Ceballos, Manuel; Falcón, Raúl M.; Núñez-Valdés, Juan; Tenorio, Ángel F. A historical perspective of Tian’s evolution algebras. (English) Zbl 1503.17032 Expo. Math. 40, No. 3, 819-843 (2022). An \(n\)-dimensional evolution algebra \(A\) over a field \(K\) is a \(K\)-algebra with a basis \( \{e_j: 1 \leq j \leq n \}\) such that \(e_i e_j=0\) for \(i \neq j\). The family \( \{e_j\}_j\) is called a natural basis of \(A\). These algebras were introduced by J. P. Tian [Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)] motivated by some laws of genetics. Each \(e_j\) corresponds to an allele. Set \(e_i^2= \sum_j p_{ij} e_j^2\) for every \(i\), then the \(p_{ij}\) are called the structural constants of \(A\). If \(K= \mathbb R\), \(p_{ij} \geq 0\) for every \(i, j\) and \(\sum_jp_{ij}=1\) for every \(i\), then \(A\) is called a Markov evolution algebra. In the paper under review the authors survey some main results about the structure and properties of evolution algebras. In particular, they consider the connections with graph theory, Markov chains and some applications in biology. Reviewer: Nadia Boudi (Rabat) Cited in 5 Documents MSC: 17D92 Genetic algebras 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras Keywords:evolution algebras; genetic algebras; historical perspective Citations:Zbl 1136.17001 PDFBibTeX XMLCite \textit{M. Ceballos} et al., Expo. Math. 40, No. 3, 819--843 (2022; Zbl 1503.17032) Full Text: DOI References: [1] Albert, A. A., Non-associative algebras: I. Fundamental concepts and isotopy, Ann. Math., 43, 685-707 (1942) · Zbl 0061.04807 [2] Alsarayreh, A.; Qaralleh, I.; Ahmad, M. Z., Derivation of three dimensional evolution algebra, JP J. Algebra Number Theory Appl., 39, 4, 425-444 (2017) · Zbl 1377.17003 [3] Bateson, W., Mendel’s Principles of Heredity, a Defence (1902), Cambridge University Press: Cambridge University Press Cambridge [4] Bateson, B., William Bateson: Naturalist (1928), Cambridge University Press: Cambridge University Press Cambridge [5] Baur, E., Zeit. Vererbungsl., 1, 330-351 (1909) [6] Bayara, J., Sur Les AlgÈbres D’Évolution (1999), Université de Ouagadougou, (Ph.D. thesis) [7] Bayara, J.; Ouattara, M.; Micali, A.; Ferreira, J. C., Sur une classe d’algèbres d’évolution, Algebras Groups Geom., 19, 3, 315-345 (2002) · Zbl 1167.17310 [8] Behn, A.; Cabrera Casado, Y.; Siles Molina, M., Isomorphisms of four dimensional perfect non-simple evolution algebras, MAMAA 2018: Associative and Non-Associative Algebras and Applications. MAMAA 2018: Associative and Non-Associative Algebras and Applications, Springer Proc. Math. Stat., 311, 3-21 (2020) · Zbl 1459.17055 [9] Bernstein, S., Principe de stationaritè et génèralisation de la loi de Mendel, C. R. Math. Acad. Sci. Paris, 177, 581-584 (1923) · JFM 49.0368.03 [10] Bernstein, S., Solution of a mathematical problem connected with the theory of heredity, Ucheniye-Zapiski N.-I. Kaf. Ukr. Otd. Mat., 1, 83-115 (1924) [11] Bertrand, M., Algèbres non associatives et algèbres gènètiques, (MÈmorial Des Sciences MathÈmatiques Fasc, vol. 162 (1966), Gauthier-Villars Editeur: Gauthier-Villars Editeur Paris) · Zbl 0147.28401 [12] Boudi, N.; Casado, Y. Cabrera.; Siles, M., Natural families in evolution algebras (2020), arXiv preprint (2020) arXiv:2006.14460 [13] Bustamante, M. D.; Mellon, P.; Velasco, M. V., Determining when an algebra is an evolution algebra, Mathematics, 8, 1349 (2020) [14] Bustamante, M. D.; Mellon, P.; Velasco, M. V., Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence, SIAM J. Matrix Anal. Appl., 41, 4, 1616-1629 (2020) · Zbl 1456.15010 [15] Cabrera Casado, Y.; Cadavid, P.; Rodiño Montoya, M.; Rodríguez, P. M., On the characterization of the space of derivations in evolution algebras, Ann. Di Mat. Pura Ed Appl., 200, 737-755 (2021) · Zbl 1481.17059 [16] Cabrera Casado, Y.; Cardoso Gonçalves, M. I.; Gonçalves, D.; Martín Barquero, D.; Martín González, C., Chains in evolution algebras, Linear Algebra Appl., 622, 104-149 (2021) · Zbl 1481.17007 [17] Cabrera Casado, Y.; Kanuni, M.; Siles, M., Basic ideals in evolution algebras, Linear Algebra Appl., 570, 148-180 (2019) · Zbl 1454.17019 [18] Cabrera Casado, Y.; Siles, M.; Velasco, M. V., Classification of three-dimensional evolution algebras, Linear Algebra Appl., 524, 68-108 (2017) · Zbl 1384.17016 [19] Cadavid, P.; Rodiño, M. L.; Rodríguez, P. M., Characterization theorems for the spaces of derivations of evolution algebras associated to graphs, Linear Multilinear Algebra, 68, 1340-1354 (2020) · Zbl 1470.17002 [20] Cadavid, P.; Rodiño, M. L.; Rodríguez, P. M., The connection between evolution algebras, random walks and graphs, J. Algebra Appl., 19, 2, Article 2050023 pp. (2020) · Zbl 1442.17028 [21] Cadavid, P.; Rodiño, M. L.; Rodríguez, P. M., On the isomorphisms between evolution algebras of graphs and random walks, Linear Multilinear Algebra, 69, 10, 1858-1877 (2021) · Zbl 1472.05134 [22] Camacho, L. M.; Gómez, J. R.; Omirov, B. A.; Turdibaev, R. M., The derivations of some evolution algebras, Linear Multilinear Algebra, 61, 3, 309-322 (2013) · Zbl 1311.17001 [23] Camacho, L. M.; Gómez, J. R.; Omirov, B. A.; Turdibaev, R. M., Some properties of evolution algebras, Bull. Korean Math. Soc., 50, 5, 1481-1494 (2013) · Zbl 1278.05120 [24] Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A., On the property of subalgebras of evolution algebras, Algebr. Represent. Theory, 22, 2, 281-296 (2019) · Zbl 1478.17033 [25] Campos, T. M.M.; Holgate, P., Algebraic isotopy in genetics, IMA J. Math. Appl. Med. Biol., 4, 215-222 (1987) · Zbl 0628.92018 [26] Cardoso Gonçalves, M. I.; Gonçalves, D.; Martín Barquero, D.; Martín González, C.; Siles Molina, M., Squares and associative representations of two dimensional evolution algebras, J. Algebra Appl., 20, 6, Article 2150090 pp. (2021) · Zbl 1471.17056 [27] Casado, Y. Cabrera.; Siles, M.; Velasco, M. V., Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra Appl., 495, 122-162 (2016) · Zbl 1395.17079 [28] Casas, J. M.; Ladra, M.; Omirov, B. A.; Rozikov, U. A., On nilpotent index and dibaricity of evolution algebras, Linear Algebra Appl., 439, 1, 90-105 (2013) · Zbl 1343.17021 [29] Casas, J. M.; Ladra, M.; Omirov, B. A.; Rozikov, U. A., On evolution algebras, Algebra Colloq., 21, 2, 331-342 (2014) · Zbl 1367.17026 [30] Casas, J. M.; Ladra, M.; Rozikov, U. A., A chain of evolution algebras, Linear Algebra Appl., 435, 4, 852-870 (2011) · Zbl 1220.17026 [31] Castle, W. E., The laws of galton and mendel and some laws governing race improvement by selection, Proc. Amer. Acad. Arts, 35, 233-242 (1903) [32] Ceballos, M.; Núñez, J.; Tenorio, A. F., Finite-dimensional evolution algebras and (pseudo)digraphs, Math. Methods Appl. Sci. (2020) · Zbl 1485.17004 [33] Celorrio, M. E.; Velasco, M. V., Classifying evolution algebras of dimensions two and three, Mathematics, 7, 1236 (2019) [34] Conseibo, A.; Savadogo, S.; Ouattara, M., Duplicate, Bernstein algebras and evolution algebras, Matematiche (Catania), 76, 1, 193-209 (2021) · Zbl 1487.17056 [35] Correns, C., Gregor Mendels Regel über das Verhalten der Nachkommenschaft der Bastarde, Berichte Der Deutschen Botanischen Gesellschaft, 18, 158-168 (1900) [36] Correns, C., Zeit Vererbungsl., 1, 291-329 (1909) [37] Costa, R., On the derivation algebra of gametic algebras for polyploidy with multiple alleles, Bol. Soc. Brasil. Mat., 13, 69-81 (1982) · Zbl 0575.17013 [38] Costa, R., On the derivation algebra of zygotic algebras for polyploidy with multiple alleles, Bol. Soc. Brasil. Mat., 14, 63-80 (1983) · Zbl 0575.17014 [39] Elduque, A.; Labra, A., Evolution algebras and graphs, J. Algebra Appl., 14, 7, Article 1550103 pp. (2015), 10 · Zbl 1356.17028 [40] Elduque, A.; Labra, A., On nilpotent evolution algebras, Linear Algebra Appl., 505, 11-31 (2016) · Zbl 1386.17006 [41] Elduque, A.; Labra, A., Evolution algebras, automorphisms, and graphs, Linear Multilinear Algebra, 69, 2, 331-342 (2021) · Zbl 1481.17005 [42] Etherington, I. M.H., Genetic algebras, Proc. Roy. Soc. Edinburgh, 59, 242-258 (1939) · JFM 66.1209.01 [43] Etherington, I. M.H., Duplication of linear algebras, Proc. Edinburgh Math. Soc., 6, 2, 222-230 (1941) · Zbl 0061.05302 [44] Falcón, O. J.; Falcón, R. M.; Núñez, J., Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field, J. Algebra, 472, 573-593 (2017) · Zbl 1407.17041 [45] Falcón, O. J.; Falcón, R. M.; Núñez, J., Algebraic computation of genetic patterns related to three-dimensional evolution algebras, Appl. Math. Comput., 319, 510-517 (2018) · Zbl 1430.13048 [46] Falcón, R. M.; Falcón, O. J.; Núñez, J., A historical perspective of the theory of isotopisms, Symmetry, 10, 1-21 (2018) · Zbl 1423.17018 [47] Fisher, R. A., On the dominance ratio, Proc. Roy. Soc. Edinburgh, 52, 321-431 (1922) [48] Ganikhodzhaev, R.; Mukhamedov, F.; Pirnapasov, A.; Qaralleh, I., On genetic Volterra algebras and their derivations, Commun. Algebra, 46, 1353-1366 (2018) · Zbl 1417.17036 [49] Glivenkov, V., Algebra mendelienne, Comptes Rendus de Acad. Des Sci. de URSS, 4, 13, 385-386 (1936) · JFM 62.0622.03 [50] Gonshör, H., Contributions to genetic algebras, Proc. Edinburgh Math. Soc., 17, 289-298 (1971) · Zbl 0247.92002 [51] Gonshör, H., Derivations in genetic algebras, Comm. Algebra, 16, 8, 1525-1542 (1988) · Zbl 0648.17012 [52] González, S.; Gutiérrez, J. C.; Martínez, C., The Bernstein problem in dimension \(5\), J. Algebra, 177, 676-697 (1995) · Zbl 0856.17028 [53] González-López, R.; Núñez, J., Endowing evolution algebras with properties of discrete structures, Port. Math., 77, 3-4, 423-443 (2020) · Zbl 1472.17099 [54] Gutiérrez, J. C., Solution of the Bernstein problem in the non-regular case, J. Algebra, 223, 109-132 (2000) · Zbl 1007.17022 [55] Hardy, G. H., Mendelian proportions in a mixed population, Science, 28, 49-50 (1908) [56] Hegazi, A. S.; Abdelwahab, H., Nilpotent evolution algebras over arbitrary fields, Linear Algebra Appl., 486, 345-360 (2015) · Zbl 1358.17041 [57] Holgate, P., Genetic algebras associated with polyploidy, Proc. Edinb. Math. Soc., 15, 1-9 (1966) · Zbl 0144.27202 [58] Holgate, P., Genetic algebras associated with sex linkage, Proc. Edinburgh Math. Soc., 17, 113-120 (1970), 1970/71 · Zbl 0259.92005 [59] Holgate, P., Characterisations of genetic algebras, J. Lond. Math. Soc., 2, 6, 169-174 (1972) · Zbl 0261.17014 [60] Holgate, P., Genetic algebras satisfying Bernstein’s stationarity principle, J. Lond. Math. Soc., 9, 613-623 (1975) · Zbl 0365.92025 [61] Holgate, P., The interpretation of derivations in genetic algebras, Linear Algebra Appl., 85, 75-79 (1987) · Zbl 0603.92013 [62] Imomkulov, A. N., Isomorphicity of two dimensional evolution algebras and evolution algebras corresponding to their idempotents, Uzbek Math. J., 2, 63-73 (2018) · Zbl 1474.13049 [63] Imomkulov, A. N., Classification of a family of three dimensional real evolution algebras, TWMS J. Pure Appl. Math., 10, 225-238 (2019) · Zbl 1454.13045 [64] Imomkulov, A. N., Behavior and dynamics of the set of absolute nilpotent and idempotent elements of chain of evolution algebras depending on the time, Bull. NUUz: Math Nat. Sc., 3, 4, 2 (2020) [65] Imomkulov, A. N., On absolute nilpotent elements of the evolution algebras corresponding to approximation of finite dimensional algebras, Bull. Inst. Math., 3, 37-41 (2020) [66] Imomkulov, A. N.; Rozikov, U. A., Approximation of an algebra by evolution algebras, Uzbek Math. J., 3, 70-84 (2020) · Zbl 1488.13067 [67] Imomkulov, A. N.; Velasco, M. V., Chain of three-dimensional evolution algebras, Filomat, 34, 10, 3175-3190 (2020) · Zbl 1499.13072 [68] Jiménez-Gestal, C.; Pérez-Izquierdo, J. M., Ternary derivations of generalized Cayley-Dickson algebras, Comm. Algebra, 31, 10, 5071-5094 (2003) · Zbl 1036.17001 [69] Khudoyberdiyev, A. Kh.; Omirov, B. A.; Qaralleh, I., Few remarks on evolution algebras, J. Algebra Appl., 14, 4, Article 1550053 pp. (2015), 16 · Zbl 1378.17052 [70] Kostitzin, V. A., Sur les coefficients mendeliens d’heredite, Comptes Rendus de Acad. Des Sci. de URSS, 206, 883-885 (1938) · JFM 64.0556.04 [71] Ladra, M.; Murodov, S. N., On new classes of chains of evolution algebras, Hacet. J. Math. Stat., 50, 1, 146-158 (2021) · Zbl 1488.17061 [72] Ladra, M.; Omirov, B.; Rozikov, U. A., Dibaric and evolution algebras in biology, Lobachevskii J. Math., 35, 198-210 (2014) · Zbl 1370.17034 [73] Ladra, M.; Rozikov, U. A., Construction of flows of finite-dimensional algebras, J. Algebra, 492, 475-489 (2017) · Zbl 1421.17007 [74] Ladra, M.; Rozikov, U. A., Flow of finite-dimensional algebras, J. Algebra, 470, 263-288 (2017) · Zbl 1406.17048 [75] Lu, J.; Li, J., Dynamics of stage-structured discrete mosquito population models, J. Appl. Anal. Comput., 1, 1, 53-67 (2011) · Zbl 1304.39009 [76] Lyubich, Y. I., Mathematical structures in population genetics (english translation), (Biomathematics, vol. 22 (1992), Springer: Springer Berlin) · Zbl 0747.92019 [77] Martín Barquero, D.; Martín González, C.; Sánchez-Ortega, J.; Vandeyar, M., Ternary mappings of triangular algebra, Aequ. Math., 95, 841-865 (2021) · Zbl 1493.16046 [78] Mendel, G., Versuche über Pflanzenhybriden, (Verhandlungen des Naturforschenden Vereines in Brünn, Vol. 4 (1866)), 3-47 [79] Micali, A.; Revoy, P., Sur les algèbres gamétiques, Proc. Edinburgh Math. Soc, 29, 2, 187-197 (1986) · Zbl 0572.17014 [80] Mukhamedov, F.; Khakimov, O.; Omirov, B.; Qaralleh, I., Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex, J. Algebra Appl., 18, 12, Article 1950233 pp. (2019), 23 · Zbl 1456.17019 [81] Mukhamedov, F.; Khakimov, O.; Qaralleh, I., Classification of nilpotent evolution algebras and extensions of their derivations, Comm. Algebra, 48, 10, 4155-4169 (2020) · Zbl 1480.17010 [82] Mukhamedov, F.; Qaralleh, I., On derivations of genetic algebras, J. Phys. Conf. Ser., 553, Article 012004 pp. (2014) [83] Murodov, Sh. N., Classification dynamics of two-dimensional chains of evolution algebras, Internat. J. Math., 25, 2, Article 1450012 pp. (2014), 23 · Zbl 1290.17029 [84] N., Murodov Sh., Classification of two-dimensional real evolution algebras and dynamics of some two-dimensional chains of evolution algebras, Uzbek. Mat. Zh., 2014, 2, 102-111 (2014) · Zbl 1290.17029 [85] Narkuziev, B. A., Evolution algebras corresponding to permutations, Uzbek. Mat. Zh., 2014, 4, 109-114 (2014) [86] Núñez, J.; Rodríguez-Arévalo, M. L.; Villar, M. T., Certain particular families of graphicable algebras, Appl. Math. Comput., 246, 1, 416-425 (2014) · Zbl 1338.05123 [87] Núñez, J.; Silvero, M.; Villar, M. T., Mathematical tools for the future: Graph theory and graphicable algebras, Appl. Math. Comput., 219, 11, 6113-6125 (2013) · Zbl 1272.05078 [88] Omirov, B. A.; Rozikov, U. A.; Tulenbayev, K. M., On real chains of evolution algebras, Linear Multilinear Algebra, 63, 3, 586-600 (2015) · Zbl 1361.17030 [89] Omirov, B. A.; Rozikov, U. A.; Velasco, M. V., A class of nilpotent evolution algebras, Comm. Algebra, 47, 4, 1556-1567 (2019) · Zbl 1437.17017 [90] Ouattara, M.; Savadogo, S., Evolution train algebras, Gulf J. Math., 8, 37-51 (2020) · Zbl 1498.17054 [91] Ouattara, M.; Savadogo, S., Power-associative evolution algebras, MAMAA 2018: Associative and Non-Associative Algebras and Applications. MAMAA 2018: Associative and Non-Associative Algebras and Applications, Springer Proc. Math. Stat., 311, 23-49 (2020) · Zbl 1460.17003 [92] Paniello, I., Evolution coalgebras, Linear Multilinear Algebra, 67, 1539-1553 (2019) · Zbl 1446.17046 [93] Paniello, I., Evolution coalgebras on chicken populations, Linear Multilinear Algebra, 68, 3, 528-540 (2020) · Zbl 1437.17018 [94] Paniello, I., Markov evolution algebras, Linear Multilinear Algebra (2020) [95] Pearson, K., Mathematical contributions to the theory of evolution. XI. On the influence of natural selection on the variability and correlation of organs, Philos. Trans. R. Soc. A, 200, 1-66 (1903) · JFM 34.0270.01 [96] Qaralleh, I., On genetic and evolution algebras, J. Phys. Conf. Ser., 819, Article 012011 pp. (2017), 6 [97] Qaralleh, I., A description of derivations of a class of nilpotent evolution algebras, Malays. J. Math. Sci., 14, 2, 295-304 (2020) · Zbl 1498.17055 [98] Qaralleh, I.; Mukhamedov, F., Volterra evolution algebras and their graphs, Linear Multilinear Algebra, 69, 2228-2244 (2021) · Zbl 1500.17028 [99] Reed, M. L., Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc., 34, 107-130 (1997) · Zbl 0876.17040 [100] Rozikov, U. A., Population Dynamics. Algebraic and Probablistic Approach, 443 (2020), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ [101] Rozikov, U. A.; Murodov, Sh. N., Dynamics of two-dimensional evolution algebras, Lobachevskii J. Math., 34, 4, 344-358 (2013) · Zbl 1297.37011 [102] Rozikov, U. A.; Tian, J. P., Evolution algebras generated by gibbs measures, Lobachevskii J. Math., 32, 4, 270-277 (2011) · Zbl 1260.46031 [103] Rozikov, U. A.; Velasco, M. V., A discrete-time dynamical system and an evolution algebra of mosquito population, J. Math. Biol., 78, 4, 1225-1244 (2019) · Zbl 1409.92210 [104] Schafer, R. D., Structure of genetic algebras, Amer. J. Math., 71, 121-135 (1949) · Zbl 0034.02004 [105] Serebrowsky, A., On the properties of the mendelian equations, Doklady A.N.SSSR., 2, 33-36 (1934), (in Russian) · JFM 60.0476.01 [106] Spillman, W. J., Application of Some of the Principles of Heredity To Plant Breeding (1909), U.S. Government Printing Office [107] Sunada, T., L-functions in geometry and some applications, Lecture Notes in Math., 1201, 266-284 (1986) · Zbl 0605.58046 [108] Tian, J. P., Evolution Algebra Theory, 145 (2004), University of California: University of California Riverside, (Thesis Ph.D.) [109] Tian, J. P., Evolution algebras and their applications, (Lecture Notes in Mathematics, Vol. 1921 (2008), Springer-Verlag: Springer-Verlag Berlín) · Zbl 1136.17001 [110] Tian, J. P., Algebraic model of non-mendelian inheritance, Discrete Contin. Dyn. Syst. Ser. S, 4, 6, 1577-1586 (2011) · Zbl 1263.92033 [111] Tian, J. P., Invitation to research of new mathematics from biology: evolution algebras, topics in functional analysis and algebra, Contemp. Math, 672, 257-272 (2016), Amer. Math. Soc · Zbl 1415.17030 [112] Tian, J. P.; Li, B.-L., Coalgebraic structure of genetic inheritance, Math. Biosci. Eng., 1, 2, 243-266 (2004) · Zbl 1061.17027 [113] Tian, J. P.; Lin, X.-S., Colored coalescent theory, Discrete Contin. Dyn. Syst., suppl, 833-845 (2005) · Zbl 1148.60049 [114] Tian, J. P.; Vojtechovsky, P., Mathematical concepts of evolution algebras in non-mendelian genetics, Quasigroups Related Systems, 14, 1, 111-122 (2006) · Zbl 1112.17001 [115] Tian, J. P.; Xiao, S. L., Continuous-time Markov process on graphs, Stoch. Anal. Appl., 24, 5, 953-972 (2006) · Zbl 1113.60071 [116] Tian, J. P.; Zou, Y. M., Finitely generated nil but not nilpotent evolution algebras, J. Algebra Appl., 13, 1, Article 1350070 pp. (2014), 10 · Zbl 1296.17011 [117] Tschermak, E.v., Über Züchtung neuer Getreiderassen mittels künstlicher Kreuzung. I. Kritisch-historische Betrachtungen, Zeitschrift FÜr Das Landwirtschaftliche Versuchswesen in Österreich, 4, 1029-1060 (1901) [118] Velasco, M. V., The jacobson radical of an evolution algebra, J. Spectr. Theory, 9, 2, 601-634 (2019) · Zbl 1415.17031 [119] Vries, H., Species and Varieties: Their Origin By Mutation (1905), The Open court publishing company: The Open court publishing company Chicago [120] Weinberg, W., Über den Nachweis der Vererbung beim Menschen, Jahresh. Ver. Vaterl. Naturkd. Wb., 64, 368-382 (1908) [121] Wörz-Busekros, A., Algebras in genetics, (Lecture Notes in Biomathematics (1980)) · Zbl 0431.92017 [122] Wright, S., Evolution in mendelian populations, Genetics, 16, 97-159 (1931) [123] Yule, G. U., Mendel’s laws and their probable relations to intra-racial heredity, New Phytol., 1, 222-238 (1902) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.