self-conjugateadj.

Brit.

/ˌsɛlfˈkɒndʒᵿɡət/, U.S.

/ˌsɛlfˈkɑndʒəɡət/

Origin: Formed within English, by derivation. Etymons: self- prefix, conjugate adj.

Etymology: < self- prefix + conjugate adj.

Mathematics.

With reference to a given relation that a mathematical entity has to itself.

1. Geometry. Designating a triangle in which, with respect to a given conic section, each side lies on the polar (polar n. 2) of the vertex to which it is opposite. Also: designating a point which intersects its own polar, or a line which intersects its own pole (pole n.² 8(b)).

ΘΚΠ

the world > relative properties > number > geometry > shape or figure > [adjective] > having specific property

hypotenusal?a1560

oblique?a1560

local1673

focal1676

octantal1777

symmetrical1794

radical1848

self-conjugate1855

quadric1856

stellated1859

periphractic1881

homoeoidal1883

tridiametral1891

one-sided1893

semi-infinite1903

simplicial1913

mirror-symmetric1952

1855 G. Salmon Treat. Conic Sections (ed. 3) 321 Let us define a self-conjugate triangle, one such that any side is the polar of the opposite vertex with regard to a given conic.

1885 J. Casey Treat. Analyt. Geom. 305 Let the conics be referred to their common self-conjugate triangle.

1889 Q. Jrnl. Pure & Appl. Math. 23 357 All tangents to the curve are self-conjugate lines.

1964 H. S. M. Coxeter Projective Geom. vii. 62 Every self-conjugate line contains just one self-conjugate point.

2004 Math. Gaz. 88 419 It is known that the diagonal point triangle EFG is self-conjugate.

2. Of a function: †(a) (in the mathematics of quaternions) a function f for which the scalar part of x⋅f (y) is the same as that of y⋅f (x), regardless of the choice of x and y (obsolete); (b) (in a Boolean algebra) a function f for which f (x)⋅y = 0 if and only if f (y)⋅x = 0 (now rare).

ΚΠ

1866 W. R. Hamilton & W. E. Hamilton Elements Quaternions §349 Where the function ϕ₀ρ is its own conjugate, or is the common self-conjugate part of ϕρ and ϕ′ρ.

1903 Philos. Trans. (Royal Soc.) A. 201 276 When the functions are self-conjugate..the two complexes combine into one.

1951 Amer. Jrnl. Math. 73 903 If..a function f is a conjugate of itself, then we call fself-conjugate.

1978 Notre Dame Jrnl. Formal Logic 19 504 In section 1 we investigate the properties of a self-conjugate function on a Boolean algebra.

3. Designating a partition (partition n. 9b) in which each row of its Young or Ferrers diagram has the same number of boxes or dots as the corresponding column. Also: designating a Young or Ferrers diagram of this kind.See also Ferrers diagram n., Young diagram n. at Young n.² 2.

ΚΠ

1882 J. J. Sylvester in Amer. Jrnl. Math. 5 264 Any one of the former contains a class of what may be termed singular partitions, in the sense that they are their own associates, or more briefly, self-conjugate in respect to the Ferrers transformation.

1882 J. J. Sylvester in Amer. Jrnl. Math. 5 278 The self-conjugate graph corresponding to any partition of unrepeated odd numbers..will be found by the following rule.

1946 Math. Tables & Other Aids Computation 2 166 It follows at once from the theorem of Sylvester that Q(n) is also the number of self-conjugate partitions of n.

1979 E. S. Page & L. B. Wilson Introd. Computational Combinatorics iv. 74 A partition whose Ferrers graph reads the same by rows and by columns is called self-conjugate.

1992 Trans. Amer. Math. Soc. 332 167 For n ≥ 8 there always exist two distinct self-conjugate Young diagrams.

2016 Trans. Amer. Math. Soc. 368 5807 The main problem for the alternating groups was reduced to the question of which Specht modules labelled by self-conjugate partitions have exactly two composition factors.

4. Designating a matrix that is the transpose of itself; (in later use also) designating a matrix that is identical to the matrix formed from its transpose by taking the complex conjugate of each entry; = Hermitian adj. Now rare.

ΚΠ

1884 Proc. London Math. Soc. 16 16 Consider any symmetrical (or, as I prefer to call it, self-conjugate) matrix, and the substitution defined by it.

1913 C. E. Cullis Matrices & Determinoids I. i. 5 A self-conjugate matrix is a square matrix which is symmetrical with respect to its leading diagonal.

1962 R. D. Richtmyer Surv. Difference Methods for Non-steady Fluid Dynamics (National Center Atmospheric Research) 8 It can always be written in the form C + iD, where C and D are Hermitian (self-conjugate) matrices.

2004 Analog Integrated Circuits & Signal Processing 40 8/1 We do not suppose that R and D are self-conjugate, but for simplicity we assume that R is invertible.

5. Designating a subgroup that contains every element of the form ghg⁻¹, where h is any element of the subgroup and g any element of the group in which it lies; = normal adj. 11.

ΚΠ

1886 Amer. Jrnl. Math. 8 268 In the language of the theory of substitutions, λ belongs to the exceptional self-conjugate sub-group of the general group of 24 permutations of the 4 roots.

1937 A. A. Albert Mod. Higher Algebra vi. 131 We call ℌ a normal divisor (or invariant subgroup, or self-conjugate subgroup) of .

1978 Gen. Relativity & Gravitation 9 444 Let π^N be any induced action of an invariant (i.e., self-conjugate) subgroup of G.

2008 M. S. Dresselhaus et al. Group Theory i. 11 The order of the factor group is the index of the self-conjugate subgroup.

Derivatives

self-ˈconjugately adv. now disused as a normal subgroup; cf. sense 5.

ΚΠ

1893 Proc. London Math. Soc. 25 12 Let K be such a sub-group of order p^β, and let J be the greatest sub-group that contains K self-conjugately.

1938 Proc. London Math. Soc. 43 512 Here the system normalizers are of order 2, and are contained self-conjugately in subgroups of order 4.

This entry has been updated (OED Third Edition, January 2018; most recently modified version published online March 2022).

单词	self-conjugate
释义	self-conjugateadj. Brit. /ˌsɛlfˈkɒndʒᵿɡət/, U.S. /ˌsɛlfˈkɑndʒəɡət/ Origin: Formed within English, by derivation. Etymons: self- prefix, conjugate adj. Etymology: < self- prefix + conjugate adj. Mathematics. With reference to a given relation that a mathematical entity has to itself. 1. Geometry. Designating a triangle in which, with respect to a given conic section, each side lies on the polar (polar n. 2) of the vertex to which it is opposite. Also: designating a point which intersects its own polar, or a line which intersects its own pole (pole n.² 8(b)). ΘΚΠ the world > relative properties > number > geometry > shape or figure > [adjective] > having specific property hypotenusal?a1560 oblique?a1560 local1673 focal1676 octantal1777 symmetrical1794 radical1848 self-conjugate1855 quadric1856 stellated1859 periphractic1881 homoeoidal1883 tridiametral1891 one-sided1893 semi-infinite1903 simplicial1913 mirror-symmetric1952 1855 G. Salmon Treat. Conic Sections (ed. 3) 321 Let us define a self-conjugate triangle, one such that any side is the polar of the opposite vertex with regard to a given conic. 1885 J. Casey Treat. Analyt. Geom. 305 Let the conics be referred to their common self-conjugate triangle. 1889 Q. Jrnl. Pure & Appl. Math. 23 357 All tangents to the curve are self-conjugate lines. 1964 H. S. M. Coxeter Projective Geom. vii. 62 Every self-conjugate line contains just one self-conjugate point. 2004 Math. Gaz. 88 419 It is known that the diagonal point triangle EFG is self-conjugate. 2. Of a function: †(a) (in the mathematics of quaternions) a function f for which the scalar part of x⋅f (y) is the same as that of y⋅f (x), regardless of the choice of x and y (obsolete); (b) (in a Boolean algebra) a function f for which f (x)⋅y = 0 if and only if f (y)⋅x = 0 (now rare). ΚΠ 1866 W. R. Hamilton & W. E. Hamilton Elements Quaternions §349 Where the function ϕ₀ρ is its own conjugate, or is the common self-conjugate part of ϕρ and ϕ′ρ. 1903 Philos. Trans. (Royal Soc.) A. 201 276 When the functions are self-conjugate..the two complexes combine into one. 1951 Amer. Jrnl. Math. 73 903 If..a function f is a conjugate of itself, then we call fself-conjugate. 1978 Notre Dame Jrnl. Formal Logic 19 504 In section 1 we investigate the properties of a self-conjugate function on a Boolean algebra. 3. Designating a partition (partition n. 9b) in which each row of its Young or Ferrers diagram has the same number of boxes or dots as the corresponding column. Also: designating a Young or Ferrers diagram of this kind.See also Ferrers diagram n., Young diagram n. at Young n.² 2. ΚΠ 1882 J. J. Sylvester in Amer. Jrnl. Math. 5 264 Any one of the former contains a class of what may be termed singular partitions, in the sense that they are their own associates, or more briefly, self-conjugate in respect to the Ferrers transformation. 1882 J. J. Sylvester in Amer. Jrnl. Math. 5 278 The self-conjugate graph corresponding to any partition of unrepeated odd numbers..will be found by the following rule. 1946 Math. Tables & Other Aids Computation 2 166 It follows at once from the theorem of Sylvester that Q(n) is also the number of self-conjugate partitions of n. 1979 E. S. Page & L. B. Wilson Introd. Computational Combinatorics iv. 74 A partition whose Ferrers graph reads the same by rows and by columns is called self-conjugate. 1992 Trans. Amer. Math. Soc. 332 167 For n ≥ 8 there always exist two distinct self-conjugate Young diagrams. 2016 Trans. Amer. Math. Soc. 368 5807 The main problem for the alternating groups was reduced to the question of which Specht modules labelled by self-conjugate partitions have exactly two composition factors. 4. Designating a matrix that is the transpose of itself; (in later use also) designating a matrix that is identical to the matrix formed from its transpose by taking the complex conjugate of each entry; = Hermitian adj. Now rare. ΚΠ 1884 Proc. London Math. Soc. 16 16 Consider any symmetrical (or, as I prefer to call it, self-conjugate) matrix, and the substitution defined by it. 1913 C. E. Cullis Matrices & Determinoids I. i. 5 A self-conjugate matrix is a square matrix which is symmetrical with respect to its leading diagonal. 1962 R. D. Richtmyer Surv. Difference Methods for Non-steady Fluid Dynamics (National Center Atmospheric Research) 8 It can always be written in the form C + iD, where C and D are Hermitian (self-conjugate) matrices. 2004 Analog Integrated Circuits & Signal Processing 40 8/1 We do not suppose that R and D are self-conjugate, but for simplicity we assume that R is invertible. 5. Designating a subgroup that contains every element of the form ghg⁻¹, where h is any element of the subgroup and g any element of the group in which it lies; = normal adj. 11. ΚΠ 1886 Amer. Jrnl. Math. 8 268 In the language of the theory of substitutions, λ belongs to the exceptional self-conjugate sub-group of the general group of 24 permutations of the 4 roots. 1937 A. A. Albert Mod. Higher Algebra vi. 131 We call ℌ a normal divisor (or invariant subgroup, or self-conjugate subgroup) of . 1978 Gen. Relativity & Gravitation 9 444 Let π^N be any induced action of an invariant (i.e., self-conjugate) subgroup of G. 2008 M. S. Dresselhaus et al. Group Theory i. 11 The order of the factor group is the index of the self-conjugate subgroup. Derivatives self-ˈconjugately adv. now disused as a normal subgroup; cf. sense 5. ΚΠ 1893 Proc. London Math. Soc. 25 12 Let K be such a sub-group of order p^β, and let J be the greatest sub-group that contains K self-conjugately. 1938 Proc. London Math. Soc. 43 512 Here the system normalizers are of order 2, and are contained self-conjugately in subgroups of order 4. This entry has been updated (OED Third Edition, January 2018; most recently modified version published online March 2022). < adj.1855
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