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invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).
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Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from
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moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this
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used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists
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for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy
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The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in
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The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and
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Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their
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The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass")
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which has a different sign for the space and time dimensions. This length is preserved under any
Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic
1777:{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}
341:), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.
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of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.
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1303:{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|\mathbf {p} _{1}+\mathbf {p} _{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\mathbf {p} _{1}\cdot \mathbf {p} _{2}\right).\end{aligned}}}
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If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by
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exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass (a.k.a.
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respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is
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If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.
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1041:{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}
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In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the
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under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by
337:. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized
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of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total
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In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle
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squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.
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energy which the system may be observed to have, when seen by various observers from various inertial frames.
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821:{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}
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The
Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2
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590:{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}
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In a two-particle collision (or a two-particle decay) the square of the invariant mass (in
2021:{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}
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The invariant mass of a system made of two massless particles whose momenta form an angle
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is the invariant mass of the system of particles, equal to the mass of the decay particle.
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the center of momentum frame (again, called the "rest frame" if the system is bound).
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that leads to
Einstein's famous conclusion about equivalence of energy and mass. See
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689:). This equation says that the invariant mass is the pseudo-Euclidean length of the
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in the rest frame of the particle, and can be calculated by the particle's
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Note that for reasons above, such a rest frame does not exist for single
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Quarks & Leptons: An
Introductory Course in Modern Particle Physics
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of the particles (includes both magnitude and direction of the momenta)
2336:(Report). Sandia National Laboratories. SAND2006-6063. Archived from
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30:"Proper mass" redirects here. For the liturgical mass proper, see
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672:{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}
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of the system moves in a straight line with a steady subluminal
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Special relativity § Relativistic dynamics and invariance
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or many photons moving in exactly the same direction) have
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of the reaction is defined as follows (in natural units):
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to the sum of the rest masses of the particles within it.
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Physics for
Scientists and Engineers, Volume 2, page 1073
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They will often also interact through one or more of the
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Motion-independent mass, equals total mass when at rest
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Development of the
Doppler Electron Velocimeter—Theory
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system even though it does not exist for each photon.
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of the system is simply the invariant mass times the
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706:relativistic version of the Pythagorean theorem
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2160:The concept of rest energy follows from the
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71:Learn how and when to remove these messages
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2354:Modell, Michael; Reid, Robert C. (1974).
265:, is the portion of the total mass of an
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233:Learn how and when to remove this message
168:Learn how and when to remove this message
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1834:. Additionally the transverse momentum,
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2316:"13.6 Relativistic Energy or E = m c^2"
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681:This invariant mass is the same in all
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713:for an elementary particle. The Dirac
331:invariant mass and are referred to as
2202:Landau, L.D.; Lifshitz, E.M. (1975).
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2358:Thermodynamics and Its Applications
1887:) then the invariant mass becomes:
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2153:. In general, only differences in
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2122:{\displaystyle E_{0}=m_{0}c^{2},}
405:Possible 4-momenta of particles.
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2293:. Georgia State University
2180:Mass in special relativity
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2206:. Butterworth Heinemann.
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1717:
1715:
1710:
1706:
1702:
1699:
1696:
1691:
1687:
1683:
1678:
1674:
1670:
1667:
1664:
1661:
1656:
1652:
1646:
1641:
1637:
1633:
1628:
1624:
1618:
1614:
1610:
1605:
1601:
1597:
1594:
1591:
1589:
1587:
1582:
1578:
1574:
1571:
1568:
1565:
1560:
1556:
1552:
1549:
1546:
1543:
1538:
1534:
1530:
1527:
1524:
1519:
1515:
1511:
1508:
1505:
1500:
1496:
1492:
1489:
1486:
1483:
1480:
1475:
1471:
1467:
1464:
1461:
1458:
1456:
1454:
1449:
1444:
1438:
1426:
1421:
1408:
1403:
1398:
1394:
1388:
1384:
1380:
1375:
1371:
1367:
1364:
1361:
1359:
1355:
1351:
1347:
1346:
1326:
1314:
1311:
1295:
1291:
1285:
1280:
1275:
1270:
1265:
1260:
1255:
1251:
1245:
1241:
1236:
1232:
1229:
1224:
1219:
1215:
1211:
1206:
1201:
1197:
1193:
1190:
1188:
1186:
1181:
1176:
1170:
1165:
1160:
1155:
1150:
1144:
1139:
1134:
1130:
1124:
1120:
1116:
1111:
1107:
1103:
1100:
1097:
1095:
1091:
1087:
1083:
1082:
1066:
1063:
1055:missing energy
1037:
1032:
1027:
1016:
1011:
1008:
998:
993:
989:
984:
979:
974:
964:
960:
957:
948:
944:
940:
935:
930:
926:
909:
908:
891:
887:
877:
865:
862:
852:
840:
817:
812:
807:
803:
799:
795:
791:
786:
781:
776:
772:
769:
765:
760:
755:
750:
744:
740:
736:
732:
711:Dirac equation
668:
663:
658:
654:
650:
645:
640:
636:
632:
627:
622:
618:
584:
579:
575:
571:
566:
561:
556:
551:
548:
543:
538:
533:
529:
523:
518:
514:
500:frame, by the
470:
464:invariant mass
455:
452:
423:
420:
365:center of mass
315:Systems whose
310:speed of light
255:intrinsic mass
247:invariant mass
241:
240:
223:
222:
192:
190:
183:
176:
175:
90:
88:
81:
76:
50:
49:
47:
40:
26:
9:
6:
4:
3:
2:
2421:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2391:
2389:
2375:
2373:0-13-914861-2
2369:
2365:
2364:Prentice-Hall
2360:
2359:
2350:
2339:
2332:
2325:
2317:
2311:
2309:
2292:
2288:
2281:
2279:
2271:
2265:
2261:
2249:
2247:0-471-88741-2
2243:
2239:
2234:
2233:
2227:
2223:
2219:
2215:
2213:0-7506-2768-9
2209:
2205:
2200:
2199:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2171:
2169:
2168:
2163:
2158:
2156:
2152:
2136:
2116:
2111:
2107:
2101:
2097:
2093:
2088:
2084:
2075:
2057:
2053:
2043:
2041:
2038:(also called
2037:
2028:
2015:
2004:
2000:
1996:
1991:
1987:
1980:
1977:
1974:
1966:
1962:
1958:
1953:
1949:
1942:
1939:
1931:
1928:
1924:
1918:
1915:
1911:
1907:
1904:
1899:
1895:
1874:
1871:
1868:
1846:
1842:
1821:
1814:
1798:
1784:
1767:
1761:
1758:
1755:
1752:
1749:
1741:
1737:
1731:
1727:
1723:
1720:
1718:
1708:
1700:
1697:
1694:
1689:
1685:
1681:
1676:
1672:
1665:
1662:
1659:
1654:
1650:
1644:
1639:
1635:
1631:
1626:
1616:
1612:
1608:
1603:
1599:
1592:
1590:
1580:
1569:
1566:
1563:
1558:
1554:
1550:
1547:
1544:
1541:
1536:
1532:
1528:
1525:
1522:
1517:
1513:
1506:
1498:
1494:
1490:
1487:
1484:
1481:
1478:
1473:
1469:
1459:
1457:
1447:
1436:
1424:
1419:
1401:
1396:
1386:
1382:
1378:
1373:
1369:
1362:
1360:
1353:
1349:
1324:
1310:
1293:
1289:
1283:
1273:
1268:
1258:
1253:
1249:
1243:
1239:
1234:
1230:
1227:
1222:
1217:
1213:
1209:
1204:
1199:
1195:
1191:
1189:
1179:
1168:
1158:
1153:
1137:
1132:
1122:
1118:
1114:
1109:
1105:
1098:
1096:
1089:
1085:
1072:
1071:natural units
1062:
1060:
1056:
1051:
1048:
1035:
1030:
1009:
1006:
991:
982:
977:
972:
962:
958:
955:
946:
942:
938:
933:
928:
924:
906:
885:
878:
863:
860:
853:
838:
831:
830:
829:
815:
810:
801:
793:
784:
779:
774:
770:
767:
763:
758:
753:
748:
742:
738:
734:
730:
718:
716:
712:
707:
701:
697:
692:
688:
684:
679:
666:
661:
643:
638:
634:
630:
625:
620:
616:
605:
600:
599:natural units
582:
564:
559:
554:
549:
546:
541:
536:
531:
527:
521:
516:
512:
503:
499:
494:
489:
481:
477:
469:
465:
461:
451:
449:
445:
440:
436:
434:
428:
415:
409:
403:
399:
396:
393:, or rays of
392:
387:
385:
380:
374:
370:
366:
362:
357:
354:
350:
349:
342:
340:
336:
335:
330:
326:
322:
318:
317:four-momentum
313:
311:
307:
303:
298:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
237:
234:
219:
216:February 2016
209:
205:
200:
196:
193:This article
191:
187:
182:
181:
172:
169:
161:
150:
147:
143:
140:
136:
133:
129:
126:
122:
119: –
118:
114:
113:Find sources:
107:
103:
97:
96:
91:This article
89:
85:
80:
79:
74:
72:
65:
64:
59:
58:
53:
48:
39:
38:
33:
19:
2357:
2349:
2338:the original
2324:
2295:. Retrieved
2291:HyperPhysics
2290:
2264:
2231:
2226:Martin, Alan
2203:
2165:
2159:
2044:
2039:
2035:
2034:
1790:
1316:
1068:
1052:
1049:
910:
719:
699:
695:
680:
603:
497:
492:
467:
463:
457:
441:
437:
432:
429:
425:
413:
407:
388:
383:
378:
363:system, the
360:
358:
346:
343:
332:
314:
305:
299:
262:
258:
254:
250:
246:
244:
229:
213:
204:You can help
194:
164:
155:
145:
138:
131:
124:
112:
100:Please help
95:verification
92:
68:
61:
55:
54:Please help
51:
2285:Nave, C.R.
2036:Rest energy
2031:Rest energy
691:four-vector
321:null vector
306:rest energy
300:Because of
285:related by
259:proper mass
2388:Categories
2196:References
685:(see also
417:is massive
158:March 2011
128:newspapers
57:improve it
2297:28 August
2256:Citations
2001:ϕ
1997:−
1988:ϕ
1981:
1975:−
1963:η
1959:−
1950:η
1943:
1872:≫
1822:η
1799:ϕ
1762:θ
1759:
1753:−
1701:θ
1698:
1666:−
1663:θ
1660:
1632:−
1570:θ
1567:
1548:θ
1545:
1402:−
1325:θ
1274:⋅
1259:−
1138:−
1061:is used.
1010:∑
1007:−
992:∑
983:−
959:∑
956:−
943:∑
886:∑
861:∑
794:∑
785:−
768:∑
644:−
565:−
433:not equal
414:the other
251:rest mass
208:talk page
63:talk page
18:Rest mass
2228:(1984).
2174:See also
2074:particle
1443:‖
1407:‖
1175:‖
1143:‖
1026:‖
988:‖
905:momentum
806:‖
790:‖
657:‖
649:‖
578:‖
570:‖
488:momentum
486:and its
448:negative
369:velocity
334:massless
279:momentum
2272:. 1997.
2149:is the
391:photons
384:minimum
361:massive
339:tachyon
289:. If a
142:scholar
2370:
2244:
2210:
2155:energy
2129:where
1057:) the
828:where
601:where
597:or in
490:
482:
480:energy
462:, the
351:. See
325:photon
304:, the
275:energy
271:system
267:object
206:. The
144:
137:
130:
123:
115:
2341:(PDF)
2334:(PDF)
2072:of a
1073:) is
395:light
319:is a
149:JSTOR
135:books
2399:Mass
2368:ISBN
2299:2023
2242:ISBN
2208:ISBN
1940:cosh
1811:and
476:mass
329:zero
277:and
263:mass
245:The
121:news
1978:cos
1756:cos
1695:cos
1651:sin
1564:cos
1542:sin
1020:out
967:out
606:= 1
498:any
458:In
408:One
269:or
104:by
2390::
2366:.
2307:^
2289:.
2277:^
2240:.
2236:.
2224:;
2170:.
1002:in
951:in
698:,
608:,
504::
450:.
257:,
253:,
249:,
66:.
2376:.
2318:.
2301:.
2250:.
2216:.
2137:c
2117:,
2112:2
2108:c
2102:0
2098:m
2094:=
2089:0
2085:E
2058:0
2054:E
2016:.
2013:)
2010:)
2005:2
1992:1
1984:(
1972:)
1967:2
1954:1
1946:(
1937:(
1932:2
1929:T
1925:p
1919:1
1916:T
1912:p
1908:2
1905:=
1900:2
1896:M
1875:m
1869:E
1847:T
1843:p
1768:.
1765:)
1750:1
1747:(
1742:2
1738:p
1732:1
1728:p
1724:2
1721:=
1709:2
1705:)
1690:2
1686:p
1682:+
1677:1
1673:p
1669:(
1655:2
1645:2
1640:2
1636:p
1627:2
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1600:p
1596:(
1593:=
1581:2
1577:]
1573:)
1559:2
1555:p
1551:,
1537:2
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1529:,
1526:0
1523:,
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1510:(
1507:+
1504:)
1499:1
1495:p
1491:,
1488:0
1485:,
1482:0
1479:,
1474:1
1470:p
1466:(
1463:[
1460:=
1448:2
1437:2
1431:p
1425:+
1420:1
1414:p
1397:2
1393:)
1387:2
1383:E
1379:+
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1370:E
1366:(
1363:=
1354:2
1350:M
1294:.
1290:)
1284:2
1279:p
1269:1
1264:p
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1235:(
1231:2
1228:+
1223:2
1218:2
1214:m
1210:+
1205:2
1200:1
1196:m
1192:=
1180:2
1169:2
1164:p
1159:+
1154:1
1149:p
1133:2
1129:)
1123:2
1119:E
1115:+
1110:1
1106:E
1102:(
1099:=
1090:2
1086:M
1036:.
1031:2
1015:p
997:p
978:2
973:)
963:E
947:E
939:(
934:=
929:2
925:W
914:W
890:p
864:E
839:W
816:,
811:2
802:c
798:p
780:2
775:)
771:E
764:(
759:=
754:2
749:)
743:2
739:c
735:W
731:(
702:)
700:p
696:E
694:(
667:.
662:2
653:p
639:2
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631:=
626:2
621:0
617:m
604:c
583:2
574:p
560:2
555:)
550:c
547:E
542:(
537:=
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528:c
522:2
517:0
513:m
493:p
484:E
471:0
468:m
379:c
348:c
236:)
230:(
218:)
214:(
171:)
165:(
160:)
156:(
146:·
139:·
132:·
125:·
98:.
73:)
69:(
34:.
20:)
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