3102:. Given a metric space, or a set and scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to another metric space such that the distance between the elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional space, two geometric figures are
360:
192:
41:
3184:
In other cases, such as "taking a random integer" or "taking a random real number", there are no probability distributions at all symmetric with respect to relabellings or to exchange of equally long subintervals. Other reasonable symmetries do not single out one particular distribution, or in other
929:. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
1151:
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the
752:
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (as matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric.
2282:
This implies the Pauli exclusion principle for fermions. In fact, the Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a
2900:
3195:
A possible symmetry for randomness with positive outcomes is that the former applies for the logarithm, i.e., the outcome and its reciprocal have the same distribution. However this symmetry does not single out any particular distribution uniquely.
2625:
1713:
886:
1843:
2731:
1396:
2979:
1549:
2430:
2778:
2786:
2345:
2315:
3162:
Symmetries may be found by solving a related set of ordinary differential equations. Solving these equations is often much simpler than solving the original differential equations.
1250:
3046:
522:
459:
292:
2477:, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity
747:
3159:, knowledge of an appropriate set of Lie symmetries allows one to explicitly calculate a set of first integrals, yielding a complete solution without integration.
2524:
3199:
For a "random point" in a plane or in space, one can choose an origin, and consider a probability distribution with circular or spherical symmetry, respectively.
2905:
The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
1587:
3148:
is a continuous symmetry of the system of differential equations. Knowledge of a Line symmetry can be used to simplify an ordinary differential equation through
805:
3125:
Isometries have been used to unify the working definition of symmetry in geometry and for functions, probability distributions, matrices, strings, graphs, etc.
1736:
2636:
3428:
1256:
3067:
We call a relation symmetric if every time the relation stands from A to B, it stands too from B to A. Note that symmetry is not the exact opposite of
133:
In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above.
45:
2279:
In quantum mechanics, bosons have representatives that are symmetric under permutation operators, and fermions have antisymmetric representatives.
3513:
3488:
3438:
2911:
3137:
is a transformation that leaves the differential equation invariant. Knowledge of such symmetries may help solve the differential equation.
3177:
In the case of a real interval of possible outcomes, symmetry with respect to interchanging sub-intervals of equal length corresponds to a
1428:
2078:
from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group
1961:
after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are
3469:
3332:
3178:
2353:
3189:
3145:
3171:
3156:
1157:
1153:
1160:
states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every
2895:{\displaystyle \langle \psi |x,x\rangle +\langle \psi |x,y\rangle +\langle \psi |y,x\rangle +\langle \psi |y,y\rangle \,}
2739:
80:
3170:
In the case of a finite number of possible outcomes, symmetry with respect to permutations (relabelings) implies a
34:
3349:
160:
30:
3213:
3115:
310:
2003:
the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a
2320:
2290:
2284:
76:
72:
1199:
3068:
1861:
1076:
variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally,
673:
666:
336:
181:
156:
3192:
that may leave the probability distribution unchanged, that is reflection in a point, for example zero.
3134:
3103:
2137:
2107:
2067:, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any
2063:, considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a
2052:
1164:
907:
896:
95:
2991:
98:
of the object onto itself which preserves the structure. This can occur in many ways; for example, if
3111:
2156:
1893:
1033:
1008:
985:
528:
470:
115:
2736:
This is zero, because the two particles have zero probability to both be in the superposition state
3118:. Up to a relation by a rigid motion, they are equal if related by a
2174:
2091:
2004:
1992:
1881:
1877:
1873:
1857:
969:
926:
914:
560:
536:
532:
344:
306:
164:
152:
148:
142:
3372:
3149:
3062:
2099:
2064:
1927:
414:
250:
599:
is finite and that the function is integrable (e.g., has no vertical asymptotes between −
3532:
3509:
3484:
3465:
3434:
3328:
3320:
3208:
2222:
2170:
2075:
2000:
1885:
716:
662:
107:
2620:{\displaystyle A(x,y)=\langle \psi |x,y\rangle =\langle \psi |(|x\rangle \otimes |y\rangle )}
1171:
can alternatively be given as a polynomial expression in the coefficients of the polynomial.
895:
is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a
3499:
3403:
3364:
2238:
2103:
1980:
1897:
1708:{\displaystyle T(v_{1},v_{2},\dots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots ,v_{\sigma r})}
1560:
1168:
922:
648:
52:
881:{\displaystyle {\begin{bmatrix}1&7&3\\7&4&-5\\3&-5&6\end{bmatrix}}}
3407:
3119:
2234:
2230:
2178:
2041:
1962:
981:
943:
892:
626:
is finite and the function is integrable (e.g., has no vertical asymptotes between −
1838:{\displaystyle T_{i_{1}i_{2}\dots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.}
2726:{\displaystyle \langle \psi |((|x\rangle +|y\rangle )\otimes (|x\rangle +|y\rangle ))\,}
527:
Geometrically, the graph of an odd function has rotational symmetry with respect to the
126:
of the set to itself which preserves the distance between each pair of points (i.e., an
3424:
3257:
2206:
2126:
2012:
918:
691:
658:
568:
540:
319:
17:
3526:
3141:
3099:
2119:
2068:
1921:
757:
699:
3457:
2262:
2130:
1974:
1853:
1391:{\displaystyle 4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}}
973:
119:
3185:
words, there is not a unique probability distribution providing maximum symmetry.
3503:
3106:
if they are related by an isometry: related by either a
3391:
3286:
2226:
2186:
2029:
1988:
1950:
1909:
1901:
1578:
1566:
1085:
911:
903:
385:
221:
2201:(because they cannot preserve the property of a number having a square root in
3233:
3107:
2025:
1965:. Thus, Galois theory studies the symmetries inherent in algebraic equations.
1926:
Given a polynomial, it may be that some of the roots are connected by various
1043:
961:
2167:
1000:
989:
703:
359:
191:
123:
103:
49:
2197:
have nontrivial field automorphisms, which however do not extend to all of
75:: the property that a mathematical object remains unchanged under a set of
3095:
3085:
2250:
2151:. When the vector space is finite-dimensional, the automorphism group of
1996:
584:
298:
127:
91:
68:
63:
2630:
is necessarily antisymmetric. To prove it, consider the matrix element:
3376:
2974:{\displaystyle \langle \psi |x,y\rangle +\langle \psi |y,x\rangle =0\,}
2056:
1905:
2493:
1726:
order symmetric tensor represented in coordinates as a quantity with
1574:
1544:{\displaystyle X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}\,}
706:(i.e., it is invariant under matrix transposition). Formally, matrix
3368:
756:
The entries of a symmetric matrix are symmetric with respect to the
40:
71:, but also in other branches of mathematics. Symmetry is a type of
358:
190:
39:
2193:) there are no nontrivial field automorphisms. Some subfields of
654:
The
Maclaurin series of an odd function includes only odd powers.
552:
1884:
of all symmetric tensors can be naturally identified with the
297:
Geometrically speaking, the graph face of an even function is
3505:
Finding
Moonshine: A Mathematician's Journey Through Symmetry
2425:{\displaystyle |\psi \rangle =\sum _{x,y}A(x,y)|x,y\rangle }
672:
The
Fourier series of a periodic odd function includes only
147:
The types of symmetry considered in basic geometry include
3327:(Felix Pahl translation ed.). Springer. p. 376.
2122:
center it can be embedded into its own automorphism group.
1953:(or rearrangements) of the roots having the property that
3464:. Princeton Science Library. Princeton University Press.
1949:. The central idea of Galois theory is to consider those
1930:. For example, it may be that for two of the roots, say
2213:, there is a unique nontrivial automorphism that sends
102:
is a set with no additional structure, a symmetry is a
2743:
2324:
2294:
2233:). Field automorphisms are important to the theory of
2147:. An automorphism is an invertible linear operator on
1011:(i.e., the number of elements) of the symmetric group
814:
3325:
Mathematical foundations of computational engineering
2994:
2914:
2789:
2742:
2639:
2527:
2356:
2323:
2293:
1739:
1718:
for every permutation σ of the symbols {1,2,...,
1590:
1431:
1259:
1202:
808:
719:
473:
417:
253:
163:, which are described more fully in the main article
3430:
799:
For example, the following 3ร3 matrix is symmetric:
992:from the set of symbols to itself. Since there are
925:with complex-valued entries, which is equal to its
3040:
2973:
2894:
2773:{\displaystyle \scriptstyle |x\rangle +|y\rangle }
2772:
2725:
2619:
2424:
2339:
2309:
2275:Symmetry in quantum mechanics: bosons and fermions
1837:
1707:
1543:
1390:
1244:
1156:are the most fundamental symmetric polynomials. A
880:
741:
516:
453:
286:
638:is infinite, but only if the integral converges.
543:about the origin. Examples of odd functions are
2492:is not a matrix but an antisymmetric rank-two
2036:is an automorphism. The automorphism group of
899:must be zero, since each is its own negative.
651:of an even function includes only even powers.
1957:algebraic equation satisfied by the roots is
610:The integral of an even function from −
8:
2961:
2941:
2935:
2915:
2888:
2868:
2862:
2842:
2836:
2816:
2810:
2790:
2766:
2752:
2713:
2699:
2679:
2665:
2640:
2611:
2597:
2575:
2569:
2549:
2419:
2365:
2333:
2303:
3280:
3278:
2435:and antisymmetry under exchange means that
988:of such permutations, which are treated as
106:map from the set to itself, giving rise to
2229:) many "wild" automorphisms (assuming the
388:-valued function of a real variable, then
317:-axis. Examples of even functions include
224:-valued function of a real variable, then
3037:
2993:
2970:
2947:
2921:
2913:
2891:
2874:
2848:
2822:
2796:
2788:
2758:
2744:
2741:
2722:
2705:
2691:
2671:
2657:
2646:
2638:
2603:
2589:
2581:
2555:
2526:
2405:
2375:
2357:
2355:
2325:
2322:
2295:
2292:
1821:
1805:
1792:
1787:
1772:
1759:
1749:
1744:
1738:
1693:
1671:
1655:
1633:
1614:
1601:
1589:
1540:
1534:
1524:
1508:
1498:
1482:
1472:
1456:
1446:
1436:
1430:
1382:
1372:
1359:
1343:
1338:
1328:
1315:
1305:
1300:
1287:
1282:
1272:
1267:
1258:
1230:
1225:
1212:
1207:
1201:
1193:, one has symmetric polynomials such as:
809:
807:
730:
718:
510:
472:
416:
252:
114:is a set of points in the plane with its
396:if the following equation holds for all
232:if the following equation holds for all
3419:
3417:
3224:
2499:Conversely, if the diagonal quantities
2340:{\displaystyle \scriptstyle |y\rangle }
2310:{\displaystyle \scriptstyle |x\rangle }
1848:The space of symmetric tensors of rank
3350:"Automorphisms of the Complex Numbers"
2082:there is a natural group homomorphism
3495:(Concise introduction for lay reader)
1422:, one has as a symmetric polynomial:
1245:{\displaystyle X_{1}^{3}+X_{2}^{3}-7}
7:
3129:Symmetries of differential equations
2241:. In the case of a Galois extension
1900:. Symmetric tensors occur widely in
1003:) possible permutations of a set of
2518:, then the wavefunction component:
1995:to itself. It is, in some sense, a
1892:. A related concept is that of the
760:. So if the entries are written as
3114:of a rigid motion and a
2287:in which one particle is in state
1969:Automorphisms of algebraic objects
208:is an example of an even function.
58:. Lie groups have many symmetries.
25:
2270:Symmetry in representation theory
917:. The corresponding object for a
618:is twice the integral from 0 to +
376:is an example of an odd function.
3146:system of differential equations
3041:{\displaystyle A(x,y)=-A(y,x)\,}
1154:elementary symmetric polynomials
3179:continuous uniform distribution
3157:ordinary differential equations
2177:to itself. In the cases of the
2011:. It is, loosely speaking, the
1088:ฯ of the subscripts 1, 2, ...,
587:of an odd function from −
517:{\displaystyle f(x)+f(-x)=0\,.}
3209:Use of symmetry in integration
3034:
3022:
3010:
2998:
2948:
2922:
2875:
2849:
2823:
2797:
2759:
2745:
2719:
2716:
2706:
2692:
2688:
2682:
2672:
2658:
2654:
2651:
2647:
2614:
2604:
2590:
2586:
2582:
2556:
2543:
2531:
2406:
2402:
2390:
2358:
2326:
2296:
1702:
1648:
1639:
1594:
1379:
1352:
906:symmetric matrix represents a
501:
492:
483:
477:
448:
439:
430:
424:
281:
272:
263:
257:
1:
3433:. New York: Springer Verlag.
3396:Symmetry: Culture and Science
3258:"Maths in a minute: Symmetry"
3172:discrete uniform distribution
2071:, but not of a ring or field.
1007:symbols, it follows that the
634:). This also holds true when
2225:, but there are infinitely (
1999:of the object, and a way of
1860:to the dual of the space of
933:Symmetry in abstract algebra
665:even function includes only
3483:. Oxford University Press.
3319:PJ Pahl, R Damrath (2001).
972:whose elements are all the
454:{\displaystyle -f(x)=f(-x)}
3549:
3392:"A definition of symmetry"
3390:Petitjean, Michel (2007).
3348:Yale, Paul B. (May 1966).
3083:
3060:
2166:A field automorphism is a
2074:A group automorphism is a
1972:
1919:
1577:that is invariant under a
1558:
1031:
941:
681:Symmetry in linear algebra
287:{\displaystyle f(x)=f(-x)}
179:
140:
86:Given a structured object
35:Bilateral (disambiguation)
28:
3190:isometry in one dimension
3075:Symmetry in metric spaces
2032:of the elements of a set
1581:of its vector arguments:
921:inner product space is a
161:glide reflection symmetry
31:Symmetry (disambiguation)
3481:Symmetry and the Monster
3214:Invariance (mathematics)
3098:-preserving map between
2261:pointwise is called the
2253:of all automorphisms of
1852:on a finite-dimensional
742:{\displaystyle A=A^{T}.}
535:remains unchanged after
309:remains unchanged after
305:-axis, meaning that its
3309:Jacobson (2009), p. 31.
3166:Symmetry in probability
2780:. But this is equal to
2317:and the other in state
2129:, an endomorphism of a
1862:homogeneous polynomials
1401:and in three variables
595:is zero, provided that
118:structure or any other
3321:"ยง7.5.5 Automorphisms"
3052:Symmetry in set theory
3042:
2975:
2896:
2774:
2727:
2621:
2426:
2341:
2311:
2205:). In the case of the
1839:
1709:
1545:
1392:
1246:
882:
743:
518:
455:
377:
288:
209:
182:Even and odd functions
176:Even and odd functions
157:translational symmetry
59:
18:Symmetry (mathematics)
3291:mathworld.wolfram.com
3238:mathworld.wolfram.com
3188:There is one type of
3135:differential equation
3080:Isometries of a space
3043:
2976:
2897:
2775:
2728:
2622:
2427:
2342:
2312:
2053:elementary arithmetic
1840:
1722:}. Alternatively, an
1710:
1546:
1393:
1247:
1165:polynomial expression
1028:Symmetric polynomials
908:self-adjoint operator
902:In linear algebra, a
897:skew-symmetric matrix
883:
744:
702:that is equal to its
519:
456:
362:
289:
194:
149:reflectional symmetry
43:
3479:Ronan, Mark (2006).
3357:Mathematics Magazine
3120:direct isometry
2992:
2912:
2787:
2740:
2637:
2525:
2462:. This implies that
2354:
2321:
2291:
2157:general linear group
1894:antisymmetric tensor
1858:naturally isomorphic
1737:
1588:
1429:
1257:
1200:
1082:symmetric polynomial
1040:symmetric polynomial
1034:Symmetric polynomial
806:
717:
686:Symmetry in matrices
471:
415:
301:with respect to the
251:
171:Symmetry in calculus
137:Symmetry in geometry
29:For other uses, see
3285:Weisstein, Eric W.
3232:Weisstein, Eric W.
2223:complex conjugation
2155:is the same as the
2100:inner automorphisms
2040:is also called the
1993:mathematical object
1928:algebraic equations
1882:graded vector space
1878:characteristic zero
1348:
1310:
1292:
1277:
1235:
1217:
990:bijective functions
980:symbols, and whose
927:conjugate transpose
915:inner product space
531:, meaning that its
165:Symmetry (geometry)
143:Symmetry (geometry)
67:occurs not only in
48:of the exceptional
3508:. Harper Collins.
3150:reduction of order
3063:Symmetric relation
3057:Symmetric relation
3038:
2971:
2892:
2770:
2769:
2723:
2617:
2422:
2386:
2337:
2336:
2307:
2306:
2009:automorphism group
1835:
1730:indices satisfies
1705:
1541:
1388:
1334:
1296:
1278:
1263:
1242:
1221:
1203:
1167:in the roots of a
878:
872:
788:, for all indices
739:
514:
451:
378:
284:
210:
122:, a symmetry is a
108:permutation groups
60:
3515:978-0-00-738087-9
3500:du Sautoy, Marcus
3490:978-0-19-280723-6
3440:978-0-387-95000-6
3110:, or a
3108:rigid motion
2371:
2265:of the extension.
2239:Galois extensions
2171:ring homomorphism
2094:is the group Inn(
2076:group isomorphism
1886:symmetric algebra
1555:Symmetric tensors
1179:In two variables
404:in the domain of
240:in the domain of
153:rotation symmetry
16:(Redirected from
3540:
3519:
3494:
3475:
3445:
3444:
3421:
3412:
3411:
3387:
3381:
3380:
3354:
3345:
3339:
3338:
3316:
3310:
3307:
3301:
3300:
3298:
3297:
3282:
3273:
3272:
3270:
3269:
3254:
3248:
3247:
3245:
3244:
3229:
3133:A symmetry of a
3047:
3045:
3044:
3039:
2980:
2978:
2977:
2972:
2951:
2925:
2901:
2899:
2898:
2893:
2878:
2852:
2826:
2800:
2779:
2777:
2776:
2771:
2762:
2748:
2732:
2730:
2729:
2724:
2709:
2695:
2675:
2661:
2650:
2626:
2624:
2623:
2618:
2607:
2593:
2585:
2559:
2513:
2491:
2476:
2461:
2431:
2429:
2428:
2423:
2409:
2385:
2361:
2346:
2344:
2343:
2338:
2329:
2316:
2314:
2313:
2308:
2299:
2237:, in particular
2235:field extensions
2179:rational numbers
1981:abstract algebra
1963:rational numbers
1948:
1898:alternating form
1844:
1842:
1841:
1836:
1831:
1830:
1829:
1828:
1813:
1812:
1800:
1799:
1779:
1778:
1777:
1776:
1764:
1763:
1754:
1753:
1714:
1712:
1711:
1706:
1701:
1700:
1679:
1678:
1663:
1662:
1638:
1637:
1619:
1618:
1606:
1605:
1571:symmetric tensor
1561:Symmetric tensor
1550:
1548:
1547:
1542:
1539:
1538:
1529:
1528:
1513:
1512:
1503:
1502:
1487:
1486:
1477:
1476:
1461:
1460:
1451:
1450:
1441:
1440:
1397:
1395:
1394:
1389:
1387:
1386:
1377:
1376:
1364:
1363:
1347:
1342:
1333:
1332:
1320:
1319:
1309:
1304:
1291:
1286:
1276:
1271:
1251:
1249:
1248:
1243:
1234:
1229:
1216:
1211:
1169:monic polynomial
968:symbols) is the
938:Symmetric groups
923:Hermitian matrix
887:
885:
884:
879:
877:
876:
748:
746:
745:
740:
735:
734:
710:is symmetric if
696:symmetric matrix
649:Maclaurin series
622:, provided that
523:
521:
520:
515:
460:
458:
457:
452:
375:
326:
293:
291:
290:
285:
207:
110:. If the object
21:
3548:
3547:
3543:
3542:
3541:
3539:
3538:
3537:
3523:
3522:
3516:
3498:
3491:
3478:
3472:
3456:
3453:
3448:
3441:
3425:Olver, Peter J.
3423:
3422:
3415:
3402:(2โ3): 99โ119.
3389:
3388:
3384:
3369:10.2307/2689301
3352:
3347:
3346:
3342:
3335:
3318:
3317:
3313:
3308:
3304:
3295:
3293:
3284:
3283:
3276:
3267:
3265:
3256:
3255:
3251:
3242:
3240:
3231:
3230:
3226:
3222:
3205:
3168:
3131:
3088:
3082:
3077:
3065:
3059:
3054:
2990:
2989:
2910:
2909:
2785:
2784:
2738:
2737:
2635:
2634:
2523:
2522:
2500:
2478:
2463:
2436:
2352:
2351:
2319:
2318:
2289:
2288:
2277:
2272:
2231:axiom of choice
2207:complex numbers
2138:linear operator
2042:symmetric group
2028:, an arbitrary
2021:
2015:of the object.
1977:
1971:
1959:still satisfied
1939:
1924:
1918:
1817:
1801:
1788:
1783:
1768:
1755:
1745:
1740:
1735:
1734:
1689:
1667:
1651:
1629:
1610:
1597:
1586:
1585:
1563:
1557:
1530:
1520:
1504:
1494:
1478:
1468:
1452:
1442:
1432:
1427:
1426:
1421:
1414:
1407:
1378:
1368:
1355:
1324:
1311:
1255:
1254:
1198:
1197:
1192:
1185:
1177:
1147:
1138:
1131:
1120:
1109:
1102:
1071:
1062:
1055:
1036:
1030:
1019:
982:group operation
959:
950:symmetric group
946:
944:Symmetric group
940:
935:
893:diagonal matrix
871:
870:
865:
857:
851:
850:
842:
837:
831:
830:
825:
820:
810:
804:
803:
787:
781:
772:
726:
715:
714:
688:
683:
644:
581:
469:
468:
413:
412:
363:
357:
318:
249:
248:
195:
189:
184:
178:
173:
145:
139:
90:of any sort, a
81:transformations
56:
38:
23:
22:
15:
12:
11:
5:
3546:
3544:
3536:
3535:
3525:
3524:
3521:
3520:
3514:
3496:
3489:
3476:
3470:
3452:
3449:
3447:
3446:
3439:
3413:
3382:
3363:(3): 135โ141.
3340:
3333:
3311:
3302:
3287:"Odd Function"
3274:
3262:plus.maths.org
3249:
3223:
3221:
3218:
3217:
3216:
3211:
3204:
3201:
3167:
3164:
3130:
3127:
3084:Main article:
3081:
3078:
3076:
3073:
3061:Main article:
3058:
3055:
3053:
3050:
3049:
3048:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2983:
2982:
2969:
2966:
2963:
2960:
2957:
2954:
2950:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2924:
2920:
2917:
2903:
2902:
2890:
2887:
2884:
2881:
2877:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2851:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2825:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2799:
2795:
2792:
2768:
2765:
2761:
2757:
2754:
2751:
2747:
2734:
2733:
2721:
2718:
2715:
2712:
2708:
2704:
2701:
2698:
2694:
2690:
2687:
2684:
2681:
2678:
2674:
2670:
2667:
2664:
2660:
2656:
2653:
2649:
2645:
2642:
2628:
2627:
2616:
2613:
2610:
2606:
2602:
2599:
2596:
2592:
2588:
2584:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2558:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2516:in every basis
2433:
2432:
2421:
2418:
2415:
2412:
2408:
2404:
2401:
2398:
2395:
2392:
2389:
2384:
2381:
2378:
2374:
2370:
2367:
2364:
2360:
2335:
2332:
2328:
2305:
2302:
2298:
2276:
2273:
2271:
2268:
2267:
2266:
2164:
2127:linear algebra
2123:
2072:
2049:
2020:
2017:
2013:symmetry group
1973:Main article:
1970:
1967:
1920:Main article:
1917:
1914:
1846:
1845:
1834:
1827:
1824:
1820:
1816:
1811:
1808:
1804:
1798:
1795:
1791:
1786:
1782:
1775:
1771:
1767:
1762:
1758:
1752:
1748:
1743:
1716:
1715:
1704:
1699:
1696:
1692:
1688:
1685:
1682:
1677:
1674:
1670:
1666:
1661:
1658:
1654:
1650:
1647:
1644:
1641:
1636:
1632:
1628:
1625:
1622:
1617:
1613:
1609:
1604:
1600:
1596:
1593:
1559:Main article:
1556:
1553:
1552:
1551:
1537:
1533:
1527:
1523:
1519:
1516:
1511:
1507:
1501:
1497:
1493:
1490:
1485:
1481:
1475:
1471:
1467:
1464:
1459:
1455:
1449:
1445:
1439:
1435:
1419:
1412:
1405:
1399:
1398:
1385:
1381:
1375:
1371:
1367:
1362:
1358:
1354:
1351:
1346:
1341:
1337:
1331:
1327:
1323:
1318:
1314:
1308:
1303:
1299:
1295:
1290:
1285:
1281:
1275:
1270:
1266:
1262:
1252:
1241:
1238:
1233:
1228:
1224:
1220:
1215:
1210:
1206:
1190:
1183:
1176:
1173:
1143:
1136:
1129:
1121:) =
1114:
1107:
1100:
1067:
1060:
1053:
1032:Main article:
1029:
1026:
1015:
955:
942:Main article:
939:
936:
934:
931:
889:
888:
875:
869:
866:
864:
861:
858:
856:
853:
852:
849:
846:
843:
841:
838:
836:
833:
832:
829:
826:
824:
821:
819:
816:
815:
813:
783:
777:
768:
750:
749:
738:
733:
729:
725:
722:
692:linear algebra
687:
684:
682:
679:
678:
677:
670:
659:Fourier series
655:
652:
643:
640:
580:
577:
525:
524:
513:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
462:
461:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
356:
353:
295:
294:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
188:
187:Even functions
185:
180:Main article:
177:
174:
172:
169:
141:Main article:
138:
135:
54:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3545:
3534:
3531:
3530:
3528:
3517:
3511:
3507:
3506:
3501:
3497:
3492:
3486:
3482:
3477:
3473:
3471:0-691-02374-3
3467:
3463:
3459:
3458:Weyl, Hermann
3455:
3454:
3450:
3442:
3436:
3432:
3431:
3426:
3420:
3418:
3414:
3409:
3405:
3401:
3397:
3393:
3386:
3383:
3378:
3374:
3370:
3366:
3362:
3358:
3351:
3344:
3341:
3336:
3334:3-540-67995-2
3330:
3326:
3322:
3315:
3312:
3306:
3303:
3292:
3288:
3281:
3279:
3275:
3263:
3259:
3253:
3250:
3239:
3235:
3228:
3225:
3219:
3215:
3212:
3210:
3207:
3206:
3202:
3200:
3197:
3193:
3191:
3186:
3182:
3180:
3175:
3173:
3165:
3163:
3160:
3158:
3153:
3151:
3147:
3143:
3142:Line symmetry
3138:
3136:
3128:
3126:
3123:
3121:
3117:
3113:
3109:
3105:
3101:
3100:metric spaces
3097:
3093:
3087:
3079:
3074:
3072:
3070:
3064:
3056:
3051:
3031:
3028:
3025:
3019:
3016:
3013:
3007:
3004:
3001:
2995:
2988:
2987:
2986:
2967:
2964:
2958:
2955:
2952:
2944:
2938:
2932:
2929:
2926:
2918:
2908:
2907:
2906:
2885:
2882:
2879:
2871:
2865:
2859:
2856:
2853:
2845:
2839:
2833:
2830:
2827:
2819:
2813:
2807:
2804:
2801:
2793:
2783:
2782:
2781:
2763:
2755:
2749:
2710:
2702:
2696:
2685:
2676:
2668:
2662:
2643:
2633:
2632:
2631:
2608:
2600:
2594:
2578:
2572:
2566:
2563:
2560:
2552:
2546:
2540:
2537:
2534:
2528:
2521:
2520:
2519:
2517:
2511:
2507:
2503:
2497:
2495:
2489:
2485:
2481:
2474:
2470:
2466:
2459:
2455:
2451:
2447:
2443:
2439:
2416:
2413:
2410:
2399:
2396:
2393:
2387:
2382:
2379:
2376:
2372:
2368:
2362:
2350:
2349:
2348:
2330:
2300:
2286:
2285:sum of states
2280:
2274:
2269:
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2212:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2169:
2165:
2162:
2158:
2154:
2150:
2146:
2142:
2139:
2135:
2132:
2128:
2124:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2070:
2069:abelian group
2066:
2062:
2058:
2055:, the set of
2054:
2050:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2022:
2018:
2016:
2014:
2010:
2007:, called the
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1976:
1968:
1966:
1964:
1960:
1956:
1952:
1946:
1942:
1937:
1933:
1929:
1923:
1922:Galois theory
1916:Galois theory
1915:
1913:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1832:
1825:
1822:
1818:
1814:
1809:
1806:
1802:
1796:
1793:
1789:
1784:
1780:
1773:
1769:
1765:
1760:
1756:
1750:
1746:
1741:
1733:
1732:
1731:
1729:
1725:
1721:
1697:
1694:
1690:
1686:
1683:
1680:
1675:
1672:
1668:
1664:
1659:
1656:
1652:
1645:
1642:
1634:
1630:
1626:
1623:
1620:
1615:
1611:
1607:
1602:
1598:
1591:
1584:
1583:
1582:
1580:
1576:
1572:
1568:
1562:
1554:
1535:
1531:
1525:
1521:
1517:
1514:
1509:
1505:
1499:
1495:
1491:
1488:
1483:
1479:
1473:
1469:
1465:
1462:
1457:
1453:
1447:
1443:
1437:
1433:
1425:
1424:
1423:
1418:
1411:
1404:
1383:
1373:
1369:
1365:
1360:
1356:
1349:
1344:
1339:
1335:
1329:
1325:
1321:
1316:
1312:
1306:
1301:
1297:
1293:
1288:
1283:
1279:
1273:
1268:
1264:
1260:
1253:
1239:
1236:
1231:
1226:
1222:
1218:
1213:
1208:
1204:
1196:
1195:
1194:
1189:
1182:
1174:
1172:
1170:
1166:
1163:
1159:
1155:
1149:
1146:
1142:
1135:
1128:
1124:
1118:
1113:
1106:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1070:
1066:
1059:
1052:
1048:
1045:
1041:
1035:
1027:
1025:
1023:
1018:
1014:
1010:
1006:
1002:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
958:
954:
951:
945:
937:
932:
930:
928:
924:
920:
916:
913:
909:
905:
900:
898:
894:
891:Every square
873:
867:
862:
859:
854:
847:
844:
839:
834:
827:
822:
817:
811:
802:
801:
800:
797:
795:
791:
786:
780:
776:
771:
767:
763:
759:
758:main diagonal
754:
736:
731:
727:
723:
720:
713:
712:
711:
709:
705:
701:
700:square matrix
697:
693:
685:
680:
675:
671:
668:
664:
660:
656:
653:
650:
646:
645:
641:
639:
637:
633:
629:
625:
621:
617:
613:
608:
606:
602:
598:
594:
590:
586:
578:
576:
574:
570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
511:
507:
504:
498:
495:
489:
486:
480:
474:
467:
466:
465:
445:
442:
436:
433:
427:
421:
418:
411:
410:
409:
407:
403:
399:
395:
391:
387:
383:
374:
370:
366:
361:
355:Odd functions
354:
352:
350:
346:
342:
338:
334:
330:
325:
323:
316:
312:
308:
304:
300:
278:
275:
269:
266:
260:
254:
247:
246:
245:
243:
239:
235:
231:
227:
223:
219:
215:
206:
202:
198:
193:
186:
183:
175:
170:
168:
166:
162:
158:
154:
150:
144:
136:
134:
131:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
84:
82:
78:
74:
70:
66:
65:
57:
51:
47:
42:
36:
32:
27:
19:
3504:
3480:
3461:
3451:Bibliography
3429:
3399:
3395:
3385:
3360:
3356:
3343:
3324:
3314:
3305:
3294:. Retrieved
3290:
3266:. Retrieved
3264:. 2016-06-23
3261:
3252:
3241:. Retrieved
3237:
3227:
3198:
3194:
3187:
3183:
3176:
3169:
3161:
3154:
3139:
3132:
3124:
3091:
3089:
3069:antisymmetry
3066:
2984:
2904:
2735:
2629:
2515:
2509:
2505:
2501:
2498:
2487:
2483:
2479:
2472:
2468:
2464:
2457:
2453:
2449:
2445:
2441:
2437:
2434:
2281:
2278:
2263:Galois group
2258:
2254:
2246:
2242:
2218:
2214:
2210:
2202:
2198:
2194:
2190:
2187:real numbers
2182:
2160:
2152:
2148:
2144:
2140:
2133:
2131:vector space
2115:
2111:
2095:
2087:
2083:
2079:
2060:
2045:
2037:
2033:
2008:
1985:automorphism
1984:
1978:
1975:Automorphism
1958:
1954:
1951:permutations
1944:
1940:
1935:
1931:
1925:
1889:
1869:
1865:
1854:vector space
1849:
1847:
1727:
1723:
1719:
1717:
1570:
1564:
1416:
1409:
1402:
1400:
1187:
1180:
1178:
1161:
1150:
1144:
1140:
1133:
1126:
1122:
1116:
1111:
1104:
1097:
1093:
1089:
1081:
1077:
1073:
1068:
1064:
1057:
1050:
1046:
1039:
1037:
1021:
1016:
1012:
1004:
997:
993:
977:
974:permutations
965:
956:
952:
949:
947:
901:
890:
798:
793:
789:
784:
778:
774:
769:
765:
761:
755:
751:
707:
695:
689:
635:
631:
627:
623:
619:
615:
611:
609:
604:
600:
596:
592:
588:
582:
572:
564:
556:
548:
544:
526:
463:
405:
401:
397:
393:
389:
381:
379:
372:
368:
364:
348:
340:
332:
328:
321:
314:
302:
296:
241:
237:
233:
229:
225:
217:
213:
211:
204:
200:
196:
146:
132:
120:metric space
111:
99:
87:
85:
62:
61:
26:
3234:"Invariant"
3112:composition
2227:uncountably
2114:. Thus, if
2030:permutation
1989:isomorphism
1910:mathematics
1902:engineering
1579:permutation
1567:mathematics
1086:permutation
1084:if for any
986:composition
579:Integrating
380:Again, let
46:root system
3408:1274.58003
3296:2019-12-06
3268:2019-12-06
3243:2019-12-06
3220:References
3116:reflection
2185:) and the
2102:and whose
2026:set theory
1864:of degree
1092:, one has
1044:polynomial
962:finite set
313:about the
311:reflection
77:operations
73:invariance
3460:(1989) .
3104:congruent
3017:−
2962:⟩
2945:ψ
2942:⟨
2936:⟩
2919:ψ
2916:⟨
2889:⟩
2872:ψ
2869:⟨
2863:⟩
2846:ψ
2843:⟨
2837:⟩
2820:ψ
2817:⟨
2811:⟩
2794:ψ
2791:⟨
2767:⟩
2753:⟩
2714:⟩
2700:⟩
2686:⊗
2680:⟩
2666:⟩
2644:ψ
2641:⟨
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2601:⊗
2598:⟩
2579:ψ
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2550:⟨
2514:are zero
2420:⟩
2373:∑
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2363:ψ
2334:⟩
2304:⟩
2168:bijective
1823:σ
1815:…
1807:σ
1794:σ
1766:…
1695:σ
1684:…
1673:σ
1657:σ
1624:…
1515:−
1489:−
1463:−
1237:−
1162:symmetric
1001:factorial
860:−
845:−
704:transpose
496:−
464:That is,
443:−
419:−
299:symmetric
276:−
124:bijection
104:bijective
50:Lie group
3533:Symmetry
3527:Category
3502:(2012).
3462:Symmetry
3427:(1986).
3203:See also
3096:distance
3092:isometry
3086:Isometry
2251:subgroup
2090:) whose
2057:integers
2019:Examples
1997:symmetry
1872:. Over
1175:Examples
773:), then
663:periodic
585:integral
537:rotation
128:isometry
92:symmetry
69:geometry
64:Symmetry
3377:2689301
2257:fixing
2173:from a
2120:trivial
2106:is the
2001:mapping
1991:from a
1938:, that
1906:physics
1158:theorem
1139:, ...,
1110:, ...,
1063:, ...,
984:is the
976:of the
919:complex
910:over a
567:), and
541:degrees
539:of 180
343:), and
220:) be a
96:mapping
3512:
3487:
3468:
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3406:
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3331:
2494:tensor
2108:center
2104:kernel
2086:โ Aut(
1987:is an
1880:, the
1874:fields
1575:tensor
960:(on a
676:terms.
669:terms.
667:cosine
642:Series
529:origin
365:ƒ
197:ƒ
116:metric
3373:JSTOR
3353:(PDF)
3144:of a
3094:is a
2475:) = 0
2448:) = โ
2217:into
2175:field
2159:, GL(
2136:is a
2098:) of
2092:image
2005:group
1983:, an
1080:is a
1072:) in
1042:is a
1009:order
970:group
698:is a
661:of a
533:graph
384:be a
307:graph
94:is a
3510:ISBN
3485:ISBN
3466:ISBN
3435:ISBN
3329:ISBN
3155:For
2249:the
2118:has
2065:ring
1934:and
1908:and
1569:, a
1415:and
1186:and
1108:ฯ(2)
1101:ฯ(1)
948:The
912:real
904:real
792:and
694:, a
674:sine
657:The
647:The
630:and
614:to +
603:and
591:to +
583:The
561:sinh
400:and
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345:cosh
236:and
230:even
222:real
212:Let
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159:and
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