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over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements. A
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377:
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235:
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of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra.
1328:
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1166:
Ring theory and algebraic geometry. Proceedings of the 5th international conference on algebra and algebraic geometry, SAGA V, León, Spain
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if the structure constants in the linear form are all non-negative. An evolution algebra is necessarily commutative and
1678:
1459:
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Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil
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168:
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21:
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In applications to genetics, these algebras often have a basis corresponding to the genetically different
688:
1655:
Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike)
1629:
641:
who showed that special train algebras are genetic algebras and genetic algebras are train algebras.
310:
106:
75:
1549:
541:
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24:) algebra used to model inheritance in genetics. Some variations of these algebras are called
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1168:, Lect. Notes Pure Appl. Math., vol. 221, New York, NY: Marcel Dekker, pp. 223–239,
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is a finite-dimensional real algebra for which all structure constants lie between 0 and 1.
610:
277:
1621:
1561:
1507:
1407:
1347:
1320:
1297:
Bernstein, S. N. (1923), "Principe de stationarité et généralisation de la loi de Mendel",
1190:. Lect. Notes Pure Appl. Math. Vol. 211. New York, NY: Marcel Dekker. pp. 35–42.
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González, S.; Martínez, C. (2001), "About
Bernstein algebras", in Granja, Ángel (ed.),
1672:
1315:, Mémorial des Sciences Mathématiques, Fasc. 162, Gauthier-Villars Éditeur, Paris,
1490:
1186:
Catalan, A. (2000). "E-ideals in
Bernstein algebras". In Costa, Roberto (ed.).
1383:
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1391:
1253:
1227:
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1476:
Reed, Mary Lynn (1997), "Algebraic structure of genetic inheritance",
71:
46:
1537:
1602:, Lecture Notes in Biomathematics, vol. 36, Berlin, New York:
660:
of the weight function is nilpotent and the principal powers of
656:
A special train algebra is a baric algebra in which the kernel
1025:{\displaystyle a^{n}+c_{1}w(a)a^{n-1}+\cdots +c_{n}w(a)^{n}=0}
1522:
Schafer, Richard D. (1949), "Structure of genetic algebras",
895:{\displaystyle x^{n}+c_{1}w(x)x^{n-1}+\cdots +c_{n}w(x)^{n}}
145:
in genetics, is a (possibly non-associative) baric algebra
101:
Baric algebras (or weighted algebras) were introduced by
1572:, Lecture Notes in Mathematics, vol. 1921, Berlin:
1370:
Etherington, I. M. H. (1941), "Special train algebras",
670:
showed that special train algebras are train algebras.
605:
evolution algebra is one defined over the reals: it is
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301:
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229:
682:, section 4) as special cases of baric algebras.
653:, section 4) as special cases of baric algebras.
1313:Algèbres non associatives et algèbres génétiques
113:is a possibly non-associative algebra over
64:
121:, called the weight, from the algebra to
59:). The study of these algebras was started by
1478:Bulletin of the American Mathematical Society
1234:. Springer Undergraduate Mathematics Series.
86:
8:
1424:"Bernstein problem in mathematical genetics"
509:
476:
449:{\displaystyle U_{e}=\{a\in \ker w:ea=a/2\}}
443:
402:
372:{\displaystyle B=Ke\oplus U_{e}\oplus Z_{e}}
1139:
679:
667:
650:
649:Special train algebras were introduced by
585:
102:
515:{\displaystyle Z_{e}=\{a\in \ker w:ea=0\}}
133:A Bernstein algebra, based on the work of
1570:Evolution algebras and their applications
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1151:
786:{\displaystyle 1+c_{1}+\cdots +c_{n}=0}
638:
522:. Although these subspaces depend on
237:. Every such algebra has idempotents
1138:Zygotic algebras were introduced by
584:Copular algebras were introduced by
7:
1372:The Quarterly Journal of Mathematics
637:Genetic algebras were introduced by
90:
81:For surveys of genetic algebras see
1657:(in Russian), Kiev: Naukova Dumka,
724:{\displaystyle c_{1},\ldots ,c_{n}}
678:Train algebras were introduced by
613:but not necessarily associative or
117:together with a homomorphism
14:
1598:Wörz-Busekros, Angelika (1980),
1525:American Journal of Mathematics
135:Sergei Natanovich Bernstein
1327:Etherington, I. M. H. (1939),
1114:
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1007:
1000:
959:
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835:
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538:Bernstein algebra is one with
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208:
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172:
1:
1491:10.1090/S0273-0979-97-00712-X
1088:defined as principal powers,
1271:-ideals in baric algebras".
16:In mathematical genetics, a
1635:Encyclopedia of Mathematics
1628:Wörz-Busekros, A. (2001) ,
1568:Tian, Jianjun Paul (2008),
1465:Encyclopedia of Mathematics
1447:Encyclopedia of Mathematics
1429:Encyclopedia of Mathematics
1267:Catalán S., Abdón (1994). "
1232:Introduction to Ring Theory
569:{\displaystyle U_{e}^{2}=0}
153:with a weight homomorphism
1700:
1311:Bertrand, Monique (1966),
1123:{\displaystyle (a^{k-1})a}
793:. The formal polynomial
105:. A baric algebra over a
731:be elements of the field
1684:Non-associative algebras
1422:Lyubich, Yu.I. (2001) ,
1653:Lyubich, Yu.I. (1983),
1384:10.1093/qmath/os-12.1.1
1336:Proc. R. Soc. Edinburgh
267:{\displaystyle e=a^{2}}
1299:C. R. Acad. Sci. Paris
1124:
1082:
1055:
1054:{\displaystyle a\in B}
1026:
917:is a train algebra if
896:
787:
725:
645:Special train algebras
570:
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450:
373:
303:
302:{\displaystyle w(a)=1}
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30:special train algebras
1125:
1083:
1081:{\displaystyle a^{k}}
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909:. The baric algebra
897:
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571:
517:
451:
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1600:Algebras in genetics
1458:Micali, A. (2001) ,
1440:Micali, A. (2001) ,
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311:Peirce decomposition
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87:Wörz-Busekros (1980)
1679:Population genetics
1460:"Bernstein algebra"
559:
76:structure constants
1329:"Genetic algebras"
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892:
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668:Etherington (1941)
592:Evolution algebras
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143:Hardy–Weinberg law
129:Bernstein algebras
103:Etherington (1939)
38:Bernstein algebras
1630:"Genetic algebra"
1613:978-0-387-09978-1
1583:978-3-540-74283-8
1374:, Second Series,
1140:Etherington (1939
1035:for all elements
680:Etherington (1939
651:Etherington (1939
615:power-associative
598:evolution algebra
586:Etherington (1939
317:corresponding to
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1358:, archived from
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1217:Tian (2008) p.20
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1208:Tian (2008) p.18
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1134:Zygotic algebras
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633:Genetic algebras
621:Gametic algebras
580:Copular algebras
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61:Ivor Etherington
57:weighted algebra
42:copular algebras
34:gametic algebras
1699:
1698:
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1647:Further reading
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1604:Springer-Verlag
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1574:Springer-Verlag
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1538:10.2307/2372100
1521:
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1442:"Baric algebra"
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627:gametic algebra
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83:Bertrand (1966)
22:non-associative
20:is a (possibly
18:genetic algebra
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1484:(2): 107–130,
1480:, New Series,
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1238:. p. 56.
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674:Train algebras
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639:Schafer (1949)
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97:Baric algebras
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1365:on 2011-07-06
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55:(also called
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1360:the original
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1273:Mat. Contemp
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1268:
1262:
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1034:
914:
913:with weight
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904:
732:
684:
677:
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664:are ideals.
661:
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607:non-negative
606:
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241:of the form
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15:
1532:: 121–135,
1342:: 242–258,
536:exceptional
165:satisfying
91:Reed (1997)
1673:Categories
1663:0593.92011
1592:1136.17001
1516:0876.17040
1416:0027.29401
1400:67.0093.04
1356:0027.29402
1285:0868.17023
1245:1852332069
1196:0968.17013
1174:1005.17021
1146:References
141:) on the
74:, and the
1640:EMS Press
1546:0002-9327
1500:0002-9904
1470:EMS Press
1452:EMS Press
1434:EMS Press
1392:0033-5606
1305:: 581–584
1254:1615-2085
1107:−
1046:∈
982:⋯
971:−
858:⋯
847:−
762:⋯
706:…
489:
483:∈
415:
409:∈
357:⊕
344:⊕
1279:: 7–12.
1230:(2000).
611:flexible
49:algebras
1622:0599179
1562:0027751
1554:2372100
1508:1414973
1408:0005111
1378:: 1–8,
1348:0000597
1321:0215885
1061:, with
309:. The
137: (
72:gametes
63: (
47:zygotic
1661:
1620:
1610:
1590:
1580:
1560:
1552:
1544:
1514:
1506:
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1406:
1398:
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1354:
1346:
1319:
1283:
1252:
1242:
1194:
1172:
534:. An
382:where
109:
51:, and
1550:JSTOR
1363:(PDF)
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