n
1654:(
1652:E
1648:n
1644:n
1642:(
1640:O
1614:(
1592:(
1574:(
1555:(
1515:6
1474:(
1460:(
1446:(
1419:(
1405:(
1386:(
1363:(
1349:(
1310:(
1288:(
1270:(
1240:(
1201:(
1139:)
1136:1
1133:+
1130:n
1127:(
1117:n
1113:n
1109:m
1105:0
1101:n
1097:k
1093:n
1089:k
1085:n
1068:)
1063:n
1060:n
1055:(
1042:n
1038:k
1034:k
1030:n
1026:k
1022:n
1015:0
1011:n
1007:n
1003:k
985:1
979:n
976:=
973:m
963:n
959:n
955:n
921:,
915:)
910:m
906:m
903:2
897:(
891:=
886:2
879:)
874:k
871:m
866:(
857:m
852:0
849:=
846:k
819:1
813:n
810:=
807:m
783:.
777:)
771:1
765:n
760:2
754:n
751:2
745:(
739:=
734:2
727:)
722:k
718:1
712:n
706:(
697:1
691:n
686:0
683:=
680:k
663:n
659:n
641:.
635:)
629:1
623:n
618:1
612:n
606:(
600:,
594:,
588:)
583:2
579:1
573:n
567:(
561:,
555:)
550:1
546:1
540:n
534:(
528:,
522:)
517:0
513:1
507:n
501:(
486:n
451:n
440:n
269:3
265:2
261:1
209:.
182:)
178:(
154:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
