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facts ... let's have at least a section number, but better a page(range) for each definition and important result. But rather than just using a single notation, I think there are books that switch from the present defintion for scalar rates to switch the sign when needed for a "matrix". And also recall the need to make things consistent for other articles that use transition rate and transition matrix.
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formulae need to use the other convention or a different notation. Is someone able to give a proper citation for a source (or sources) that definitely do use a single convention for defining the required rates and matrices? I did hide some text that looked definitely wrong with the convention stated.
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contain a section titled "Historical notes" (p.52) where a quote from
Youschkevitch suggests Markov's own work was limited to the discrete case ("For him the only real examples of the chains were literary texts, where the two states denoted the vowels and consonants"), but I haven't made any further
2443:
The example given in the article, describing a three-state space consisting of {Bull Market, Bear Market, Recession}, seems rather strange. In their ordinary meaning, the "Recession" state is not mutually distinct from the other two. It is certainly possible to have a
Recession during a Bull Market
352:
This is incorrect. Markov chains can have a non-discrete state space. However the word "chain" is often reserved for discrete time! A search in google scholar of "continuous time markov chain" yields one third of the result compared to "continuous time markov process". The article should definitely
974:
There are two problems: (i) to get a consistent formulation here for "Continuous time Markov process", and (ii) to provide definition, used in other articles and linked through to here, for "transition rate" and "transition rate matrix" where these would need to be of the "usual" form. It would be
923:
There is no good reason (as discussed below) to stick to the present convention, except the need to have a good source for inline citations for the various results that are quoted. A bald "reference" to a complete book at the end of an article is hardly helpful to anyone hoping to check particular
943:
The one inline citation given uses the sign convention as in the start of this article and has formulae agreeing with some of those here (at present), but a lot of the others don't appear. Some edits and reversion indicate that there may be more than one convention around, and it may be that some
441:
Good catch. When I wrote the EMC section, I hadn't heard that referred to as a "jump process" before, but from a quick search of papers, they look like they're used at least somewhat interchangeably, so I merged the two sections. I'm not familiar with the recent literature or subtleties between
1736:
notation, but rather a separate variable instead. The only notable exceptions that springs to mind are econophysics and computational finance (also tangentially related to my work) -- I've seen both notations used there, which may be due to the number of physicists in the field. I lean toward
959:
When I initially contributed to this article, I used the
Stewart reference (currently at the bottom). If I recall correctly, he consistently uses a single convention for CTMCs throughout. Note that the convention for DTMCs and CTMCs is different by definition (at least as far as I am aware).
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It shouldn't. I've changed it. I think some people may reserve the word "chain" for processes with discrete state spaces and this article may at some point have been moved from "continuous-time Markov chain". (Check the edit history, maybe?)
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which seems like an instructive path (for a rusty physicist like me at least) to understand the origin of this nice result. As I said, I am rusty, and I think I may have swapped columns and rows in the state rate matrix, in which case
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Somewhat in line with the original concern in this thread I have a concern about the adding of all transition probabilities to give one. I do not have the original Parzen reference cited here, but it occurs pretty obvious to me
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for a few examples. The elements in CTMCs are rates, not probabilities, and the diagonal elements are sort of an inflow to balance the outflow (maybe not the best way to think about it, but it's one analogy).
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is just because I am a physicist thenΒ :-). As I understand the theory, it does not have to be continuous with time. It could be continuous with some other variable. On the other hand the name of the article is
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be renamed! The name chain does not make sense for something that moves in continuous time on a contiuous space. To call a poisson-process a chain would be ok but a brownian motion is not a chain of events.
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article is presently (except possibly in the lead) only about discrete-time and discrete-space. The present version of this article is only about continuous-time discrete-space. And the present version of
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The state space is still assumed to be discrete in the article. A Markov process on a discrete space is generally referred to as a Markov chain. The more general definition should be given here. --
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726:. there's a comment in the main text explaining "no minus sign here if 1-qh in conditional prob above, as the q must be positive", which I believe to be correct (see also later where there's
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I dont know definition of conservative Q matrix, but the equation seems to be wrong, because it implies negative probabilties. maybe there is missing 1 and it should look like
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This looks wrong to me. In the example, 3 is the product of 6 and 1/2, with 6 being the rate rather the holding time (which is a random variable with a mean of 1/rate).
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good to give several different citation that agree on a defintion for these things (or possibly introduce an auxiliary matrix with negative values on the diagonal).
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400:. A possibility is to rename the present version of this article to sometime like "continuous time Markov chain" or "Markov chain (continuous time)" and to expand
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Each non-diagonal value can be computed as the product of the original state's holding time with the probability from the jump chain of moving to the given state.
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or Bear Market. It would be better to define the three-state space as {Bull Market, Bear Market, Stagnant Market} or {Bull Market, Bear Market, Mixed Market}. β
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I would make that change myself, but it requires re-drawing the figure, and I don't have an easy way to do that. Incidentally, the figure also violates the
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It seems to me that the terms "jump process" and "embedded Markov chain" relate to the same things; maybe the corresponding paragraphs should be merged?
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901:; this is the convention I've seen in recent papers I've read on markov processes. Is there a good reason to stick with the convention used here?
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The diagonal of the Q matrix is supposed to be negative for CTMCs, and the formula you were questioning is, in fact, correct in the article. See
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You are correct. It should also be the expected holding time, rather than the holding time (which is a random variable). I've just fixed this.
2377:, which mean you need to do one diagonalization, and then you can easily calculate the transition state probability matrix for arbitrary times
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are given as zero, but the second definition will have them as 1. I believe the first definition is correct, so we want instead that
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these two terms, so if anyone knows of any subtle differences between EMC and jump process, please edit accordingly. I see that
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And there is no simple elementwise expression, which is the source of complexity of calculating these. So it seems wrong...
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412:, "continuous time Markov chain" (renamed from here) and to an entirely new version of "Continuous-time Markov process".
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In the section about
Embedded Markov chains, there seems to be a discrepancy between the definition given in terms of
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in this case represents two different variables, not a difference equation. Also, to the best of my knowledge,
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I'm not exactly in OR either, but my research touches on it on occasion. I don't tend to see much use of the
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refers to a finance term, but it doesn't sound like it's the same as what is discussed in this article.
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is the transition probability matrix of the EMC, we should be trying to find the stationary states of
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2051:{\displaystyle d\mathbf {p} =\mathbf {p} (t+dt)-\mathbf {p} (t)=\mathbf {Q} \cdot \mathbf {p} (t)dt}
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is a small step in time. A little later, when writing down the differential equation, we also meet
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for that content in the latter page, and it must not be deleted as long as the latter page exists.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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2235:(This is something I am still trying to understand, but isn't the best approach to diagonalize
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2116:{\displaystyle {\frac {\partial \mathbf {p} }{\partial t}}=\mathbf {Q} \cdot \mathbf {p} }
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guideline that says "Numbers between β1 and +1 require a leading zero (0.02, not .02)". β
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2370:{\displaystyle {\boldsymbol {\Pi }}(t)=\mathbf {U} \exp(\mathbf {D} t)\mathbf {U} ^{-1}}
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should be a row vector instead and multiplied from the left, but with the same result.
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but seem to be suffering some awkward font issues. I'm unsure why at the moment.
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I just noticed that there's a similar example for the discrete-time case in the
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1946:{\displaystyle \mathbf {p} (t+dt)=(\mathbf {I} +\mathbf {Q} dt)\mathbf {p} (t)}
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seems like too much abuse of notation for a math article, especially because
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Yep, you are right, it is incorrect. I have taken it out of the article.
2296:{\displaystyle \mathbf {Q} t=\mathbf {U} (\mathbf {D} t)\mathbf {U} ^{-1}}
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Note: I have not checked to see if this change affects the definition of
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not be easier to understand, or is that just because I am a physicist? --
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Is that not correct, or am I just very tired and in an unfamiliar domain?
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1434:{\displaystyle \Pr(X(t+h)=i|X(t)=i)=1-\sum _{i\neq j}q_{ij}h+o(h),\,}
3281:{\displaystyle P(t)=\sum _{k=0}^{\infty }{\frac {t^{k}Q^{k}}{k!}}.}
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1272:{\displaystyle \Pr(X(t+h)=j|X(t)=i)=\delta _{ij}+q_{ij}h+o(h),\,}
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only has definitions suitable for discrete-space; similarly for
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for the steady states of the discrete time Markov process. If
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This is something that needs to be sorted out properly. The
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You're right about getting zeroes being correct, but need
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Thanks for these suggestions, I've now updated the image,
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http://www.encyclopediaofmath.org/index.php/Markov_process
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I also personally find it notationally confusing to use
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Strange example: {Bull Market, Bear Market, Recession}
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to properly cover continous-(multivariate)-space. Or
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Markov process, and it is explicitly mentioned that
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1112:{\displaystyle \Pr(X(t+h)=j|X(t)=i)=q_{ij}h+o(h)}
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2137:This is triggered by a question I raised at the
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2896:Can anyone shed light on the history of CTMCs?
1551:{\displaystyle \sum _{j}\Pr(X(t+h)=j|X(t)=i)=1}
540:{\displaystyle q_{ii}=1-\sum _{j\neq i}q_{ij},}
3195:By definition of matrix exponential, however:
2062:and by dividing this difference equation with
1455:In order to have the probablities add up to 1
894:{\displaystyle q_{ii}=-\sum _{j\neq i}q_{ij}}
719:{\displaystyle q_{ii}=\sum _{j\neq i}q_{ij}.}
8:
408:could be a simple overview with pointers to
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2165:{\displaystyle {\boldsymbol {\Pi }}(t)}
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2530:Well spotted, I've updated that too.
1807:I guess my notational preference for
304:Undo the name change, see this page:
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195:This article is within the scope of
90:This article is within the scope of
2876:to get them. I'll change this now.
1649:Maybe someone was tacitly assuming
661:This line has been changed back to
49:It is of interest to the following
18:Talk:Continuous-time Markov process
3380:Mid-importance Statistics articles
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829:{\displaystyle p(t+h)=e^{Qh}p(t)}
773:to have a transition matrix s.t.
215:Knowledge:WikiProject Mathematics
2920:Formula for P(t) seems incorrect
2685:{\displaystyle S=-D_{Q}^{-1}Q-I}
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2498:It looks good to me! Thanks. β
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3327:Rate instead of holding time.
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209:and see a list of open tasks.
104:and see a list of open tasks.
3390:C-Class mathematics articles
2381:as a simple matrix product?)
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970:05:24, 21 January 2010 (UTC)
954:10:45, 20 January 2010 (UTC)
762:{\displaystyle e^{-q_{ii}r}}
279:Continuous-time Markov chain
3375:C-Class Statistics articles
2965:{\displaystyle P(t)=e^{tQ}}
2391:) 07:31, 2 July 2010 (UTC)
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2139:Mathematics Reference Desk
1859:{\displaystyle \partial t}
1782:{\displaystyle t+\Delta t}
1133:Should have an additional
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436:13:48, 25 April 2007 (UTC)
3301:14:10, 21 July 2013 (UTC)
2975:has elements of the form
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2902:Markov Chain course notes
2398:is not diagonalizable. --
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911:15:22, 7 April 2010 (UTC)
836:, which is only valid if
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2423:16:47, 2 July 2010 (UTC)
2408:10:08, 2 July 2010 (UTC)
1823:{\displaystyle \Delta t}
1803:04:12, 2 July 2010 (UTC)
1753:{\displaystyle \Delta t}
1729:{\displaystyle \Delta t}
1707:12:00, 1 July 2010 (UTC)
1667:04:49, 1 July 2010 (UTC)
1605:{\displaystyle \Delta t}
241:project's priority scale
2483:I haven't had problems.
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198:WikiProject Mathematics
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1269:
1268:
1109:
891:
877:
826:
759:
716:
699:
618:
604:
537:
520:
190:Mathematics portal
45:content assessment
3311:comment added by
3273:
3087:
2741:{\displaystyle S}
2410:
2176:will be just the
2095:
1476:
1379:
862:
684:
589:
505:
313:comment added by
297:
296:
255:
254:
251:
250:
247:
246:
146:
145:
142:
141:
16:(Redirected from
3402:
3323:
3287:
3285:
3284:
3279:
3274:
3272:
3264:
3263:
3262:
3253:
3252:
3242:
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3234:
3191:
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3183:
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3172:
3149:
3148:
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3098:
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3078:
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2997:
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2849:
2773:
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2765:
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2739:
2723:
2721:
2720:
2715:
2691:
2689:
2688:
2683:
2671:
2663:
2634:
2632:
2631:
2626:
2620:
2612:
2587:
2585:
2584:
2579:
2577:
2576:
2392:
2376:
2374:
2373:
2368:
2366:
2365:
2357:
2345:
2331:
2314:
2302:
2300:
2299:
2294:
2292:
2291:
2283:
2271:
2263:
2252:
2232:
2230:
2229:
2224:
2216:
2190:
2171:
2169:
2168:
2163:
2152:
2122:
2120:
2119:
2114:
2112:
2104:
2096:
2094:
2086:
2085:
2076:
2057:
2055:
2054:
2049:
2032:
2024:
2007:
1981:
1973:
1952:
1950:
1949:
1944:
1933:
1919:
1911:
1882:
1865:
1863:
1862:
1857:
1829:
1827:
1826:
1821:
1788:
1786:
1785:
1780:
1759:
1757:
1756:
1751:
1735:
1733:
1732:
1727:
1696:
1694:
1693:
1688:
1634:
1632:
1631:
1626:
1611:
1609:
1608:
1603:
1557:
1555:
1554:
1549:
1520:
1484:
1440:
1438:
1437:
1432:
1407:
1406:
1393:
1348:
1278:
1276:
1275:
1270:
1245:
1244:
1229:
1228:
1192:
1118:
1116:
1115:
1110:
1090:
1089:
1053:
900:
898:
897:
892:
890:
889:
876:
855:
854:
835:
833:
832:
827:
813:
812:
768:
766:
765:
760:
758:
757:
753:
752:
725:
723:
722:
717:
712:
711:
698:
680:
679:
627:
625:
624:
619:
617:
616:
603:
582:
581:
566:
565:
546:
544:
543:
538:
533:
532:
519:
495:
494:
325:
276:
264:
263:
257:
223:
222:
219:
216:
213:
192:
187:
186:
176:
169:
168:
163:
155:
148:
136:importance scale
118:
117:
114:
111:
108:
87:
80:
79:
74:
66:
59:
42:
36:
35:
27:
21:
3410:
3409:
3405:
3404:
3403:
3401:
3400:
3399:
3365:
3364:
3329:
3306:
3265:
3254:
3244:
3243:
3200:
3199:
3161:
3153:
3137:
3112:
3107:
3106:
3079:
3050:
3049:
3010:
2985:
2980:
2979:
2949:
2929:
2928:
2922:
2894:
2833:
2829:
2828:
2811:
2810:
2750:
2749:
2730:
2729:
2694:
2693:
2641:
2640:
2590:
2589:
2565:
2560:
2559:
2556:
2441:
2352:
2305:
2304:
2278:
2243:
2242:
2181:
2180:
2143:
2142:
2087:
2077:
2070:
2069:
1961:
1960:
1873:
1872:
1845:
1844:
1809:
1808:
1762:
1761:
1739:
1738:
1715:
1714:
1673:
1672:
1614:
1613:
1591:
1590:
1471:
1470:
1395:
1309:
1308:
1233:
1217:
1153:
1152:
1135:Kronecker delta
1078:
1014:
1013:
878:
843:
838:
837:
801:
775:
774:
741:
733:
728:
727:
700:
668:
663:
662:
605:
573:
554:
549:
548:
521:
483:
478:
477:
459:
398:Markov property
308:
302:
272:
261:
220:
217:
214:
211:
210:
188:
181:
161:
115:
112:
109:
106:
105:
72:
43:on Knowledge's
40:
23:
22:
15:
12:
11:
5:
3408:
3406:
3398:
3397:
3392:
3387:
3382:
3377:
3367:
3366:
3363:
3362:
3328:
3325:
3289:
3288:
3277:
3271:
3268:
3261:
3257:
3251:
3247:
3238:
3233:
3230:
3227:
3223:
3219:
3216:
3213:
3210:
3207:
3193:
3192:
3181:
3178:
3171:
3168:
3164:
3160:
3156:
3152:
3147:
3144:
3140:
3136:
3133:
3130:
3127:
3122:
3119:
3115:
3100:
3099:
3085:
3082:
3075:
3070:
3067:
3063:
3057:
3053:
3044:
3039:
3036:
3033:
3029:
3025:
3020:
3017:
3013:
3009:
3006:
3003:
3000:
2995:
2992:
2988:
2973:
2972:
2959:
2956:
2952:
2948:
2945:
2942:
2939:
2936:
2921:
2918:
2893:
2890:
2889:
2888:
2865:
2860:
2857:
2852:
2848:
2845:
2842:
2839:
2836:
2832:
2827:
2824:
2821:
2818:
2763:
2760:
2757:
2737:
2713:
2710:
2707:
2704:
2701:
2681:
2678:
2675:
2670:
2667:
2662:
2658:
2654:
2651:
2648:
2624:
2619:
2616:
2611:
2607:
2603:
2600:
2597:
2575:
2572:
2568:
2555:
2552:
2551:
2550:
2549:
2548:
2547:
2546:
2545:
2544:
2543:
2542:
2440:
2437:
2436:
2435:
2434:
2433:
2432:
2431:
2430:
2429:
2428:
2427:
2426:
2425:
2364:
2361:
2356:
2351:
2348:
2344:
2340:
2337:
2334:
2330:
2326:
2323:
2320:
2317:
2313:
2290:
2287:
2282:
2277:
2274:
2270:
2266:
2262:
2258:
2255:
2251:
2222:
2219:
2215:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2189:
2161:
2158:
2155:
2151:
2135:
2125:
2124:
2123:
2111:
2107:
2103:
2099:
2093:
2090:
2084:
2080:
2060:
2059:
2058:
2047:
2044:
2041:
2038:
2035:
2031:
2027:
2023:
2019:
2016:
2013:
2010:
2006:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1980:
1976:
1972:
1968:
1955:
1954:
1953:
1942:
1939:
1936:
1932:
1928:
1925:
1922:
1918:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1881:
1855:
1852:
1819:
1816:
1778:
1775:
1772:
1769:
1749:
1746:
1725:
1722:
1711:
1710:
1709:
1686:
1683:
1680:
1624:
1621:
1601:
1598:
1578:
1577:
1576:
1575:
1574:
1573:
1565:
1564:
1563:
1562:
1561:
1560:
1559:
1558:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1519:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1483:
1479:
1461:
1460:
1459:
1458:
1457:
1456:
1448:
1447:
1446:
1445:
1444:
1443:
1442:
1441:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1405:
1402:
1398:
1392:
1389:
1386:
1382:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1347:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1299:
1298:
1297:
1296:
1295:
1294:
1286:
1285:
1284:
1283:
1282:
1281:
1280:
1279:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1243:
1240:
1236:
1232:
1227:
1224:
1220:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1191:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1143:
1142:
1141:
1140:
1139:
1138:
1126:
1125:
1124:
1123:
1122:
1121:
1120:
1119:
1108:
1105:
1102:
1099:
1096:
1093:
1088:
1085:
1081:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1052:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1004:
1003:
1002:
1001:
1000:
999:
990:
989:
988:
987:
941:
940:
939:
938:
937:
936:
926:193.62.153.194
916:
915:
914:
913:
888:
885:
881:
875:
872:
869:
865:
861:
858:
853:
850:
846:
825:
822:
819:
816:
811:
808:
804:
800:
797:
794:
791:
788:
785:
782:
756:
751:
748:
744:
740:
736:
715:
710:
707:
703:
697:
694:
691:
687:
683:
678:
675:
671:
615:
612:
608:
602:
599:
596:
592:
588:
585:
580:
576:
572:
569:
564:
561:
557:
536:
531:
528:
524:
518:
515:
512:
508:
504:
501:
498:
493:
490:
486:
463:67.233.152.184
458:
455:
454:
453:
429:
428:
427:
426:
425:
424:
406:Markov process
402:Markov process
394:Markov process
380:
379:
378:
377:
359:
357:
356:
355:
354:
347:
346:
315:213.103.216.40
301:
298:
295:
294:
289:now serves to
265:
253:
252:
249:
248:
245:
244:
233:
227:
226:
224:
207:the discussion
194:
193:
177:
165:
164:
156:
144:
143:
140:
139:
132:Mid-importance
128:
122:
121:
119:
102:the discussion
88:
76:
75:
73:Midβimportance
67:
55:
54:
48:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3407:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3372:
3370:
3361:
3357:
3353:
3349:
3348:
3347:
3346:
3342:
3338:
3337:92.110.219.57
3333:
3326:
3324:
3322:
3318:
3314:
3310:
3303:
3302:
3298:
3294:
3275:
3269:
3266:
3259:
3255:
3249:
3245:
3231:
3228:
3225:
3221:
3217:
3211:
3205:
3198:
3197:
3196:
3179:
3176:
3169:
3166:
3162:
3158:
3154:
3150:
3145:
3142:
3138:
3134:
3128:
3120:
3117:
3113:
3105:
3104:
3103:
3102:which equals
3083:
3080:
3073:
3068:
3065:
3061:
3055:
3051:
3037:
3034:
3031:
3027:
3023:
3018:
3015:
3011:
3007:
3001:
2993:
2990:
2986:
2978:
2977:
2976:
2957:
2954:
2950:
2946:
2940:
2934:
2927:
2926:
2925:
2919:
2917:
2916:
2912:
2908:
2903:
2899:
2898:Richard Weber
2891:
2887:
2883:
2879:
2863:
2858:
2855:
2850:
2843:
2837:
2834:
2830:
2825:
2822:
2819:
2816:
2808:
2807:
2806:
2804:
2800:
2796:
2792:
2787:
2785:
2781:
2777:
2761:
2758:
2755:
2735:
2727:
2708:
2705:
2702:
2679:
2676:
2673:
2668:
2665:
2660:
2656:
2652:
2649:
2646:
2638:
2622:
2617:
2614:
2609:
2605:
2601:
2598:
2595:
2573:
2570:
2566:
2553:
2541:
2537:
2533:
2529:
2528:
2527:
2523:
2519:
2515:
2511:
2510:
2509:
2505:
2501:
2497:
2496:
2495:
2491:
2487:
2484:
2482:
2476:
2475:
2474:
2470:
2466:
2462:
2458:
2457:
2456:
2455:
2451:
2447:
2438:
2424:
2420:
2416:
2415:Halcyonhazard
2412:
2411:
2409:
2405:
2401:
2397:
2396:
2390:
2386:
2382:
2380:
2362:
2359:
2346:
2335:
2332:
2324:
2318:
2288:
2285:
2272:
2256:
2253:
2240:
2239:
2217:
2206:
2203:
2200:
2194:
2179:
2175:
2156:
2140:
2136:
2133:
2132:
2126:
2105:
2097:
2091:
2068:
2067:
2065:
2061:
2045:
2042:
2036:
2025:
2017:
2011:
2000:
1994:
1991:
1988:
1985:
1974:
1966:
1959:
1958:
1956:
1937:
1923:
1920:
1912:
1901:
1895:
1892:
1889:
1886:
1871:
1870:
1869:
1853:
1842:
1838:
1833:
1817:
1806:
1805:
1804:
1800:
1796:
1795:Halcyonhazard
1792:
1791:Michael Hardy
1776:
1770:
1767:
1747:
1723:
1712:
1708:
1704:
1700:
1684:
1681:
1678:
1670:
1669:
1668:
1664:
1660:
1659:Michael Hardy
1656:
1652:
1648:
1647:
1646:
1642:
1638:
1622:
1619:
1599:
1588:
1584:
1583:
1582:
1581:
1580:
1579:
1571:
1570:
1569:
1568:
1567:
1566:
1545:
1542:
1536:
1533:
1527:
1521:
1513:
1510:
1504:
1501:
1498:
1492:
1481:
1477:
1469:
1468:
1467:
1466:
1465:
1464:
1463:
1462:
1454:
1453:
1452:
1451:
1450:
1449:
1426:
1420:
1414:
1411:
1408:
1403:
1400:
1396:
1390:
1387:
1384:
1380:
1376:
1373:
1370:
1364:
1361:
1355:
1349:
1341:
1338:
1332:
1329:
1326:
1320:
1307:
1306:
1305:
1304:
1303:
1302:
1301:
1300:
1292:
1291:
1290:
1289:
1288:
1287:
1264:
1258:
1252:
1249:
1246:
1241:
1238:
1234:
1230:
1225:
1222:
1218:
1214:
1208:
1205:
1199:
1193:
1185:
1182:
1176:
1173:
1170:
1164:
1151:
1150:
1149:
1148:
1147:
1146:
1145:
1144:
1136:
1132:
1131:
1130:
1129:
1128:
1127:
1103:
1097:
1094:
1091:
1086:
1083:
1079:
1075:
1069:
1066:
1060:
1054:
1046:
1043:
1037:
1034:
1031:
1025:
1012:
1011:
1010:
1009:
1008:
1007:
1006:
1005:
996:
995:
994:
993:
992:
991:
986:
982:
978:
973:
972:
971:
967:
963:
962:Halcyonhazard
958:
957:
956:
955:
951:
947:
935:
931:
927:
922:
921:
920:
919:
918:
917:
912:
908:
904:
886:
883:
879:
873:
870:
867:
863:
859:
856:
851:
848:
844:
820:
814:
809:
806:
802:
798:
792:
789:
786:
780:
772:
754:
749:
746:
742:
738:
734:
713:
708:
705:
701:
695:
692:
689:
685:
681:
676:
673:
669:
660:
659:
658:
654:
650:
649:Halcyonhazard
645:
642:
641:
640:
639:
635:
631:
630:Stepan Roucka
613:
610:
606:
600:
597:
594:
590:
586:
583:
578:
574:
570:
567:
562:
559:
555:
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3307:β Preceding
3304:
3290:
3194:
3101:
2974:
2923:
2907:Gareth Jones
2895:
2878:Gareth Jones
2805:Christopher
2795:76.120.32.59
2790:
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2786:Christopher
2776:76.120.32.59
2725:
2636:
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2532:Gareth Jones
2514:Markov chain
2486:Gareth Jones
2478:
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430:
410:Markov chain
389:Markov chain
367:129.7.128.30
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309:β Preceding
303:
274:Markov chain
269:this version
237:Mid-priority
236:
196:
162:Midβpriority
131:
91:
51:WikiProjects
2518:BarrelProof
2500:BarrelProof
2465:BarrelProof
2446:BarrelProof
1835:Continuous-
771:very useful
547:instead of
300:Terminology
212:Mathematics
203:mathematics
159:Mathematics
3369:Categories
2905:progress.
2516:article. β
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107:Statistics
98:statistics
70:Statistics
2303:and then
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457:citations
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2385:Slaunger
1957:as then
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1637:Slaunger
977:Melcombe
946:Melcombe
414:Melcombe
311:unsigned
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903:Flies 1
473:Michal
287:history
239:on the
134:on the
41:C-class
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