1698:
1175:
1693:{\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}}
3412:
27:
866:
627:
1014:
711:
2930:, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.
1790:
with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called
2265:
2497:
2404:
423:
2315:
1075:
1170:
514:
648:
2123:
343:
2170:
449:
2908:
in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state.
885:
2638:. Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is,
2815:
IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in
2048:
1180:
2847:
639:. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an
861:{\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}}
3432:
489:
2933:
Another involution used in computers is an order-2 bitwise permutation. For example. a colour value stored as integers in the form
2671:
1019:
3062:
2670:
The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the
1977:, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise
2177:
3369:
3098:
3072:
1803:
20:
2415:
2322:
3437:
3119:
2873:. Since the converse of the converse is the original relation, the conversion operation is an involution on the
348:
3048:
2270:
493:
2530:, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the
3396:
3361:
3151:
2886:
1787:
1730:
1127:
2822:
The involutiveness of negation is an important characterization property for logics and the corresponding
3416:
3391:
2827:
2823:
2766:
2664:
3201:
716:
2527:
1783:
1703:
644:
2093:
1763:
meet any line (not through a vertex) in three pairs of an involution. This theorem has been called
1719:
302:
185:
1093:
3197:
708:, then its graph is its own reflection. Some basic examples of involutions include the functions
201:
3386:
2795:
Generally in non-classical logics, negation that satisfies the law of double negation is called
1838:. Except for in characteristic 2, such operators are diagonalizable for a given basis with just
3237:
2882:
2131:
622:{\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.}
189:
151:
65:
3088:
2874:
2859:
2690:
428:
293:
220:
3257:
3180:
3129:
2839:
2599:
2576:
2090:, an (anti-)involution is defined by the following axioms: if we consider a transformation
1993:
1965:
1776:
636:
252:
227:
104:
3379:
3305:
Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions".
2851:
2808:
868:
These may be composed in various ways to produce additional involutions. For example, if
8:
3345:
2816:
2799:. In algebraic semantics, such a negation is realized as an involution on the algebra of
2595:
2568:
2507:
2063:
1982:
1978:
1760:
1723:
669:
665:
289:
3353:
3332:
3314:
3168:
2926:
This predates binary computers; practically all mechanical cipher machines implement a
2804:
2531:
2087:
2003:
3365:
3094:
3068:
3044:
2927:
2901:
2870:
2644:
1986:
1815:
1737:
1009:{\displaystyle f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}}
209:
165:
133:
3360:, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI:
3336:
3375:
3324:
3249:
3205:
3160:
2905:
2862:, and other pairs of important varieties of algebras (resp. corresponding logics).
2754:
2635:
2561:
2503:
2067:
193:
137:
72:
2904:
with a given value for one parameter is an involution on the other parameter. XOR
2743:
of involutions subject only to relations involving powers of pairs of elements of
2663:
is an involution if and only if it can be written as a finite product of disjoint
136:
is a trivial example of an involution. Examples of nontrivial involutions include
3269:
3216:
3176:
3125:
2866:
2835:
2831:
2775:
1846:
s on the diagonal of the corresponding matrix. If the operator is orthogonal (an
1726:. Performing a reflection twice brings a point back to its original coordinates.
1715:
217:
3328:
1767:'s Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the
2750:
2075:
1919:, and that is the identity on all other basis vectors. It can be checked that
3426:
2916:
2878:
2734:
2572:
497:
2749:. Coxeter groups can be used, among other things, to describe the possible
281:
The number of involutions, including the identity involution, on a set with
3114:
3093:, Springer Science & Business Media, Problem 1.11(a), p. 27,
3010:
2591:
2564:
that is its own inverse function. Examples of involutions in common rings:
2017:
1749:
479:
3283:
2803:. Examples of logics that have involutive negation are Kleene and Bochvar
3349:
3015:
2912:
2812:
2800:
2660:
2553:
632:
475:
471:
467:
49:
1752:
of period 2, that is, a projectivity that interchanges pairs of points.
3172:
3142:
2855:
2843:
2071:
640:
463:
459:
455:
205:
181:
1116:
Other nonlinear examples can be constructed by wrapping an involution
3319:
2920:
2547:
2535:
2056:
1974:
1764:
126:
3164:
631:
The number of fixed points of an involution on a finite set and its
2770:
1733:; not a reflection in the above sense, and so, a distinct example.
197:
1964:
For a specific basis, any linear operator can be represented by a
654:
1798:
Another type of involution occurring in projective geometry is a
511:
can also be expressed by non-recursive formulas, such as the sum
44:
that, when applied twice, brings one back to the starting point.
3411:
3005:
2911:
Two special cases of this, which are also involutions, are the
1756:
Any projectivity that interchanges two points is an involution.
3240:(Cambridge and London: Harvard and Heinemann), pp. 610–3
3067:(2nd ed.), W. W. Norton & Company, Inc, p. 426,
3020:
1786:, it has another, and consists of the correspondence between
213:
26:
2409:
An anti-involution does not obey the last axiom but instead
1714:
A simple example of an involution of the three-dimensional
484:
2898:
2081:
1877:
are basis elements. There exists a linear transformation
3234:
3343:
1820:
In linear algebra, an involution is a linear operator
2418:
2325:
2273:
2180:
2134:
2096:
1853:
For example, suppose that a basis for a vector space
1178:
1130:
1022:
888:
714:
517:
431:
351:
305:
3124:, Reading, Mass.: Addison-Wesley, pp. 48, 65,
2915:operation which is XOR with an all-ones value, and
2826:. For instance, involutive negation characterizes
2491:
2398:
2309:
2259:
2164:
2117:
1692:
1164:
1069:
1016:is an involution, and more generally the function
1008:
860:
682:. This is due to the fact that the inverse of any
621:
443:
417:
337:
2652:was defined more broadly, and accordingly so was
3424:
2260:{\displaystyle f(x_{1}+x_{2})=f(x_{1})+f(x_{2})}
1992:The definition of involution extends readily to
3145:(1990), "A one-sentence proof that every prime
2838:arises by adding the law of double negation to
2526:. Taken as an axiom, it leads to the notion of
655:Involution throughout the fields of mathematics
2889:involution, it is preserved under conversion.
2492:{\displaystyle f(x_{1}x_{2})=f(x_{2})f(x_{1})}
2399:{\displaystyle f(x_{1}x_{2})=f(x_{1})f(x_{2})}
686:function will be its reflection over the line
21:Involution (disambiguation) § Mathematics
3307:Computers & Mathematics with Applications
276:
560:
547:
3304:
2885:. While this ordering is reversed with the
2842:. The same relationship holds also between
1702:Other elementary involutions are useful in
2919:encryption, which is an XOR with a secret
668:of an involution (on the real numbers) is
418:{\displaystyle a_{n}=a_{n-1}+(n-1)a_{n-2}}
3318:
2310:{\displaystyle f(\lambda x)=\lambda f(x)}
500:with a given number of cells. The number
454:The first few terms of this sequence are
3221:The Geometrical Work of Girard Desargues
3086:
2602:2; that is, an involution is an element
1092:. (This is the self-inverse subset of
1070:{\displaystyle g(x)={\frac {x+b}{cx-1}}}
704:. If, in particular, the function is an
659:
25:
3060:
1850:), it is orthonormally diagonalizable.
1759:The three pairs of opposite sides of a
1165:{\displaystyle f:=h^{-1}\circ g\circ h}
3425:
3281:
3149:≡ 1 (mod 4) is a sum of two squares",
3141:
2672:classification of finite simple groups
2581:taking the transpose in a matrix ring.
2082:Quaternion algebra, groups, semigroups
2049:Involutions are related to idempotents
1743:
1736:These transformations are examples of
3192:
3190:
3113:
2760:
1709:
251:is an involution if and only if they
120:
2755:generalizations to higher dimensions
2502:This former law is sometimes called
496:, and they also count the number of
2892:
117:twice produces the original value.
13:
3298:
3187:
2648:. By the end of the 19th century,
1981:is an independent involution, the
14:
3449:
3404:
3122:, Volume 3: Sorting and Searching
2773:in classical logic satisfies the
1809:
696:. This can be seen by "swapping"
3433:Algebraic properties of elements
3410:
2850:(and so correspondingly between
2560:is customarily taken to mean an
2039:is the identity homomorphism on
492:); these numbers are called the
16:Function that is its own inverse
3275:
3262:
3236:, Volume II, number 362 in the
3120:The Art of Computer Programming
3090:The Elements of Operator Theory
2769:is an involution. Accordingly,
2765:The operation of complement in
2585:
1771:of Euclid in Volume VII of the
1653:
1613:
1077:is an involution for constants
3284:"The Mechanization of Ciphers"
3243:
3226:
3210:
3198:Elementary Projective Geometry
3135:
3107:
3080:
3054:
3033:
2737:are groups generated by a set
2541:
2486:
2473:
2467:
2454:
2445:
2422:
2393:
2380:
2374:
2361:
2352:
2329:
2304:
2298:
2286:
2277:
2254:
2241:
2232:
2219:
2210:
2184:
2153:
2150:
2144:
2138:
2112:
2106:
2100:
1663:
1657:
1623:
1617:
1576:
1570:
1541:
1535:
1510:
1504:
1457:
1451:
1421:
1415:
1374:
1368:
1299:
1293:
1263:
1257:
1233:
1227:
1192:
1186:
1032:
1026:
987:
981:
978:
952:
946:
940:
937:
911:
905:
899:
820:
814:
777:
771:
735:
729:
607:
592:
396:
384:
1:
3362:American Mathematical Society
3223:, (New York: Springer), p. 54
3152:American Mathematical Monthly
3026:
2834:. Correspondingly, classical
2809:Łukasiewicz many-valued logic
2118:{\displaystyle x\mapsto f(x)}
1826:on a vector space, such that
1731:reflection through the origin
643:of elements has at least one
338:{\displaystyle a_{0}=a_{1}=1}
3232:Ivor Thomas (editor) (1980)
3087:Kubrusly, Carlos S. (2011),
2575:, and its equivalent in the
2125:then it is an involution if
1704:solving functional equations
649:Fermat's two squares theorem
647:. This can be used to prove
30:An involution is a function
7:
3392:Encyclopedia of Mathematics
3329:10.1016/j.camwa.2006.10.029
2999:
2598:is an involution if it has
2031:is called an involution if
1124:and its inverse, producing
10:
3454:
3270:"A Course on Group Theory"
3202:Cambridge University Press
3061:Russell, Bertrand (1903),
2693:if there is an involution
2545:
1813:
277:Involutions on finite sets
18:
3064:Principles of mathematics
3041:Calculus: Single Variable
2528:semigroup with involution
2165:{\displaystyle f(f(x))=x}
1782:If an involution has one
1120:in an arbitrary function
113:. Equivalently, applying
3254:Introduction to Geometry
3039:Robert Alexander Adams,
2961:, resulting in the form
2059:in a one-to-one manner.
2055:is invertible then they
3358:The book of involutions
2877:. Binary relations are
2869:, every relation has a
2172:(it is its own inverse)
1989:is also an involution.
444:{\displaystyle n>1.}
288:elements is given by a
216:transformation and the
3438:Functions and mappings
3272:. p. 10, section 1.13.
3238:Loeb Classical Library
3219:and J. J. Gray (1987)
2776:law of double negation
2493:
2400:
2311:
2261:
2166:
2119:
1729:Another involution is
1694:
1166:
1094:Möbius transformations
1071:
1010:
862:
623:
564:
445:
419:
339:
45:
3282:Goebel, Greg (2018).
3258:John Wiley & Sons
3196:A.G. Pickford (1909)
2875:category of relations
2824:varieties of algebras
2577:split-complex numbers
2546:Further information:
2506:. It also appears in
2494:
2401:
2312:
2262:
2167:
2120:
2074:are special types of
1973:. Every matrix has a
1848:orthogonal involution
1814:Further information:
1695:
1167:
1106:, then normalized to
1072:
1011:
863:
660:Real-valued functions
624:
531:
446:
420:
340:
294:Heinrich August Rothe
221:polyalphabetic cipher
62:self-inverse function
29:
3419:at Wikimedia Commons
3346:Merkurjev, Alexander
3288:Classical Cryptology
2840:intuitionistic logic
2416:
2323:
2271:
2178:
2132:
2094:
1955:is an involution of
1859:is chosen, and that
1777:Pappus of Alexandria
1176:
1128:
1020:
886:
712:
515:
429:
349:
303:
125:Any involution is a
19:For other uses, see
3354:Tignol, Jean-Pierre
2817:t-norm fuzzy logics
2805:three-valued logics
2569:complex conjugation
2538:as the involution.
2064:functional analysis
1983:conjugate transpose
1979:complex conjugation
1788:harmonic conjugates
1761:complete quadrangle
1748:An involution is a
1744:Projective geometry
290:recurrence relation
239:of two involutions
166:complex conjugation
58:involutory function
3344:Knus, Max-Albert;
2761:Mathematical logic
2642:was taken to mean
2594:, an element of a
2532:full linear monoid
2489:
2396:
2307:
2257:
2162:
2115:
2088:quaternion algebra
2078:with involutions.
1738:affine involutions
1710:Euclidean geometry
1690:
1688:
1162:
1067:
1006:
858:
856:
633:number of elements
619:
441:
415:
335:
210:reciprocal ciphers
121:General properties
46:
3415:Media related to
2949:, could exchange
2928:reciprocal cipher
2902:bitwise operation
2871:converse relation
2852:Łukasiewicz logic
2786:is equivalent to
2751:regular polyhedra
2645:permutation group
2068:Banach *-algebras
1996:. Given a module
1987:Hermitian adjoint
1816:Involutory matrix
1649:
1609:
1608:
1528:
1493:
1408:
1357:
1220:
1065:
1004:
849:
795:
614:
558:
494:telephone numbers
3445:
3414:
3400:
3382:
3340:
3322:
3292:
3291:
3279:
3273:
3266:
3260:
3250:H. S. M. Coxeter
3247:
3241:
3230:
3224:
3214:
3208:
3206:Internet Archive
3194:
3185:
3183:
3139:
3133:
3132:
3115:Knuth, Donald E.
3111:
3105:
3103:
3084:
3078:
3077:
3058:
3052:
3037:
2995:
2976:
2960:
2954:
2948:
2893:Computer science
2867:binary relations
2865:In the study of
2854:and fuzzy logic
2832:Heyting algebras
2828:Boolean algebras
2791:
2785:
2767:Boolean algebras
2748:
2742:
2730:
2708:
2698:
2688:
2682:
2636:identity element
2633:
2627:
2617:
2607:
2562:antihomomorphism
2525:
2504:antidistributive
2498:
2496:
2495:
2490:
2485:
2484:
2466:
2465:
2444:
2443:
2434:
2433:
2405:
2403:
2402:
2397:
2392:
2391:
2373:
2372:
2351:
2350:
2341:
2340:
2316:
2314:
2313:
2308:
2266:
2264:
2263:
2258:
2253:
2252:
2231:
2230:
2209:
2208:
2196:
2195:
2171:
2169:
2168:
2163:
2124:
2122:
2121:
2116:
2054:
2044:
2038:
2037:
2030:
2024:
2016:
2010:
2001:
1972:
1960:
1954:
1948:
1942:
1936:
1918:
1909:
1900:
1891:
1882:
1876:
1867:
1858:
1845:
1841:
1837:
1832:
1825:
1699:
1697:
1696:
1691:
1689:
1682:
1681:
1650:
1648:
1634:
1610:
1601:
1600:
1591:
1587:
1529:
1521:
1498:
1494:
1492:
1478:
1440:
1439:
1409:
1407:
1396:
1385:
1362:
1358:
1356:
1349:
1348:
1338:
1331:
1330:
1320:
1282:
1281:
1221:
1219:
1218:
1203:
1171:
1169:
1168:
1163:
1149:
1148:
1123:
1119:
1112:
1105:
1091:
1084:
1080:
1076:
1074:
1073:
1068:
1066:
1064:
1050:
1039:
1015:
1013:
1012:
1007:
1005:
997:
977:
976:
964:
963:
936:
935:
923:
922:
898:
897:
881:
874:
867:
865:
864:
859:
857:
850:
848:
831:
813:
812:
796:
788:
770:
769:
728:
727:
703:
699:
695:
681:
672:across the line
628:
626:
625:
620:
615:
613:
585:
584:
574:
566:
563:
559:
551:
545:
527:
526:
510:
487:
450:
448:
447:
442:
424:
422:
421:
416:
414:
413:
380:
379:
361:
360:
344:
342:
341:
336:
328:
327:
315:
314:
287:
272:
250:
244:
238:
194:circle inversion
179:
178:
163:
149:
116:
112:
102:
95:
71:that is its own
70:
43:
3453:
3452:
3448:
3447:
3446:
3444:
3443:
3442:
3423:
3422:
3407:
3385:
3372:
3301:
3299:Further reading
3296:
3295:
3280:
3276:
3267:
3263:
3248:
3244:
3231:
3227:
3215:
3211:
3195:
3188:
3165:10.2307/2323918
3140:
3136:
3112:
3108:
3101:
3085:
3081:
3075:
3059:
3055:
3038:
3034:
3029:
3002:
2978:
2962:
2956:
2950:
2934:
2895:
2887:complementation
2787:
2780:
2763:
2744:
2738:
2710:
2700:
2694:
2684:
2678:
2629:
2619:
2609:
2603:
2588:
2550:
2544:
2511:
2480:
2476:
2461:
2457:
2439:
2435:
2429:
2425:
2417:
2414:
2413:
2387:
2383:
2368:
2364:
2346:
2342:
2336:
2332:
2324:
2321:
2320:
2272:
2269:
2268:
2248:
2244:
2226:
2222:
2204:
2200:
2191:
2187:
2179:
2176:
2175:
2133:
2130:
2129:
2095:
2092:
2091:
2084:
2076:Banach algebras
2052:
2040:
2033:
2032:
2026:
2020:
2012:
2006:
1997:
1968:
1956:
1950:
1944:
1938:
1920:
1917:
1911:
1908:
1902:
1899:
1893:
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1878:
1875:
1869:
1866:
1860:
1854:
1843:
1839:
1828:
1827:
1821:
1818:
1812:
1746:
1716:Euclidean space
1712:
1687:
1686:
1677:
1673:
1666:
1651:
1638:
1633:
1626:
1611:
1596:
1592:
1586:
1579:
1564:
1563:
1544:
1530:
1520:
1513:
1499:
1482:
1477:
1473:
1460:
1445:
1444:
1435:
1431:
1424:
1410:
1397:
1386:
1384:
1377:
1363:
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1326:
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1287:
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1266:
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1202:
1195:
1179:
1177:
1174:
1173:
1141:
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1129:
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1125:
1121:
1117:
1107:
1097:
1086:
1082:
1078:
1051:
1040:
1038:
1021:
1018:
1017:
996:
972:
968:
959:
955:
931:
927:
918:
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893:
889:
887:
884:
883:
876:
869:
855:
854:
835:
830:
823:
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800:
787:
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765:
761:
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757:
738:
723:
719:
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673:
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657:
580:
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483:
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403:
399:
369:
365:
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347:
346:
323:
319:
310:
306:
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301:
300:
282:
279:
256:
246:
240:
230:
202:complementation
174:
169:
155:
141:
123:
114:
108:
100:
79:
68:
31:
24:
17:
12:
11:
5:
3451:
3441:
3440:
3435:
3421:
3420:
3406:
3405:External links
3403:
3402:
3401:
3383:
3370:
3341:
3313:(1): 137–143.
3300:
3297:
3294:
3293:
3274:
3268:John S. Rose.
3261:
3242:
3225:
3209:
3186:
3134:
3106:
3099:
3079:
3073:
3053:
3031:
3030:
3028:
3025:
3024:
3023:
3018:
3013:
3008:
3001:
2998:
2986:(RGB)) = RGB,
2894:
2891:
2762:
2759:
2735:Coxeter groups
2665:transpositions
2587:
2584:
2583:
2582:
2579:
2543:
2540:
2500:
2499:
2488:
2483:
2479:
2475:
2472:
2469:
2464:
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2456:
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2367:
2363:
2360:
2357:
2354:
2349:
2345:
2339:
2335:
2331:
2328:
2318:
2317:(it is linear)
2306:
2303:
2300:
2297:
2294:
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2276:
2256:
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2240:
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2102:
2099:
2083:
2080:
1915:
1906:
1897:
1888:
1873:
1864:
1811:
1810:Linear algebra
1808:
1796:
1795:
1780:
1757:
1745:
1742:
1711:
1708:
1685:
1680:
1676:
1672:
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1569:
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1543:
1540:
1537:
1534:
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1527:
1524:
1519:
1516:
1514:
1512:
1509:
1506:
1503:
1500:
1497:
1491:
1488:
1485:
1481:
1476:
1472:
1469:
1466:
1463:
1461:
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1456:
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1314:
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1308:
1305:
1303:
1301:
1298:
1295:
1292:
1289:
1288:
1285:
1280:
1276:
1272:
1269:
1267:
1265:
1262:
1259:
1256:
1253:
1251:
1248:
1245:
1242:
1239:
1237:
1235:
1232:
1229:
1226:
1223:
1217:
1213:
1209:
1206:
1201:
1198:
1196:
1194:
1191:
1188:
1185:
1182:
1181:
1161:
1158:
1155:
1152:
1147:
1144:
1140:
1136:
1133:
1085:which satisfy
1063:
1060:
1057:
1054:
1049:
1046:
1043:
1037:
1034:
1031:
1028:
1025:
1003:
1000:
995:
992:
989:
986:
983:
980:
975:
971:
967:
962:
958:
954:
951:
948:
945:
942:
939:
934:
930:
926:
921:
917:
913:
910:
907:
904:
901:
896:
892:
853:
847:
844:
841:
838:
834:
829:
826:
824:
822:
819:
816:
811:
807:
803:
802:
799:
794:
791:
786:
783:
781:
779:
776:
773:
768:
764:
760:
759:
756:
753:
750:
747:
744:
741:
739:
737:
734:
731:
726:
722:
718:
717:
661:
658:
656:
653:
635:have the same
618:
612:
609:
606:
603:
600:
597:
594:
591:
588:
583:
579:
573:
570:
562:
557:
554:
549:
544:
541:
538:
534:
530:
525:
521:
505:
498:Young tableaux
452:
451:
440:
437:
434:
412:
409:
406:
402:
398:
395:
392:
389:
386:
383:
378:
375:
372:
368:
364:
359:
355:
334:
331:
326:
322:
318:
313:
309:
286:= 0, 1, 2, ...
278:
275:
122:
119:
97:
96:
15:
9:
6:
4:
3:
2:
3450:
3439:
3436:
3434:
3431:
3430:
3428:
3418:
3413:
3409:
3408:
3398:
3394:
3393:
3388:
3384:
3381:
3377:
3373:
3371:0-8218-0904-0
3367:
3363:
3359:
3355:
3351:
3347:
3342:
3338:
3334:
3330:
3326:
3321:
3316:
3312:
3308:
3303:
3302:
3289:
3285:
3278:
3271:
3265:
3259:
3256:, pp. 244–8,
3255:
3251:
3246:
3239:
3235:
3229:
3222:
3218:
3213:
3207:
3203:
3199:
3193:
3191:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3153:
3148:
3144:
3138:
3131:
3127:
3123:
3121:
3116:
3110:
3102:
3100:9780817649982
3096:
3092:
3091:
3083:
3076:
3074:9781440054167
3070:
3066:
3065:
3057:
3050:
3046:
3042:
3036:
3032:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3003:
2997:
2993:
2989:
2985:
2981:
2974:
2970:
2966:
2959:
2953:
2946:
2942:
2938:
2931:
2929:
2924:
2922:
2918:
2917:stream cipher
2914:
2909:
2907:
2903:
2900:
2890:
2888:
2884:
2880:
2876:
2872:
2868:
2863:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2836:Boolean logic
2833:
2829:
2825:
2820:
2818:
2814:
2810:
2806:
2802:
2798:
2793:
2790:
2784:
2778:
2777:
2772:
2768:
2758:
2756:
2752:
2747:
2741:
2736:
2732:
2729:
2725:
2721:
2717:
2713:
2707:
2703:
2697:
2692:
2691:strongly real
2687:
2681:
2675:
2673:
2668:
2666:
2662:
2657:
2655:
2651:
2647:
2646:
2641:
2637:
2632:
2626:
2622:
2616:
2612:
2606:
2601:
2597:
2593:
2580:
2578:
2574:
2573:complex plane
2570:
2567:
2566:
2565:
2563:
2559:
2555:
2549:
2539:
2537:
2533:
2529:
2523:
2519:
2515:
2509:
2505:
2481:
2477:
2470:
2462:
2458:
2451:
2448:
2440:
2436:
2430:
2426:
2419:
2412:
2411:
2410:
2388:
2384:
2377:
2369:
2365:
2358:
2355:
2347:
2343:
2337:
2333:
2326:
2319:
2301:
2295:
2292:
2289:
2283:
2280:
2274:
2249:
2245:
2238:
2235:
2227:
2223:
2216:
2213:
2205:
2201:
2197:
2192:
2188:
2181:
2174:
2159:
2156:
2147:
2141:
2135:
2128:
2127:
2126:
2109:
2103:
2097:
2089:
2079:
2077:
2073:
2069:
2065:
2060:
2058:
2050:
2046:
2043:
2036:
2029:
2023:
2019:
2015:
2009:
2005:
2000:
1995:
1990:
1988:
1984:
1980:
1976:
1971:
1967:
1962:
1959:
1953:
1947:
1941:
1935:
1931:
1927:
1923:
1914:
1905:
1896:
1887:
1881:
1872:
1863:
1857:
1851:
1849:
1836:
1831:
1824:
1817:
1807:
1806:of period 2.
1805:
1801:
1793:
1792:double points
1789:
1785:
1781:
1778:
1774:
1770:
1766:
1762:
1758:
1755:
1754:
1753:
1751:
1741:
1739:
1734:
1732:
1727:
1725:
1721:
1717:
1707:
1705:
1700:
1683:
1678:
1674:
1670:
1668:
1660:
1654:
1645:
1642:
1639:
1635:
1630:
1628:
1620:
1614:
1605:
1602:
1597:
1593:
1588:
1583:
1581:
1573:
1567:
1560:
1557:
1554:
1551:
1548:
1546:
1538:
1532:
1525:
1522:
1517:
1515:
1507:
1501:
1495:
1489:
1486:
1483:
1479:
1474:
1470:
1467:
1464:
1462:
1454:
1448:
1441:
1436:
1432:
1428:
1426:
1418:
1412:
1404:
1401:
1398:
1393:
1390:
1387:
1381:
1379:
1371:
1365:
1359:
1353:
1350:
1345:
1341:
1335:
1332:
1327:
1323:
1316:
1312:
1309:
1306:
1304:
1296:
1290:
1283:
1278:
1274:
1270:
1268:
1260:
1254:
1249:
1246:
1243:
1240:
1238:
1230:
1224:
1215:
1211:
1207:
1204:
1199:
1197:
1189:
1183:
1159:
1156:
1153:
1150:
1145:
1142:
1138:
1134:
1131:
1114:
1110:
1104:
1100:
1095:
1089:
1061:
1058:
1055:
1052:
1047:
1044:
1041:
1035:
1029:
1023:
1001:
998:
993:
990:
984:
973:
969:
965:
960:
956:
949:
943:
932:
928:
924:
919:
915:
908:
902:
894:
890:
879:
872:
851:
845:
842:
839:
836:
832:
827:
825:
817:
809:
805:
797:
792:
789:
784:
782:
774:
766:
762:
754:
751:
748:
745:
742:
740:
732:
724:
720:
707:
694:
690:
685:
680:
676:
671:
667:
652:
650:
646:
642:
638:
634:
629:
616:
610:
604:
601:
598:
595:
589:
586:
581:
577:
571:
568:
555:
552:
542:
539:
536:
532:
528:
523:
519:
508:
504:
499:
495:
491:
486:
481:
477:
473:
469:
465:
461:
457:
438:
435:
432:
410:
407:
404:
400:
393:
390:
387:
381:
376:
373:
370:
366:
362:
357:
353:
332:
329:
324:
320:
316:
311:
307:
299:
298:
297:
295:
291:
285:
274:
271:
267:
263:
259:
254:
249:
243:
237:
233:
229:
224:
222:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
177:
172:
167:
162:
158:
153:
152:reciprocation
148:
144:
139:
135:
130:
128:
118:
111:
106:
94:
90:
86:
82:
78:
77:
76:
74:
67:
63:
59:
55:
51:
42:
38:
34:
28:
22:
3390:
3387:"Involution"
3357:
3350:Rost, Markus
3320:math/0506034
3310:
3306:
3287:
3277:
3264:
3253:
3245:
3233:
3228:
3220:
3212:
3156:
3150:
3146:
3137:
3118:
3109:
3089:
3082:
3063:
3056:
3040:
3035:
3011:Automorphism
2994:(BGR)) = BGR
2991:
2987:
2983:
2979:
2972:
2968:
2964:
2957:
2951:
2944:
2940:
2936:
2932:
2925:
2910:
2896:
2864:
2858:), IMTL and
2821:
2801:truth values
2796:
2794:
2788:
2782:
2774:
2764:
2745:
2739:
2733:
2727:
2723:
2719:
2715:
2711:
2709:(where
2705:
2701:
2695:
2685:
2679:
2676:
2669:
2658:
2653:
2649:
2643:
2639:
2630:
2624:
2620:
2614:
2610:
2604:
2592:group theory
2589:
2586:Group theory
2557:
2551:
2521:
2517:
2513:
2501:
2408:
2085:
2061:
2047:
2041:
2034:
2027:
2021:
2018:endomorphism
2013:
2007:
1998:
1991:
1969:
1963:
1957:
1951:
1945:
1939:
1933:
1929:
1925:
1921:
1912:
1903:
1901:, and sends
1894:
1885:
1879:
1870:
1861:
1855:
1852:
1847:
1834:
1829:
1822:
1819:
1799:
1797:
1791:
1772:
1768:
1750:projectivity
1747:
1735:
1728:
1713:
1701:
1115:
1108:
1102:
1098:
1087:
877:
870:
705:
692:
688:
683:
678:
674:
663:
630:
506:
502:
453:
283:
280:
269:
265:
261:
257:
247:
241:
235:
231:
225:
212:such as the
188:, half-turn
175:
170:
160:
156:
146:
142:
134:identity map
131:
124:
109:
98:
92:
88:
84:
80:
61:
57:
53:
47:
40:
36:
32:
3217:J. V. Field
3016:Idempotence
2913:bitwise NOT
2848:BL-algebras
2844:MV-algebras
2813:fuzzy logic
2683:of a group
2677:An element
2661:permutation
2556:, the word
2554:ring theory
2542:Ring theory
2072:C*-algebras
1949:. That is,
1883:that sends
1804:correlation
1784:fixed point
1172:, such as:
645:fixed point
228:composition
50:mathematics
3427:Categories
3417:Involution
3380:0955.16001
3159:(2): 144,
3143:Zagier, D.
3049:0321307143
3027:References
2797:involutive
2753:and their
2699:with
2689:is called
2654:involution
2608:such that
2558:involution
2057:correspond
1802:that is a
1773:Collection
1722:through a
1720:reflection
706:involution
641:odd number
482:(sequence
206:set theory
186:reflection
182:arithmetic
54:involution
3397:EMS Press
2921:keystream
2883:inclusion
2548:*-algebra
2536:transpose
2293:λ
2281:λ
2101:↦
1975:transpose
1765:Desargues
1643:−
1603:−
1555:
1487:
1471:
1402:−
1351:−
1313:
1247:−
1208:−
1157:∘
1151:∘
1143:−
1059:−
994:−
966:∘
925:∘
843:−
749:−
670:symmetric
599:−
561:⌋
548:⌊
533:∑
408:−
391:−
374:−
296:in 1800:
292:found by
127:bijection
3356:(1998),
3337:45639619
3117:(1973),
3051:, p. 165
3043:, 2006,
3000:See also
2881:through
2771:negation
2628:, where
1937:for all
1844:−1
1800:polarity
218:Beaufort
198:geometry
190:rotation
138:negation
99:for all
66:function
35: :
3399:, 2001
3252:(1969)
3181:1041893
3173:2323918
3130:0445948
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