hypercyclicadj.

/ˌhʌɪpəˈsʌɪklɪk/

Etymology: < hyper- prefix 1c(a) + cyclic adj.

1. Algebra. Of a group: containing a nested series of normal subgroups of which the smallest is trivial, such that the quotient groups of consecutive pairs in this series are all cyclic. Also applied to other mathematical entities having similar properties.

ΘΚΠ

the world > relative properties > number > mathematical number or quantity > numerical arrangement > [adjective] > of sets > in abstract algebra > of groups

reducible1585

transitive1861

primitive1888

simple1888

special1888

cyclic1889

intransitive1889

solvable1892

finite1893

perfect1898

Abelian1900

soluble1902

proper1906

trivial1915

equivalent1948

hypercyclic1968

sporadic1968

1968 W. H. Caldwell in Pacific Jrnl. Math. 24 29 The object of this paper is to provide characterizations for certain rings R having the property that each cyclic right R-module has a cyclic injective hull. Such rings will be called hypercyclic.

1969 U. Schoenwaelder in Pacific Jrnl. Math. 31 198 A group G is called hypercyclic, if every epimorphic image, not 1, of G has a cyclic, normal subgroup, not 1.

1975 Mathematische Zeitschr. 144 283 A group G is hypercyclic if and only if G has an ascending series 1 = G₀ ⊆ G₁ ⊆... ⊆ G_λ ⊆ G_λ+1 ⊆... ⊆ G_p = G of normal subgroups G_λ such that for ordinals λ, which are not limit ordinals, G_λ/G_λ-1 is a cyclic group.

1988 Archiv der Math. LI. 13 For minimax groups the properties ‘locally nilpotent’, ‘locally supersoluble’ and ‘locally FC-nilpotent’ coincide with the properties ‘hypercentral’, ‘hypercyclic’ and ‘FC-nilpotent’, respectively.

2. Mathematics. Of the nature of or consisting of a hypercycle or hypercycles (hypercycle n. 2).

ΚΠ

1977 Naturwissenschaften 64 541/2 Only hypercyclic organizations are able to fulfil these requirements.

1981 Sci. Amer. Apr. 86/2 In a hypercyclic system a closed loop of catalytic couplings connects self-replicative cycles.

1986 Jrnl. Theoret. Biol. 118 3 The hypercycle..can be unstable when the fluctuations are anomalously enhanced under the influence of environmental fluctuations on the hypercyclic organization.

This entry has not yet been fully updated (first published 1993; most recently modified version published online June 2018).

单词	hypercyclic
释义	hypercyclicadj. /ˌhʌɪpəˈsʌɪklɪk/ Etymology: < hyper- prefix 1c(a) + cyclic adj. 1. Algebra. Of a group: containing a nested series of normal subgroups of which the smallest is trivial, such that the quotient groups of consecutive pairs in this series are all cyclic. Also applied to other mathematical entities having similar properties. ΘΚΠ the world > relative properties > number > mathematical number or quantity > numerical arrangement > [adjective] > of sets > in abstract algebra > of groups reducible1585 transitive1861 primitive1888 simple1888 special1888 cyclic1889 intransitive1889 solvable1892 finite1893 perfect1898 Abelian1900 soluble1902 proper1906 trivial1915 equivalent1948 hypercyclic1968 sporadic1968 1968 W. H. Caldwell in Pacific Jrnl. Math. 24 29 The object of this paper is to provide characterizations for certain rings R having the property that each cyclic right R-module has a cyclic injective hull. Such rings will be called hypercyclic. 1969 U. Schoenwaelder in Pacific Jrnl. Math. 31 198 A group G is called hypercyclic, if every epimorphic image, not 1, of G has a cyclic, normal subgroup, not 1. 1975 Mathematische Zeitschr. 144 283 A group G is hypercyclic if and only if G has an ascending series 1 = G₀ ⊆ G₁ ⊆... ⊆ G_λ ⊆ G_λ+1 ⊆... ⊆ G_p = G of normal subgroups G_λ such that for ordinals λ, which are not limit ordinals, G_λ/G_λ-1 is a cyclic group. 1988 Archiv der Math. LI. 13 For minimax groups the properties ‘locally nilpotent’, ‘locally supersoluble’ and ‘locally FC-nilpotent’ coincide with the properties ‘hypercentral’, ‘hypercyclic’ and ‘FC-nilpotent’, respectively. 2. Mathematics. Of the nature of or consisting of a hypercycle or hypercycles (hypercycle n. 2). ΚΠ 1977 Naturwissenschaften 64 541/2 Only hypercyclic organizations are able to fulfil these requirements. 1981 Sci. Amer. Apr. 86/2 In a hypercyclic system a closed loop of catalytic couplings connects self-replicative cycles. 1986 Jrnl. Theoret. Biol. 118 3 The hypercycle..can be unstable when the fluctuations are anomalously enhanced under the influence of environmental fluctuations on the hypercyclic organization. This entry has not yet been fully updated (first published 1993; most recently modified version published online June 2018). < adj.1968
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