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saying that the photon's four-momentum is an eigenstate. Likewise for a photon with the opposite momentum — that is, with momentum antiparallel to the implicit boost direction of the
Lorentz transformation — it will be red-shifted rather than blue-shifted but otherwise this too is a Doppler shift and an eigenstate. If we do mention any of this in the article, I'm thinking it should be exceedingly brief. —
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another spatial dimension, introducing the famous light cones, what would the independent eigenvectors be in this case? As far as I understand, an n-dimensional space can only have n linearly-independent eigenvectors (in this case n=3), which does not seem to be enough to represent the vectors where the speed does not change (light cones).
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The eigenstructure is a significant algebraic feature of any linear transformation, so it is odd that it is not mentioned in an article on the most important linear transformation in physics. This case, in particular, needs some discussion as it involves the apparent paradox (to
Galilean thinkers) of
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Yes, a photon moving in the same direction as the implicit boost direction of the
Lorentz transformation will have its energy and momentum scaled by the same multiplicative factor — an amount determined by the Doppler shift. Because it is the same scaling factor for the momentum and energy, that's
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which, on inspection, is incorrect. Note the physical units within the parenthesis must evaluate to time, but the last term has physical units that evaluate to time over velocity. This is due to the squaring of the speed of light in the denominator of the last term, which is apparently a typo. I
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More of a technical question, but how does the eigenvector interpretation generalise to higher dimensions? For example, if we have two dimensions (space and time), we find two linearly independent eigenvectors whose direction (speed) does not change, c and -c. So far so good. However, if we add
291:
The
Lorentz Transformation article is very math oriented. But you failed to mention the Geometric Algebra (GA) approach , which is much easier than the tensorial method. For example , I could show you , based on many papers by Hestenes, Doran , ... that all the very basic results of Special
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Many thanks for your advice. I have restarted Mac OS 14.6.1 and
Firefox 129.0.2 but still get a dozen yellow banners with "Math output error" in red. The rest of the math renders OK. Chrome renders all the math OK but I get the same yellow banners on Safari. Best wishes
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to be time, not the actual physical units of distance squared divided by time. In the morning, the error seems as clearly false as it seemed to be clear in the middle of the night. Sorry, and thank you, DVdm, for the quick reversion. --
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transforming between frames that are both traveling at the same speed c, and the limiting processes that are involved because of that. However, I will leave additions to this topic to editors with more experience than myself.
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Maybe "... two frames with the origins coinciding at (0, 0, 0, 0)" would satisfy the original comment, while emphasizing spacetime. On the other hand, given that this is the definition of an origin it is an odd statement.
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It is a simple matter to find the eigenvalues Sqrt and its inverse, where b is beta, and eigenstates ( and ) of the
Lorentz transformation. Some commentators on web forums claim that the eigenvalues are related to the
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The coordinates are for spacetime, so the origin is defined as (0, 0, 0, 0), which includes t=0. Saying two frames have the same origin at t=0 is like saying they have the same origin at x=0, or y=0, or
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There are lots of Math output errors (Firefox 129.0.2), but renders OK with Chrome 128.0.6613.85. Could somebody identify the problem? I am reluctant to meddle myself due to inexperience.
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Whoops, it seemed so obvious in the middle of the night, and I did a quick web search at the time to verify the equation that seemed to validate what I thought at the time. I was reading
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I have restored the original formatting. Users can say which version they prefer. I have not mastered the alignment of matrices so maybe it can be done better by some clever person.
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The
Clifford (geometric) algebra approach to spacetime physics advocated by Hestenes, Baylis and others would be worth a separate article. There does exist a rather unsatisfactory
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My concern is that the current text encourages people to think in terms of 3D space with time somehow separate, rather than thinking in term of 4D spacetime.
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A spatial vector orthogonal to the implicit boost direction will have an eigenvalue of 1. There will not be length contraction of this vector. —
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The Tex for section 6.2 (contravariant vectors) is not showing the "-" on the "-1" in the exponents. It just looks like a " 1"
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of the subject that would be a guide what NOT to do in writing a
Knowledge article on physics using geometric algebra.
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595:{\displaystyle 1/\qquad e_{0}^{2}=f_{0}^{2}=1\qquad e_{1}^{2}=f_{1}^{2}=-1\qquad e_{0}\cdot e_{1}=f_{0}\cdot f_{1}=0}
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A note to say that I got satisfactory rendering results after changing my
Appearance Skin from legacy to default.
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Relativity , time dilation , length contraction , time and length units , are contained in a single GA relation
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I think it should be mentioned that the two frames have the same origin at t=0 for the first equation
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I don't see the render errors with
Firefox. Perhaps refresh the page, restart your browser, etc. —
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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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It is now an easy task to deduce the Lorentz transformations and demonstrate the reciprocity.
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If you were using chrome, it is probably poor setups in handling MathML. See discussion in
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I came to this page to comment that the phrase "two frames with the origins coinciding at
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Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering
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See Hestenes and Doran for free on : geocalc.clas.asu.edu and geometry.mrao.cam.ac.uk
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But sorry , I will not do so , because that could be qualified original research !
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Middle of the night is a fun place alright, but sometimes a dangerous one
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The reciprocity is obvious with (3) and the fact that we can imagine the
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is the euclidian velocity wich represents the movement of the f system :
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1590:{\displaystyle 7/\qquad 1=f_{0}e_{0}e_{0}f_{0}=\gamma ^{2}(1-|v|^{2})}
695:{\displaystyle 2/\qquad f_{0}e_{0}=f_{0}\cdot e_{0}+f_{0}\wedge e_{0}}
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For whom are you writing, man???????? Only for yourself, I guess.
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2263:{\displaystyle t'=\gamma \cdot \left(t-{\frac {vx}{c^{2}}}\right)}
2169:{\displaystyle t'=\gamma \cdot \left(t-{\frac {vx}{c^{2}}}\right)}
1041:{\displaystyle 4/\qquad f_{0}\wedge e_{0}e_{0}=f_{0}-\gamma e_{0}}
794:{\displaystyle f_{0}=f_{0}\cdot e_{0}e_{0}+f_{0}\wedge e_{0}e_{0}}
2346:{\displaystyle ct'=\gamma \cdot \left(ct-{\frac {vx}{c}}\right).}
1236:{\displaystyle 5/\qquad v=(f_{0}\wedge e_{0})/(f_{0}\cdot e_{0})}
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I have added the expression for the general boost direction.
1471:{\displaystyle 6/\qquad f_{0}e_{0}=\gamma =\gamma (1+v)}
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Knowledge level-4 vital articles in Physical sciences
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851:{\displaystyle 3/\qquad f_{0}\cdot e_{0}=\gamma }
2505:I just want to say that I love this picture :)
2916:Knowledge vital articles in Physical sciences
2856:: proper link to given source, looks, remove
426:{\displaystyle (e_{0},e_{1})and(f_{0},f_{1})}
8:
2589:. Not really needed, but won't do any harm:
1320:{\displaystyle 5bis/\qquad |v|=v/e_{1}e_{0}}
318:→== Examples, please!!!!!!!!!!!!!!!!!!!! ==
2946:B-Class physics articles of High-importance
2931:B-Class vital articles in Physical sciences
2652:. Should this be discussed in the article?
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2028:"Spacetime physics with geometric algebra"
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2488:I see it with Firefox under Windows. --
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349:We consider two references frames
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167:This article is supported by
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126:and see a list of open tasks.
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2357:See the literature. I have
2064:Baylis, William E. (2012).
2032:American Journal of Physics
2000:Prokaryotic Caspase Homolog
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2179:have removed that square.
1821:{\displaystyle \tau f_{0}}
1745:{\displaystyle \tau f_{0}}
903:We notice that the vector
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295:e’0 e0= e’0 . e0 + e’0∧e0
158:project's importance scale
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2688:15:33, 11 July 2024 (UTC)
2662:22:52, 10 July 2024 (UTC)
2361:the correct equation. -
2102:The first equation reads
880:by (3) and we guess that
341:19:42, 8 April 2019 (UTC)
274:This page is archived by
170:the relativity task force
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2936:B-Class physics articles
2560:First equation condition
2098:Corrected First Equation
2090:06:20, 12 May 2019 (UTC)
2026:Hestenes, David (2003).
2008:23:11, 11 May 2019 (UTC)
2832:General boost direction
1989:08:24, 2 May 2019 (UTC)
1718:17:26, 1 May 2019 (UTC)
1696:{\displaystyle \gamma }
893:{\displaystyle \gamma }
873:{\displaystyle \gamma }
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2222:
2205:
2200:
2199:
2149:
2141:
2132:
2128:
2111:
2106:
2105:
2100:
2075:
2074:
2073:
2063:
2062:
2058:
2048:
2046:
2025:
2024:
2020:
1949:
1948:
1924:
1911:
1903:
1902:
1862:
1861:
1858:and you find :
1838:
1830:
1829:
1808:
1800:
1799:
1775:
1762:
1754:
1753:
1732:
1724:
1723:
1685:
1684:
1650:
1638:
1600:
1599:
1572:
1545:
1532:
1522:
1512:
1502:
1481:
1480:
1431:
1418:
1397:
1384:
1356:
1346:
1331:
1330:
1307:
1297:
1249:
1248:
1220:
1207:
1186:
1173:
1149:
1148:
1127:
1126:
1097:
1081:
1071:
1058:
1050:
1049:
1028:
1012:
999:
989:
976:
961:
960:
937:
932:
931:
910:
905:
904:
882:
881:
862:
861:
832:
819:
804:
803:
781:
771:
758:
745:
735:
722:
709:
704:
703:
682:
669:
656:
643:
630:
620:
605:
604:
576:
563:
550:
537:
437:
436:
410:
397:
372:
359:
351:
350:
346:OK I will try.
302:
289:
280:
219:
213:
191:
154:High-importance
137:
134:
131:
128:
127:
105:
98:
79:High‑importance
78:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2969:
2967:
2959:
2958:
2953:
2948:
2943:
2938:
2933:
2928:
2923:
2918:
2913:
2908:
2898:
2897:
2894:
2893:
2892:
2891:
2833:
2830:
2829:
2828:
2827:
2826:
2825:
2824:
2753:
2750:
2749:
2748:
2747:
2746:
2708:
2707:
2706:
2650:Doppler effect
2644:
2641:
2640:
2639:
2638:
2637:
2627:Subbookkeepper
2622:
2619:
2615:
2566:31.164.189.193
2561:
2558:
2557:
2556:
2526:98.128.172.242
2509:This picture!
2506:
2503:
2502:
2501:
2500:
2499:
2472:Pauli matrices
2445:
2442:
2441:
2440:
2439:
2438:
2437:
2436:
2391:
2388:
2374:
2373:
2355:
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2342:
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2332:
2328:
2325:
2319:
2316:
2313:
2309:
2305:
2302:
2299:
2295:
2292:
2288:
2275:
2272:
2271:
2270:
2258:
2250:
2246:
2241:
2238:
2232:
2229:
2225:
2221:
2218:
2215:
2211:
2208:
2164:
2156:
2152:
2147:
2144:
2138:
2135:
2131:
2127:
2124:
2121:
2117:
2114:
2099:
2096:
2095:
2094:
2093:
2092:
2072:
2071:
2056:
2017:
2016:
2012:
2011:
2010:
1965:
1962:
1959:
1956:
1936:
1931:
1927:
1923:
1918:
1914:
1910:
1888:
1885:
1882:
1879:
1873:
1869:
1845:
1841:
1837:
1828:on the vector
1815:
1811:
1807:
1787:
1782:
1778:
1774:
1769:
1765:
1761:
1739:
1735:
1731:
1692:
1668:
1664:
1660:
1657:
1653:
1647:
1642:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1611:
1607:
1586:
1581:
1576:
1571:
1567:
1563:
1560:
1557:
1552:
1548:
1544:
1539:
1535:
1529:
1525:
1519:
1515:
1509:
1505:
1501:
1498:
1492:
1488:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1438:
1434:
1430:
1425:
1421:
1417:
1413:
1409:
1404:
1400:
1396:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1353:
1349:
1342:
1338:
1314:
1310:
1304:
1300:
1295:
1291:
1288:
1284:
1280:
1276:
1269:
1265:
1262:
1259:
1256:
1232:
1227:
1223:
1219:
1214:
1210:
1206:
1202:
1198:
1193:
1189:
1185:
1180:
1176:
1172:
1169:
1166:
1160:
1156:
1134:
1112:
1109:
1104:
1100:
1096:
1093:
1088:
1084:
1078:
1074:
1070:
1065:
1061:
1057:
1035:
1031:
1027:
1024:
1019:
1015:
1011:
1006:
1002:
996:
992:
988:
983:
979:
972:
968:
944:
940:
917:
913:
889:
869:
847:
844:
839:
835:
831:
826:
822:
815:
811:
788:
784:
778:
774:
770:
765:
761:
757:
752:
748:
742:
738:
734:
729:
725:
721:
716:
712:
689:
685:
681:
676:
672:
668:
663:
659:
655:
650:
646:
642:
637:
633:
627:
623:
616:
612:
591:
588:
583:
579:
575:
570:
566:
562:
557:
553:
549:
544:
540:
534:
531:
528:
523:
518:
514:
510:
505:
500:
496:
490:
487:
482:
477:
473:
469:
464:
459:
455:
448:
444:
422:
417:
413:
409:
404:
400:
396:
393:
390:
387:
384:
379:
375:
371:
366:
362:
358:
300:
288:
285:
282:
281:
272:
271:
268:
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266:
265:
260:
255:
250:
245:
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235:
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222:
221:
220:
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211:
209:
208:
195:
183:
182:
179:
178:
175:
174:
165:
162:
161:
150:
144:
143:
141:
124:the discussion
111:
110:
107:Physics portal
94:
82:
81:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2968:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2903:
2901:
2889:
2885:
2881:
2877:
2876:
2875:
2871:
2867:
2863:
2859:
2858:wp:repeatlink
2855:
2851:
2850:
2849:
2847:
2843:
2839:
2831:
2822:
2818:
2814:
2810:
2809:
2807:
2803:
2799:
2794:
2793:
2792:
2788:
2784:
2780:
2777:
2771:
2770:
2769:
2767:
2763:
2759:
2752:Format errors
2751:
2745:
2741:
2737:
2733:
2730:
2724:
2723:
2722:
2718:
2714:
2709:
2704:
2700:
2696:
2691:
2690:
2689:
2685:
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2677:
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2665:
2663:
2659:
2655:
2651:
2642:
2636:
2632:
2628:
2623:
2620:
2616:
2613:
2609:
2605:
2604:
2603:
2599:
2595:
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2571:
2567:
2559:
2555:
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2542:
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2535:
2531:
2527:
2523:
2514:
2510:
2504:
2498:
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2486:
2485:
2481:
2477:
2473:
2469:
2468:
2467:
2465:
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2457:
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2435:
2431:
2427:
2415:
2414:
2413:
2410:
2406:
2389:
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2376:
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2372:
2368:
2364:
2360:
2356:
2340:
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2326:
2323:
2317:
2314:
2311:
2307:
2303:
2300:
2297:
2293:
2290:
2286:
2279:
2278:
2276:
2273:
2256:
2248:
2244:
2239:
2236:
2230:
2227:
2223:
2219:
2216:
2213:
2209:
2206:
2198:
2197:
2196:The equation
2195:
2194:
2193:
2192:
2189:
2185:
2180:
2176:
2162:
2154:
2150:
2145:
2142:
2136:
2133:
2129:
2125:
2122:
2119:
2115:
2112:
2103:
2097:
2091:
2087:
2083:
2079:
2078:
2077:
2076:
2067:
2060:
2057:
2045:
2041:
2037:
2033:
2029:
2022:
2019:
2015:
2009:
2005:
2001:
1997:
1993:
1992:
1991:
1990:
1986:
1982:
1977:
1960:
1957:
1929:
1925:
1921:
1916:
1912:
1899:
1886:
1883:
1880:
1877:
1871:
1867:
1859:
1843:
1839:
1835:
1813:
1809:
1805:
1780:
1776:
1772:
1767:
1763:
1737:
1733:
1729:
1720:
1719:
1715:
1711:
1707:
1704:
1690:
1681:
1666:
1662:
1658:
1655:
1645:
1635:
1627:
1624:
1618:
1615:
1609:
1605:
1597:
1579:
1569:
1561:
1558:
1550:
1546:
1542:
1537:
1533:
1527:
1523:
1517:
1513:
1507:
1503:
1499:
1496:
1490:
1486:
1478:
1462:
1459:
1456:
1450:
1447:
1436:
1432:
1428:
1423:
1419:
1411:
1402:
1398:
1394:
1389:
1385:
1378:
1375:
1369:
1366:
1361:
1357:
1351:
1347:
1340:
1336:
1328:
1312:
1308:
1302:
1298:
1293:
1289:
1286:
1278:
1267:
1263:
1260:
1257:
1254:
1246:
1243:
1225:
1221:
1217:
1212:
1208:
1200:
1191:
1187:
1183:
1178:
1174:
1167:
1164:
1158:
1154:
1146:
1132:
1123:
1110:
1107:
1102:
1098:
1094:
1086:
1082:
1076:
1072:
1068:
1063:
1059:
1033:
1029:
1025:
1022:
1017:
1013:
1009:
1004:
1000:
994:
990:
986:
981:
977:
970:
966:
958:
942:
938:
915:
911:
901:
887:
867:
858:
845:
842:
837:
833:
829:
824:
820:
813:
809:
801:
786:
782:
776:
772:
768:
763:
759:
755:
750:
746:
740:
736:
732:
727:
723:
719:
714:
710:
687:
683:
679:
674:
670:
666:
661:
657:
653:
648:
644:
640:
635:
631:
625:
621:
614:
610:
602:
589:
586:
581:
577:
573:
568:
564:
560:
555:
551:
547:
542:
538:
532:
529:
526:
521:
516:
512:
508:
503:
498:
494:
488:
485:
480:
475:
471:
467:
462:
457:
453:
446:
442:
434:
415:
411:
407:
402:
398:
391:
388:
385:
377:
373:
369:
364:
360:
347:
344:
342:
338:
334:
330:
329:Koitus~nlwiki
326:
319:
316:
315:
311:
307:
299:
296:
293:
286:
277:
270:
269:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
225:
224:
223:
206:
205:
200:
199:
189:
188:
172:
171:
164:
163:
159:
155:
149:
146:
145:
142:
125:
121:
117:
116:
108:
102:
97:
95:
92:
88:
87:
83:
76:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2835:
2775:
2755:
2728:
2672:
2646:
2611:
2607:
2585:
2563:
2520:— Preceding
2518:
2508:
2490:motorfingers
2476:Cuzkatzimhut
2450:— Preceding
2447:
2405:motorfingers
2184:motorfingers
2181:
2177:
2104:
2101:
2065:
2059:
2047:. Retrieved
2035:
2031:
2021:
2013:
1978:
1900:
1860:
1721:
1708:
1705:
1682:
1598:
1479:
1329:
1247:
1244:
1147:
1124:
959:
902:
859:
802:
603:
435:
348:
345:
323:— Preceding
320:
317:
303:
297:
294:
290:
287:Why not GA ?
202:
196:
168:
153:
113:
51:WikiProjects
34:
2854:some tweaks
2643:Eigenstates
2444:Section 6.2
2274:is correct.
1327:a scalar .
276:ClueBot III
2900:Categories
2880:Xxanthippe
2838:Xxanthippe
2813:Xxanthippe
2798:Xxanthippe
2758:Xxanthippe
2695:Xxanthippe
2654:Xxanthippe
2014:References
860:We define
75:Relativity
39:is rated
2787:contribs
2779:uantling
2740:contribs
2732:uantling
2684:contribs
2676:uantling
2534:contribs
2522:unsigned
2452:unsigned
2359:restored
2082:Chessfan
1981:Chessfan
1710:Chessfan
337:contribs
325:unsigned
306:Chessfan
198:Archives
2862:wp:CALC
2713:Viktaur
2038:: 691.
156:on the
129:Physics
120:Physics
70:Physics
41:B-class
2546:petite
2049:11 May
702:→
47:scale.
1048:→
204:Index
28:This
2884:talk
2870:talk
2866:DVdm
2864:. -
2842:talk
2817:talk
2802:talk
2783:talk
2762:talk
2736:talk
2717:talk
2699:talk
2680:talk
2658:talk
2631:talk
2618:z=0.
2598:talk
2594:DVdm
2592:. -
2586:Done
2570:talk
2550:talk
2530:talk
2494:Talk
2480:talk
2460:talk
2430:talk
2426:DVdm
2409:Talk
2367:talk
2363:DVdm
2188:Talk
2086:talk
2051:2019
2004:talk
1985:talk
1714:talk
333:talk
310:talk
148:High
2182:--
2040:doi
2902::
2886:)
2872:)
2848:.
2844:)
2819:)
2808:.
2804:)
2789:)
2785:|
2768:.
2764:)
2742:)
2738:|
2719:)
2701:)
2686:)
2682:|
2664:.
2660:)
2633:)
2600:)
2572:)
2552:)
2536:)
2532:•
2492::
2482:)
2474:.
2462:)
2432:)
2424:-
2407::
2369:)
2318:−
2304:⋅
2301:γ
2231:−
2220:⋅
2217:γ
2186::
2137:−
2126:⋅
2123:γ
2088:)
2036:71
2034:.
2030:.
2006:)
1987:)
1979:--
1976:.
1958:−
1887:τ
1884:γ
1806:τ
1730:τ
1716:)
1691:γ
1656:−
1628:−
1616:γ
1562:−
1547:γ
1451:γ
1429:⋅
1395:∧
1370:γ
1218:⋅
1184:∧
1095:⋅
1069:∧
1026:γ
1023:−
987:∧
888:γ
868:γ
846:γ
830:⋅
769:∧
733:⋅
680:∧
654:⋅
574:⋅
548:⋅
530:−
339:)
335:•
312:)
304:--
207:)
73::
2890:.
2882:(
2868:(
2840:(
2823:.
2815:(
2800:(
2781:(
2776:Q
2760:(
2734:(
2729:Q
2715:(
2705:.
2697:(
2678:(
2673:Q
2656:(
2629:(
2612:t
2610:=
2608:t
2596:(
2568:(
2548:(
2528:(
2478:(
2458:(
2428:(
2390:x
2387:v
2365:(
2341:.
2337:)
2331:c
2327:x
2324:v
2315:t
2312:c
2308:(
2298:=
2294:′
2291:t
2287:c
2257:)
2249:2
2245:c
2240:x
2237:v
2228:t
2224:(
2214:=
2210:′
2207:t
2163:)
2155:2
2151:c
2146:x
2143:v
2134:t
2130:(
2120:=
2116:′
2113:t
2084:(
2053:.
2042::
2002:(
1983:(
1964:)
1961:v
1955:(
1935:)
1930:1
1926:e
1922:,
1917:0
1913:e
1909:(
1881:=
1878:t
1872:/
1868:9
1844:0
1840:e
1836:t
1814:0
1810:f
1786:)
1781:1
1777:e
1773:,
1768:0
1764:e
1760:(
1738:0
1734:f
1712:(
1667:2
1663:/
1659:1
1652:)
1646:2
1641:|
1636:v
1632:|
1625:1
1622:(
1619:=
1610:/
1606:8
1585:)
1580:2
1575:|
1570:v
1566:|
1559:1
1556:(
1551:2
1543:=
1538:0
1534:f
1528:0
1524:e
1518:0
1514:e
1508:0
1504:f
1500:=
1497:1
1491:/
1487:7
1466:)
1463:v
1460:+
1457:1
1454:(
1448:=
1445:]
1442:)
1437:0
1433:e
1424:0
1420:f
1416:(
1412:/
1408:)
1403:0
1399:e
1390:0
1386:f
1382:(
1379:+
1376:1
1373:[
1367:=
1362:0
1358:e
1352:0
1348:f
1341:/
1337:6
1313:0
1309:e
1303:1
1299:e
1294:/
1290:v
1287:=
1283:|
1279:v
1275:|
1268:/
1264:s
1261:i
1258:b
1255:5
1231:)
1226:0
1222:e
1213:0
1209:f
1205:(
1201:/
1197:)
1192:0
1188:e
1179:0
1175:f
1171:(
1168:=
1165:v
1159:/
1155:5
1133:v
1111:0
1108:=
1103:0
1099:e
1092:)
1087:0
1083:e
1077:0
1073:e
1064:0
1060:f
1056:(
1034:0
1030:e
1018:0
1014:f
1010:=
1005:0
1001:e
995:0
991:e
982:0
978:f
971:/
967:4
943:0
939:e
916:0
912:f
843:=
838:0
834:e
825:0
821:f
814:/
810:3
787:0
783:e
777:0
773:e
764:0
760:f
756:+
751:0
747:e
741:0
737:e
728:0
724:f
720:=
715:0
711:f
688:0
684:e
675:0
671:f
667:+
662:0
658:e
649:0
645:f
641:=
636:0
632:e
626:0
622:f
615:/
611:2
590:0
587:=
582:1
578:f
569:0
565:f
561:=
556:1
552:e
543:0
539:e
533:1
527:=
522:2
517:1
513:f
509:=
504:2
499:1
495:e
489:1
486:=
481:2
476:0
472:f
468:=
463:2
458:0
454:e
447:/
443:1
421:)
416:1
412:f
408:,
403:0
399:f
395:(
392:d
389:n
386:a
383:)
378:1
374:e
370:,
365:0
361:e
357:(
331:(
308:(
278:.
263:8
258:7
253:6
248:5
243:4
238:3
233:2
228:1
201:(
173:.
160:.
53::
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