106:
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1402:, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the
1338:) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane
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defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.
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example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract
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These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In
793:
This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case
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that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the
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The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.
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In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a
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Given that there is a single I in the starting string MI, and one that is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).
341:. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all
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of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the
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1595:, a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.
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is invariant under some transformations. This one is invariant under horizontal and vertical translation, as well as rotation by 180° (but not under reflection).
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is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying
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of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.
707:. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:
58:. In particular, all this article confuses "invariance" (a property) and "an invariant" (a mathematical object that is left invariant under a
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on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.
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214:. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an
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tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the
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More sophisticated invariants generally have to be provided manually. In particular, when verifying an imperative program using
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Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the
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of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.
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to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a
249:. The discovery of invariants is an important step in the process of classifying mathematical objects.
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2436:, S.D. Swierstra (1991). "Iteratie en invariatie", Programmeren en Correctheid. Academic Service.
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For example, triangles such that all three sides are equal are congruent under rigid motions, via
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that are invariant under the transformation. They may, depending on the application, be called
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is a condition that is true at the beginning and the end of every iteration of a loop.
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is a good example of a logical problem where determining an invariant is of use for an
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of objects of any kind, there is a number to which we always arrive, regardless of the
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does not have this same property, as distance is not invariant under multiplication.
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The notion of invariance is formalized in three different ways in mathematics: via
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is an equation that remains true for all values of its variables. There are also
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Thirdly, if one is studying an object which varies in a family, as is common in
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Douc, Randal; Moulines, Eric; Priouret, Pierre; Soulier, Philippe (2018).
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An example derivation (with superscripts indicating the applied rules) is
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are similar: they can be transformed into each other and the ratio of the
2260:"Invariant Synthesis for Programs Manipulating Lists with Unbounded Data"
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used. Typical example properties are single integer variable ranges like
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that is always held to be true during a certain phase of execution of a
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are unchanged, "invariant" under the group action, or under an element
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MI → MII → MIIII → MUI → MUIUI → MUIUIU → MUIUIUUIUIU → MUIUIIUIU → ...
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1846:// computed invariant: ICount % 3 == 1 || ICount % 3 == 2
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in terms of coordinate charts – invariants must be unchanged under
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Number of I's is unchanged. If the invariant held, it still does.
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Number of I's is unchanged. If the invariant held, it still does.
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Any three consecutive I's (III) may be replaced with a single U (
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For other uses of the word "invariant" in computer science, see
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with respect to that transformation. For example, objects with
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cannot be 0, and hence the "while"-loop will never terminate.
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A simple example of invariance is expressed in our ability to
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that remain true when the values of their variables change.
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Invariants are used in diverse areas of mathematics such as
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Property that is not changed by mathematical transformations
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1954:"Invariant Definition (Illustrated Mathematics Dictionary)"
354:
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Invariants are especially useful when reasoning about the
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do not change with rotation of the coordinate system (see
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are defined as transformations of the plane that preserve
190:
of mathematical objects) which remains unchanged after
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The number of I's in the string is not a multiple of 3
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The string after the M may be completely duplicated (M
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the same quantity to both numbers. On the other hand,
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Automatic invariant detection in imperative programs
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2559:"Applet: Visual Invariants in Sorting Algorithms"
2254:Bouajjani, A.; Drǎgoi, C.; Enea, C.; Rezine, A.;
1406:shows that this is a complete set of invariants.
1322:Frequently one will have a group acting on a set
711:If a string ends with an I, a U may be appended (
1603:interpretation tool will be able to detect that
1550:in their code to make invariants explicit. Some
1176:{\displaystyle x\in S\Leftrightarrow T(x)\in S.}
424:is invariant under many mathematical operations.
398:is invariant under a linear change of variables.
2415:Introduction To Modern Algebra, Revised Edition
522:are invariant under orthogonal transformations.
547:is unchanged after the addition of a constant.
2295:"An axiomatic basis for computer programming"
1307:on a mathematical object (or set of objects)
869:−3 is not either. The invariant still holds.
782:following invariant interesting to consider:
353:is invariant (denoted by the Greek letter π (
8:
2125:Representations of Finite and Compact Groups
2097:Gödel, Escher, Bach: An Eternal Golden Braid
554:of a transformation are the elements in the
409:of a topological object are invariant under
68:. There might be a discussion about this on
2217:
1580:, relations between several variables like
1253:. Unsourced material may be challenged and
444:Equiareal map § Linear transformations
2533:Probability Theory: A comprehensive course
951:
947:
847:is not either. The invariant still holds.
2274:
2128:. American Mathematical Soc. p. 16.
2009:"Invariant – Encyclopedia of Mathematics"
1273:Learn how and when to remove this message
1188:
1138:
1091:, in which case the eigenvectors span an
976:{\displaystyle x\in S\implies T(x)\in S.}
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670:
634:
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589:
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566:are invariant under certain translations.
150:Learn how and when to remove this message
88:Learn how and when to remove this message
2193:
2181:
2068:
1465:, as discussed for Euler characteristic.
798:
752:Any two consecutive U's may be removed (
113:This article includes a list of general
1945:
539:is invariant under translations of the
218:is a property that is constant on each
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1019:about the circle's center. Further, a
427:Euclidean distance is invariant under
2205:
1557:have a special syntax for specifying
1364:Dual to the notion of invariants are
1203:is measurable, invariant sets form a
500:is invariant under a change of basis.
269:in which we count the objects in the
7:
2003:
2001:
1975:
1973:
1913:Mathematical constants and functions
1251:adding citations to reliable sources
1114:is an invariant line, though if the
1007:. (Some authors use the terminology
2080:
2041:Knot Theory Week 2: Tricolorability
1357:More importantly, one may define a
2351:A First Course In Abstract Algebra
1543:, all rely heavily on invariants.
1289:, presentations, and deformation.
661:is invariant under changes of the
434:Euclidean area is invariant under
119:it lacks sufficient corresponding
25:
2032:Qiao, Xiaoyu (January 20, 2015).
1525:correctness of a computer program
1030:An invariant set of an operation
317:of distances are invariant under
2093:Hofstadter, Douglas R. (1999) ,
1223:
528:is invariant under translations.
360:Some more complicated examples:
333:. These transformations produce
206:is an invariant with respect to
104:
41:
1872:Invariant differential operator
1584:, and modulus information like
1375:which formalizes the notion of
843:is not a multiple of 3, then 2×
1493:Invariants in computer science
1446:The most common examples are:
1311:then one may ask which points
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1155:
1149:
1023:is invariant as a set under a
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955:
948:
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599:{\textstyle \int _{M}K\,d\mu }
337:shapes, which is the basis of
1:
2488:Billingsley, Patrick (1995).
2153:A Course in Modern Geometries
2377:Blaisdell Publishing Company
1598:In the context of the above
1477:Unchanged under perturbation
1293:Unchanged under group action
503:The principal invariants of
2477:Encyclopedia of Mathematics
2276:10.1007/978-3-642-14295-6_8
1410:Independent of presentation
1391:, one might seek to find a
2598:
2568:by William Braynen in 1997
2399:Holt, Rinehart and Winston
2349:Fraleigh, John B. (1976),
2013:www.encyclopediaofmath.org
1499:invariant (disambiguation)
1496:
1452:presentation of a manifold
1393:complete set of invariants
1087:is an invariant set under
1053:that are stable under the
606:of the Gaussian curvature
543:. Hence the variance of a
451:projective transformations
429:orthogonal transformations
32:Invariant (disambiguation)
29:
2494:. John Wiley & Sons.
2353:(2nd ed.), Reading:
2303:Communications of the ACM
2149:Judith Cederberg (1989).
1933:Young–Deruyts development
1045:that are so important in
457:of three or more points,
2582:Mathematical terminology
2371:Herstein, I. N. (1964),
1609:
1404:AAA similarity criterion
1079:, then the line through
865:is not a multiple of 3,
537:probability distribution
461:of three or more lines,
295:between two points on a
2491:Probability and Measure
2413:McCoy, Neal H. (1968),
1923:Symmetry in mathematics
1693:// non-terminating loop
1570:Abstract interpretation
1470:presentation of a group
1463:manifold decompositions
1438:in which case it is an
1389:classification problems
1209:invariant sigma-algebra
931:under the mapping when
134:more precise citations.
2530:Klenke, Achim (2020).
2393:Kay, David C. (1969),
1546:Programmers often use
1297:Firstly, if one has a
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1177:
1095:which is stable under
995:, even though the set
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679:
655:
620:
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564:translational symmetry
496:. In other words, the
492:are invariant under a
396:degree of a polynomial
171:
2316:10.1145/363235.363259
2034:"Tricolorability.pdf"
1986:mathworld.wolfram.com
1928:Topological invariant
1908:Mathematical constant
1893:Invariants of tensors
1555:programming languages
1531:, the methodology of
1487:differential geometry
1456:change of coordinates
1198:
1178:
1122:has no fixed points.
1067:linear transformation
978:
680:
656:
654:{\displaystyle (M,g)}
626:of a two-dimensional
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509:Invariants of tensors
165:
2157:. Springer. p.
1529:optimizing compilers
1508:, an invariant is a
1416:Euler characteristic
1247:improve this section
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1137:
1013:pointwise invariant,
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811:Effect on invariant
688:Gauss–Bonnet theorem
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633:
610:
573:
498:spectrum of a matrix
376:are invariant under
239:discrete mathematics
216:equivalence relation
54:confusing or unclear
30:For other uses, see
2071:, pp. 166–167)
1980:Weisstein, Eric W.
1888:Invariant (physics)
1877:Invariant estimator
1541:program correctness
1055:inner automorphisms
1041:. For example, the
1034:is also said to be
705:impossibility proof
628:Riemannian manifold
490:linear endomorphism
449:Some invariants of
378:complex conjugation
184:mathematical object
182:is a property of a
66:clarify the article
2564:2022-02-24 at the
2450:Weisstein, Eric W.
2230:Douc et al. (2018)
2220:, pp. 313–314
2218:Billingsley (1995)
1958:www.mathsisfun.com
1593:the Hoare calculus
1582:0<=i-j<2*n-1
1533:design by contract
1483:algebraic geometry
1193:
1173:
1127:probability theory
1108:screw displacement
1093:invariant subspace
1009:setwise invariant,
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675:
651:
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299:is not changed by
172:
2543:978-3-030-56401-8
2522:978-3-319-97703-4
2373:Topics In Algebra
2244:, p. 494-495
2168:978-1-4757-3831-5
2135:978-0-8218-7196-6
1883:Invariant measure
1516:. For example, a
1510:logical assertion
1424:topological space
1283:
1282:
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1196:{\displaystyle T}
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678:{\displaystyle g}
663:Riemannian metric
619:{\displaystyle K}
273:. The quantity—a
220:equivalence class
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16:(Redirected from
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2395:College Geometry
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2326:. Archived from
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2293:(October 1969).
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1574:abstract domains
1559:class invariants
1539:for determining
1527:. The theory of
1514:computer program
1506:computer science
1468:Invariants of a
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1271:
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1215:Formal statement
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1043:normal subgroups
999:is fixed in the
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526:Lebesgue measure
422:dynamical system
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135:
130:this article by
121:inline citations
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2566:Wayback Machine
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2419:Allyn and Bacon
2412:
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2387:
2370:
2365:
2348:
2344:
2339:
2338:
2330:
2310:(10): 576–580.
2297:
2291:Hoare, C. A. R.
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2101:, Basic Books,
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2087:
2083:, pp. 219)
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2053:
2051:
2050:on May 25, 2024
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1867:Graph invariant
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1585:
1581:
1578:0<=x<1024
1577:
1567:
1552:object oriented
1502:
1495:
1479:
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1295:
1279:
1268:
1262:
1259:
1244:
1228:
1217:
1185:
1184:
1135:
1134:
1057:of the ambient
1021:conical surface
933:
932:
902:
794:for all rules:
697:
686:. This is the
667:
666:
631:
630:
608:
607:
576:
571:
570:
545:random variable
516:singular values
494:change of basis
407:homology groups
385:tricolorability
275:cardinal number
255:
212:Euclidean plane
196:transformations
156:
145:
139:
136:
126:Please help to
125:
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2554:
2553:External links
2551:
2549:
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2521:
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2500:
2485:
2464:
2445:
2430:
2410:
2390:
2386:978-1114541016
2385:
2368:
2363:
2355:Addison-Wesley
2345:
2343:
2340:
2337:
2336:
2333:on 2016-03-04.
2282:
2256:Sighireanu, M.
2246:
2234:
2222:
2210:
2208:, p. 183)
2198:
2194:Herstein (1964
2186:
2184:, p. 103)
2182:Fraleigh (1976
2174:
2167:
2141:
2134:
2114:
2107:
2085:
2073:
2069:Fraleigh (1976
2061:
2024:
1997:
1969:
1944:
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1941:
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1930:
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1915:
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1905:
1903:Knot invariant
1900:
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1880:
1874:
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1858:
1856:
1853:
1610:
1566:
1563:
1537:formal methods
1518:loop invariant
1494:
1491:
1478:
1475:
1474:
1473:
1466:
1459:
1411:
1408:
1400:SSS congruence
1371:also known as
1319:of the group.
1294:
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1231:
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1142:
1131:ergodic theory
1063:linear algebra
983:Note that the
972:
969:
966:
963:
960:
957:
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950:
946:
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940:
911:of the domain
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463:conic sections
447:
432:
425:
416:The number of
414:
399:
392:
381:
374:complex number
370:absolute value
305:multiplication
254:
251:
243:conformal maps
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2514:
2513:
2512:Markov Chains
2507:
2503:
2501:0-471-00710-2
2497:
2493:
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2460:
2459:
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2446:
2443:
2442:90-6233-681-7
2439:
2435:
2432:J.D. Fokker,
2431:
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2400:
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2364:0-201-01984-1
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2247:
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2242:Klenke (2020)
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2199:
2196:, p. 42)
2195:
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2127:
2126:
2122:Barry Simon.
2118:
2115:
2110:
2108:0-465-02656-7
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2025:
2014:
2010:
2004:
2002:
1998:
1987:
1983:
1976:
1974:
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1959:
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1949:
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1906:
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1901:
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1894:
1891:
1889:
1886:
1884:
1881:
1879:in statistics
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1875:
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1471:
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1460:
1457:
1453:
1449:
1448:
1447:
1444:
1441:
1440:intrinsically
1437:
1436:presentation,
1434:of choice of
1433:
1429:
1425:
1421:
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1292:
1290:
1288:
1287:group actions
1277:
1274:
1266:
1263:February 2010
1256:
1252:
1248:
1242:
1241:
1237:
1232:This section
1230:
1226:
1221:
1220:
1214:
1212:
1210:
1206:
1205:sigma-algebra
1190:
1183:When the map
1170:
1167:
1164:
1158:
1152:
1146:
1143:
1140:
1132:
1128:
1123:
1121:
1118:is non-zero,
1117:
1113:
1109:
1105:
1100:
1098:
1094:
1090:
1086:
1082:
1078:
1075:
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998:
994:
990:
986:
970:
967:
964:
958:
952:
944:
941:
938:
930:
929:invariant set
926:
922:
918:
915:of a mapping
914:
910:
907:
900:Invariant set
899:
897:
893:
882:
879:
876:
873:
872:
868:
864:
860:
857:
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851:
850:
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838:
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829:
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815:
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759:
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748:
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736:
732:
729:
725:
721:
718:
714:
710:
709:
708:
706:
702:
694:
689:
685:
672:
664:
645:
642:
639:
629:
613:
593:
590:
586:
581:
577:
569:The integral
568:
565:
561:
557:
553:
549:
546:
542:
538:
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530:
527:
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517:
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460:
456:
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430:
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419:
415:
412:
411:homeomorphism
408:
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386:
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375:
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361:
358:
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348:
347:circumference
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102:
101:
92:
89:
81:
71:
70:the talk page
67:
61:
57:
55:
50:This article
48:
39:
38:
33:
19:
18:Invariant set
2536:. Springer.
2532:
2515:. Springer.
2511:
2490:
2475:
2456:
2414:
2397:, New York:
2394:
2372:
2350:
2328:the original
2307:
2301:
2285:
2266:
2249:
2237:
2232:, p. 99
2225:
2213:
2201:
2189:
2177:
2152:
2144:
2124:
2117:
2096:
2088:
2076:
2064:
2052:. Retrieved
2045:the original
2040:
2027:
2016:. Retrieved
2012:
1989:. Retrieved
1985:
1961:. Retrieved
1957:
1948:
1597:
1590:
1568:
1545:
1522:
1503:
1480:
1445:
1439:
1435:
1431:
1420:cell complex
1413:
1397:
1385:
1372:
1367:coinvariants
1365:
1363:
1358:
1356:
1352:rigid motion
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1321:
1316:
1312:
1308:
1301:
1296:
1284:
1269:
1260:
1245:Please help
1233:
1124:
1119:
1103:
1101:
1096:
1088:
1084:
1080:
1076:
1069:
1047:group theory
1038:
1036:stable under
1035:
1031:
1029:
1012:
1008:
1004:
996:
988:
928:
924:
920:
916:
912:
908:
903:
894:
890:
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862:
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767:
761:
757:
753:
746:
742:
738:
734:
727:
723:
716:
712:
698:
665:
552:fixed points
482:eigenvectors
455:collinearity
418:fixed points
359:
339:trigonometry
327:translations
309:
290:
286:inequalities
279:
256:
224:
179:
173:
146:
137:
118:
84:
78:January 2024
75:
64:Please help
60:group action
51:
2472:"Invariant"
2468:Popov, V.L.
2453:"Invariant"
2375:, Waltham:
2206:McCoy (1968
1982:"Invariant"
1432:independent
1074:eigenvector
486:eigenvalues
474:determinant
467:cross-ratio
459:concurrency
440:determinant
438:which have
436:linear maps
331:reflections
297:number line
176:mathematics
132:introducing
2434:H. Zantema
2417:, Boston:
2342:References
2018:2019-12-05
1991:2019-12-05
1963:2019-12-05
1702:RandomRule
1636:RandomRule
1548:assertions
1377:congruence
1350:); then a
1328:associated
1112:screw axis
1049:are those
1027:of space.
465:, and the
263:finite set
208:isometries
192:operations
140:April 2015
115:references
56:to readers
2482:EMS Press
2470:(2001) ,
2458:MathWorld
2324:207726175
2267:Proc. CAV
2081:Kay (1969
1600:MU puzzle
1381:perimeter
1234:does not
1165:∈
1150:⇔
1144:∈
1051:subgroups
1025:homothety
1001:power set
965:∈
949:⟹
942:∈
701:MU puzzle
695:MU puzzle
594:μ
578:∫
560:symmetric
541:real line
403:dimension
366:real part
323:rotations
180:invariant
168:wallpaper
2576:Category
2562:Archived
2427:68-15225
2407:69-12075
2258:(2010).
1855:See also
1630:volatile
1615:MUPuzzle
1605:ICount%3
1461:Various
1428:manifold
1359:function
1017:rotation
991:are not
985:elements
779:property
533:variance
453:include
442:±1 (see
368:and the
351:diameter
319:scalings
293:distance
282:identity
261:. For a
253:Examples
231:topology
227:geometry
204:triangle
2054:May 25,
1373:orbits,
1255:removed
1240:sources
1072:has an
1065:, if a
505:tensors
349:to the
343:circles
335:similar
235:algebra
210:of the
128:improve
52:may be
2540:
2519:
2498:
2440:
2425:
2405:
2383:
2361:
2322:
2165:
2132:
2105:
1825:UCount
1798:UCount
1786:ICount
1759:UCount
1747:ICount
1720:UCount
1696:switch
1675:ICount
1657:UCount
1645:ICount
1586:y%4==0
1535:, and
1305:acting
1207:, the
1110:, the
927:is an
906:subset
556:domain
520:matrix
484:, and
315:ratios
311:Angles
301:adding
247:angles
186:(or a
117:, but
2331:(PDF)
2320:S2CID
2298:(PDF)
2263:(PDF)
2048:(PDF)
2037:(PDF)
1940:Notes
1837:break
1810:break
1771:break
1732:break
1669:while
1426:(the
1418:of a
1299:group
1116:pitch
1106:is a
1102:When
1061:. In
1059:group
993:fixed
535:of a
518:of a
488:of a
478:trace
420:of a
389:knots
372:of a
267:order
259:count
202:of a
188:class
178:, an
2538:ISBN
2517:ISBN
2496:ISBN
2438:ISBN
2423:LCCN
2403:LCCN
2381:ISBN
2359:ISBN
2163:ISBN
2130:ISBN
2103:ISBN
2056:2024
1816:case
1777:case
1738:case
1711:case
1621:void
1612:void
1485:and
1450:The
1330:set
1238:any
1236:cite
1129:and
1083:and
1011:vs.
808:#U's
805:#I's
802:Rule
715:I →
699:The
550:The
531:The
514:The
472:The
405:and
401:The
394:The
383:The
364:The
357:)).
329:and
313:and
291:The
237:and
200:area
2312:doi
2271:doi
2159:174
1642:int
1633:int
1504:In
1342:as
1249:by
1125:In
1003:of
987:of
861:If
839:If
737:III
726:→ M
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387:of
280:An
271:set
194:or
174:In
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2480:,
2474:,
2455:.
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2357:,
2318:.
2308:12
2306:.
2300:.
2269:.
2265:.
2161:.
2039:.
2011:.
2000:^
1984:.
1972:^
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1789:-=
1762:*=
1750:*=
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1684:!=
1561:.
1309:X,
1211:.
1099:.
923:→
919::
904:A
880:−2
877:+0
858:+1
855:−3
836:×2
833:×2
822:+1
819:+0
762:xy
760:→
756:UU
741:→
728:xx
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480:,
476:,
446:).
355:pi
325:,
321:,
233:,
229:,
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2546:.
2525:.
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2058:.
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1994:.
1966:.
1849:}
1843:}
1840:;
1834:;
1831:2
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1819:4
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1687:0
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1191:T
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1097:T
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953:T
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
