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to a base measure. For the most common, discrete, entropy the base measure is the counting measure. For "continuous" probability distributions the entropy is computed with respect to the
Lebesgue measure on the real line. In principe, however, you can choose any base measure. If you change the base measure, the entropy changes as well. (This seems counter-intuitive, since most people think that entropy is some inherent quantity.) In this respect the article is sloppy and does not mention with respect to which base measure the integral is computed and at the same time pretends that the measure space can be arbitrary. Either stick to the Lebesgue measure on the real line or do it in full generality.
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will have a (continuous) distribution of voltages around 0 and around 1. If the distribution is very sloppy, and I compute entropy with the integral formula, I get a nice positive entropy. But if I sharpen up the oscillator, so that it hits the voltages 0 and 1 more and more precisely (more and more sharply peaked at 0 and 1), instead of being sloppy, the "continuous entropy" will go negative -- more and more negative, the sharper the peaks get. This is very disconcerting if one was hoping to get the entropy resembling that of a discrete coin toss out of the system.
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mathematical definition of information, but it is more important all types of people with different background understand the problems, concepts, and solutions. My vote is to keep simple (this is after all
Knowledge, not a contest on how to make it precise with measure theory). The number of pages allowed is unlimited in internet. Please use a separate section on measure theory based definitions. In fact, by separating out this way, one will appreciate the beauty of measure theory for conceptualing problems and solving them elegantly.
417:- it is probably possible to give such a unique definition, but it would add unnecessary complication to the article; say, a Computer Science student does not normally need to study measure theory (I have a certain informal knowledge of it), as not all probability student need to know it. Beyond that, Shannon entropy is almost always used in the discrete version even because that's the natural application; beyond that, the two versions have different properties (the other article dismiss differential entropy as being of little use).--
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678:(1-k/2)*psi(k/a)*(k/2). It can be derived from the general formula for "Gamma" distributions (next lines in the same table) for case scale parameter O=2 and shape parameter k=k/2. Besides correct formula for the entropy is given at wiki-page for "Gamma distribution" in the section "Information entropy". So I ask anybody who is a specialist in the theme and may prove my supposal - is there really an error? If yes, could you please correct it.
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2785:, but perhaps there is something to discuss, so I bring it up here. The undone edit spoke of the limit of a normal distribution as the variance increases without bound and how that is similar to a uniform distribution and how the uniform distribution on the real line maximizes differential entropy. Unfortunately, the uniform distribution on the real line is at best an
734:* Neumann shows that maximization of differential entropy under a constraint on expected model entropy is equivalent to maximization of relative entropy with a particular reference measure. That reference measure satisfies the demand from Jaynes that (up to a multiplicative factor) the reference measure also had to be the "total ignorance" prior.
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I am an avg CS theory student, far from an expert. I was trying to find a general definition of entropy but I could not find anything. After a bit of thinking, I realized that entropy (differential or not) of a probability distribution (equivalently, of a random variable) must be defined with respect
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I removed the erroneous proof. It can still be seen above. I have proved the result by using variational calculus together with the fact that differential entropy is a strictly concave function (when functions differing in a set of measure zero are taken as equivalent). That takes 5 pages as a Latex
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is nearly zero outside of that interval, but the formula itself doesn't really mean anything. It is essentially unrelated to the ability to transmit information (which is the basis of entropy), and in fact, the entropy of any continuous distribution would always be infinite unless it is quantized. I
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Anyway, this article *completely* glosses over the hard problem of defining the continuous entropy. Consider a digital electronic oscillator, which nominally outputs two voltages: 0 Volts and 1 Volt. Now, if the oscillator is running at very high speed, it typically won't hit exactly 0 and 1, it
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I actually added this statement some time ago to the article (anonymously under my IP address), and it was recently marked as needing a citation or reference. I made the statement a little weaker by changing "quite general transformations" to "linear transformations", and added Reza's book as a
737:* Last not least, in physics there are frequently returning claims that differential entropy is useful, or in some settings even more powerful than relative entropy (e.g. Garbaczewski ). It would seem strange to me if such things do not find their counterparts in probability / information theory.
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Note that the continuous mutual information I(X;Y) has the distinction of retaining its fundamental significance as a measure of discrete information since it is actually the limit of the discrete mutual information of partitions of X and Y as these partitions become finer and finer. Thus it is
2484:(it is a differential cross entropy). In particular, the Kullback-Leilber divergence is not a difference of two differential entropies. Therefore, the first part of the proof does not apply. Note also that you are not using any of the assumptions: the first part works for any pdfs f and g.
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I am a practicing
Statistician with Ph.D in statistics, and I think there is a need for the definitions the way the original developer developed the thought processes about problems, concepts, and solutions. Academically I have been trained in measure theory. It is not important for single
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I added a warning to the top of the page redirecting people to LDDP a little more strongly than the previous versions had. I think this is the correct treatment for this subject. This page is worthy of note, and worth keeping, because differential entropy is used (incorrectly) all over the
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Hmm. At the top of this talk page, there is discussion of measure theory, but there is not a drop of measure theory to be found in this article. FWIW, at least some textbooks on probability *start* with defining measure theory (I own one such), so I'm not sure why this should be so scary.
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i am not sure how familiar the avg information theorist is with measure theory and integration. the article does use language such as "almost everywhere", "random variable", etc, so it must not be foreign. in that case, the division between "differential" and "non-differential" entropy is
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In the properties section, it is stated that the entropy of a transformed RV can be calculated by adding the expected value of the log of the
Jacobian to the entropy of the original RV. I believe, however, that this is only true for bijective transforms, but not for the general case.
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2789:. The article does not discuss these at all, and it is not at all clear from the present article how differential entropy would apply to an improper distribution. If someone has a source that discusses this, it might merit adding material to the article.
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The normal distribution maximizes differential entropy for a distribution with a given mean and a given variance. Perhaps it should be emphasized that there are distributions for which the mean and variance do not exist, such as the Cauchy distribution.
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reference. However, it is still true that I(X;Y) is invariant under any bijective, continuous (and thus monotonic) transformations of the continuous spaces X and Y. This fact needs a reference, though.
2064:{\displaystyle ={h}_{f\left(x\right)}\left(X\right)+{\underset {=-{h}_{g\left(x\right)}\left(X\right)}{\underbrace {\int _{-\infty }^{\infty }f\left(x\right)\log \left(g\left(x\right)\right)dx)} }}}
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invariant under quite general transformations of X and Y, and still represents the amount of discrete information that can be transmitted over a channel that admits a continuous space of values.
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Maybe the requirement that the two have the same variance was dropped? If not, they won't cancel and you end up with the ratio of standard deviations in the answer. Or am I missing something?
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The current proof on that a normal distribution maximizes differential entropy seems erroneous to me. First, it does not use normality anywhere. Second, the last integral is actually not
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This is an actual problem which does show up, and is partly solvable by using bins, and doing bin-counting. But this hardly merits the la-dee-da treatment given here! </rant: -->
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unnecessary, and quite misleading. one can simply use a single definition for entropy, for any random variable on a arbitrary measure space, and surely it has been done somewhere.
1468:{\displaystyle =\int _{-\infty }^{\infty }f\left(x\right){\underset {\log \left(x\right)\leq x-1}{\underbrace {\log \left({\frac {g\left(x\right)}{f\left(x\right)}}\right)} }}dx}
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Regarding "Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not." I propose
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Matthew Brand: "Structure
Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction". Neural Comp. vol. 11 (1999), pp. 1155-1182. Preprint:
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This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class.
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It seems to me there is else one error: in the table "Table of differential entropies" of the section "Differential entropies for various distributions", line for "
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I believe there is an error in the example of a uniform(0,1/2) distribution, the integral should evaluate to log(1/2), I do not have access to tex to correct it.
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Mutual information is invariant under diffeomorphisms of X and Y, which is not hard to show. Don't know though whether the same is true of homeomorphisms. --
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Garbaczewski: "Differential Entropy and Dynamics of Uncertainty". Journal of Statistical Physics, vol. 123 (2006), no. 2, pp. 315-355. Preprint:
539:. As a result, this is useful in that it is part of the LDDP in the cases where the measure is constant in x over some interval, and the probability
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think this article needs stronger pointers to LDDP, and the LDDP articles need to be much more clear about the relationship with discrete entropy.
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I believe there is an error in the differential entropy for multivariate
Gaussian (it should be in minus sign). No access to tex to correct it.
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The first part of your proof shows that
Kullback-Leibler divergence is non-negative. That is correct. The error is in that what you define as
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Since differential entropy is translation invariant we can assume that f(x) has the same mean as g(x). I've added this note to the proof.
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The problem here is that differential entropy doesn't really mean much. Among other things, it's not even dimensionally correct. The term
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I think it would be too quick to call differential entropy "of little use" in information theory. There are some hints to the opposite:
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The normal distribution maximizes the entropy for a fixed variance and MEAN. In the final step of the proof, it says that
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Edwin T. Jaynes, "Probability Theory: The Logic of
Science", Corrected reprint, Cambridge University Press 2004, p. 377.
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1877:{\displaystyle \int _{-\infty }^{\infty }f\left(x\right)\log \left({\frac {g\left(x\right)}{f\left(x\right)}}\right)dx}
1771:{\displaystyle \int _{-\infty }^{\infty }f\left(x\right)\log \left({\frac {g\left(x\right)}{f\left(x\right)}}\right)dx}
1076:{\displaystyle \int _{-\infty }^{\infty }g\left(x\right)xdx=0,\ \int _{-\infty }^{\infty }g\left(x\right){x}^{2}dx=1}
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PDF. I wonder whether the proof would be worthy as its own article? Are you aware of shorter routes to a proof?
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I do use properties of the Normal Distribution, please, let me know as I intend to get it back to the page. --
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731:* Brand suggested minimization of posterior differential entropy as a criterium for model selection.
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literature, so it's worth being aware of it, and understanding its properties, such as they are.
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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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on Knowledge. If you would like to participate, please visit the project page, where you can join
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2637:{\displaystyle \int _{-\infty }^{\infty }f(x){\frac {(x-\mu )^{2}}{2\sigma ^{2}}}={\frac {1}{2}}}
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Information entropy#Extending discrete entropy to the continuous case: differential entropy
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Information entropy#Extending_discrete_entropy_to_the_continuous_case:_differential_entropy
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This property is given in the article, thought it would be valuable to add a full proof.
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Adding those missing steps would be much appreciated:) After that the proof is fine. --
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It is not "unnecessary treatment */. It is very much a necessary treatment. Thanks
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I'm gonna enjoy watching you die, so let me it with my own eyes.(Darth Vader)
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Normal Distribution Maximizes The Differential Entropy For a Given Variance
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https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf
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I can incorporate this into article if you think it will be useful. --
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I don't see the error in the proof. Could you point me to the error?
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The expression in the proof should give you two different sigmas:
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While not being an expert on the subject, I do not agree, see
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Maximization in the normal distribution - Proof is incorrect
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Applying the inequality we showed prviously we get:
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https://en.wikipedia.org/Differential_entropy#Proof
2745:{\displaystyle {\frac {\sigma _{g}}{2\sigma _{f}}}}
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642:—The preceding
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1009:
1006:
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987:
983:
970:
966:
956:
955:second moment
939:
936:
933:
929:
921:
915:
912:
895:
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862:
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839:<rant: -->
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832:
831:
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611:93.173.58.105
608:
599:Error in Text
598:
596:
595:
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580:
576:
572:
552:
546:
538:
519:
516:
512:
487:
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461:198.160.96.25
458:
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437:
433:
423:
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416:
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50:
45:
41:
35:
27:
23:
18:
17:
2825:
2807:
2804:MathML error
2780:
2757:— Preceding
2754:
2702:
2688:
2668:
2646:
2537:
2518:
2500:
2486:
2436:
2422:
2419:
2405:
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2387:
2357:
2330:
2167:
1672:
1122:
875:
866:
847:
843:
838:
819:
804:
799:
784:
720:Blaisorblade
709:
681:— Preceding
675:
671:
667:
663:
661:
639:
605:— Preceding
602:
582:
473:
452:
448:
436:129.97.84.19
426:
419:Blaisorblade
401:
382:
378:
360:(dead link)
355:
330:
290:
226:Low-priority
225:
185:
151:Low‑priority
120:
80:
40:WikiProjects
2808:in section
2334:—Preceding
687:Ashc~ukwiki
668:Chi-squared
627:5.176.76.35
455:—Preceding
430:—Preceding
201:Mathematics
192:mathematics
148:Mathematics
2844:Categories
96:Statistics
87:statistics
59:Statistics
672:plus sign
2759:unsigned
2336:unsigned
957:to be 1:
808:Deepmath
695:contribs
683:unsigned
644:unsigned
607:unsigned
586:Vertigre
571:Vertigre
457:unsigned
432:unsigned
2791:𝕃eegrc
1778:we get:
743:Webtier
405:Mct mht
333:on the
306:Physics
297:Physics
253:Physics
228:on the
123:on the
30:C-class
2425:Royi A
2314:Royi A
36:scale.
2521:Kaba3
2489:Kaba3
2408:Kaba3
2390:Kaba3
909:be a
850:linas
822:Kaba3
385:Kaba3
2832:talk
2818:talk
2795:talk
2767:talk
2695:talk
2681:talk
2669:f(x)
2525:talk
2511:talk
2493:talk
2429:talk
2412:talk
2394:talk
2344:talk
2318:talk
2312:. --
1084:Let
878:Let
854:talk
826:talk
812:talk
747:talk
691:talk
652:talk
631:talk
615:talk
590:talk
575:talk
465:talk
440:talk
389:talk
366:talk
2677:PAR
1973:log
1819:log
1713:log
1428:log
1371:log
1257:log
1117:PDF
914:PDF
325:Low
220:Low
115:Mid
2846::
2834:)
2820:)
2812:--
2797:)
2769:)
2731:σ
2717:σ
2697:)
2683:)
2655:μ
2610:σ
2591:μ
2588:−
2562:∞
2557:∞
2554:−
2550:∫
2527:)
2513:)
2495:)
2487:--
2431:)
2414:)
2406:--
2396:)
2388:--
2385:.
2346:)
2320:)
2248:≤
2211:−
2117:−
2022:−
2013:⏟
1976:
1954:∞
1949:∞
1946:−
1942:∫
1822:
1800:∞
1795:∞
1792:−
1788:∫
1716:
1694:∞
1689:∞
1686:−
1682:∫
1630:−
1611:∞
1606:∞
1603:−
1599:∫
1560:−
1500:∞
1495:∞
1492:−
1488:∫
1484:≤
1451:−
1445:≤
1431:
1422:⏟
1374:
1347:∞
1342:∞
1339:−
1335:∫
1260:
1238:∞
1233:∞
1230:−
1226:∫
1222:−
1173:‖
1136:−
1031:∞
1026:∞
1023:−
1019:∫
979:∞
974:∞
971:−
967:∫
873:.
856:)
828:)
814:)
749:)
697:)
693:•
654:)
633:)
617:)
592:)
577:)
467:)
442:)
391:)
383:--
368:)
2830:(
2816:(
2793:(
2765:(
2735:f
2727:2
2721:g
2693:(
2679:(
2630:2
2627:1
2622:=
2614:2
2606:2
2599:2
2595:)
2585:x
2582:(
2576:)
2573:x
2570:(
2567:f
2523:(
2509:(
2491:(
2472:g
2450:g
2446:h
2427:(
2410:(
2392:(
2371:g
2367:h
2342:(
2316:(
2299:)
2296:x
2293:(
2289:f
2286:=
2282:)
2279:x
2276:(
2272:g
2251:0
2244:)
2241:X
2238:(
2231:)
2228:x
2225:(
2221:g
2216:h
2207:)
2204:X
2201:(
2194:)
2191:x
2188:(
2184:f
2179:h
2150:)
2147:X
2144:(
2137:)
2134:x
2131:(
2127:g
2122:h
2113:)
2110:X
2107:(
2100:)
2097:x
2094:(
2090:f
2085:h
2080:=
2055:)
2052:X
2049:(
2042:)
2039:x
2036:(
2032:g
2027:h
2019:=
2009:)
2006:x
2003:d
1999:)
1994:)
1991:x
1988:(
1984:g
1980:(
1969:)
1966:x
1963:(
1959:f
1933:+
1929:)
1926:X
1923:(
1916:)
1913:x
1910:(
1906:f
1901:h
1896:=
1872:x
1869:d
1865:)
1858:)
1855:x
1852:(
1848:f
1842:)
1839:x
1836:(
1832:g
1826:(
1815:)
1812:x
1809:(
1805:f
1766:x
1763:d
1759:)
1752:)
1749:x
1746:(
1742:f
1736:)
1733:x
1730:(
1726:g
1720:(
1709:)
1706:x
1703:(
1699:f
1656:0
1653:=
1650:x
1647:d
1643:)
1640:x
1637:(
1633:f
1626:)
1623:x
1620:(
1616:g
1595:=
1574:x
1571:d
1567:)
1563:1
1553:)
1550:x
1547:(
1543:f
1537:)
1534:x
1531:(
1527:g
1520:(
1515:)
1512:x
1509:(
1505:f
1463:x
1460:d
1454:1
1448:x
1441:)
1438:x
1435:(
1417:)
1410:)
1407:x
1404:(
1400:f
1394:)
1391:x
1388:(
1384:g
1378:(
1362:)
1359:x
1356:(
1352:f
1331:=
1310:x
1307:d
1303:)
1296:)
1293:x
1290:(
1286:g
1280:)
1277:x
1274:(
1270:f
1264:(
1253:)
1250:x
1247:(
1243:f
1219:=
1195:=
1191:)
1186:)
1183:x
1180:(
1176:g
1169:)
1166:x
1163:(
1159:f
1155:(
1149:L
1146:K
1141:D
1102:)
1099:x
1096:(
1092:f
1071:1
1068:=
1065:x
1062:d
1057:2
1052:x
1046:)
1043:x
1040:(
1036:g
1013:,
1010:0
1007:=
1004:x
1001:d
998:x
994:)
991:x
988:(
984:g
940:)
937:x
934:(
930:g
916:.
896:)
893:x
890:(
886:g
852:(
824:(
810:(
745:(
689:(
676:+
658:.
650:(
629:(
613:(
588:(
573:(
556:)
553:x
550:(
547:p
520:x
517:d
513:1
491:)
488:x
485:(
482:f
463:(
438:(
387:(
364:(
337:.
232:.
127:.
42::
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