Grassmannn.

Brit.

/ˈɡrasmən/,

/ˈɡrɑːsmən/, U.S.

/ˈɡræsmən/,

/ˈɡrɑsmən/

Origin: From a proper name. Etymon: proper name Grassmann.

Etymology: < the name of Hermann Grassmann (1809–77), German mathematician and linguist.

1. Grassmann's Law n. (also with lower-case initial in second element) [compare German Grassmanns Gesetz (1880 or earlier)] Linguistics a phonetic law formulated by Hermann Grassmann which states that where two aspirated plosives occurred in neighbouring syllables in Indo-European, dissimilation took place and the first of the two lost its aspiration in Greek and Sanskrit.E.g. the case cited in quot. 1886, where an earlier χ (/kh/) became unaspirated κ ( /k/) owing to the influence of the following θ ( /th/).This law was formulated to explain apparent exceptions to Grimm's law (see Grimm's law n. at law n.¹ 17c(c)) and published in 1863 ( Zeitschrift für vergleichende Sprachforschung 12 81–138).

ΘΚΠ

the mind > language > linguistics > study of speech sound > speech sound > sound changes > [noun] > rule or law > specific

Grimm's law1838

Verner's law1878

Grassmann's Law1886

Lachmann's law1889

the law of Hobson-Jobson1898

Sievers1934

1886 Academy 20 Mar. 206/3 F. Holthausen also gives some interesting etymologies—e.g.,..κανθύλη, which stands (by Grassmann's law) for χανθύλη.

1891 F. M. Müller Lect. Sci. Lang. II. v. 270 The following are the principal words in which, according to Grassmann's Law, double media in Gothic can be accounted for: Sk. [= Sanskrit] Gardh [etc.].

1939 L. H. Gray Found. Lang. 70 A noteworthy form of incontiguous dissimilation is seen in the operation of Grassmann's Law.

1960 S. Potter Lang. in Mod. World x. 152 Grassmann's law cannot fail to warm the heart of every true-born philologist.

2002 Glotta 78 25 At first sight the theory that in Greek stops are aspirated before /s/ appears to cause difficulties for the understanding of Grassmann's Law.

2. Mathematics and Physics. attributive and in the genitive. Designating mathematical concepts and structures introduced by or associated with Hermann Grassmann.

a. Grassmann product n. = exterior product n. at exterior adj. and n. Additions.Sometimes used to refer solely to exterior products that are defined on vector spaces.

ΚΠ

1902 Proc. Royal Irish Acad. 1900–02 6 17 We have the Grassmann ‘product’ when we divide by Ω.

1959 Nuovo Cimento 14 Suppl. No. 1 180 The Grassmann product of two equal elements is zero.

2011 E. Zeidler Quantum Field Theory III. ii. 143 If a, b ∈ E₃, then the Grassmann product..is called a bivector.

b. Grassmann manifold n. (also Grassmann's manifold) a Grassmannian (Grassmannian n.) which has been equipped with the structure of a differentiable manifold.

ΚΠ

1930 Jrnl. London Math. Soc. 5 34 On representations of the S_k's of S_n and of the Grassmann manifolds G(k, n).

1981 Jrnl. Geom. 17 51 The space (S, R) is isomorphic with the Grassmann's Manifold representing the lines of a Galois Space.

2017 G. Aubrun & S. J. Szarek Alice & Bob meet Banach App. B. 316 Each Grassmann manifold carries a natural probability measure which can be constructed in two different but equivalent ways.

c. Grassmann algebra n. (also Grassmann's algebra) = exterior algebra n. at exterior adj. and n. Additions.Sometimes used solely to refer to exterior algebras defined on vector spaces.

ΚΠ

1891 Nature 28 May 80/1 The ‘Ausdehnungslehre’ affords a system of geometrical algebra..which for convenience I shall call Grassmann's algebra of points.]

1941 Amer. Jrnl. Math. 63 788 We construct over ℒ a Grassmann algebra, in which we denote the operation of multiplication by the sign □.

1960 Ann. Math. 72 460 For more details the reader may consult standard books..concerning Grassmann algebra.

2011 M. Combescure & D. Robert Coherent States & Applic. Math. Physics xi. 310 After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system.

2014 E. C. Banks Realistic Empiricism of Mach, James, & Russell vi. 169 This tradition..reaches its intellectual apex in Grassmann algebra.

d. Grassmann variety n. a Grassmannian regarded as an algebraic variety (algebraic variety n. at algebraic adj. Compounds).

ΚΠ

1949 Ann. Math. 50 77 It might also aid in extending our present results to general Grassmann varieties and similar spaces.

2007 C. Procesi Lie Groups xiii. 533 This theorem is used to prove the projective normality of the Grassmann variety.

e. Grassmann variable n. any of the elements of an exterior algebra (exterior algebra n. (a) at exterior adj. and n. Additions).Grassmann variables have the property that xy = −yx and x² = 0 (where multiplication is defined using the exterior product).

ΚΠ

1974 Physical Rev. D. 494 In terms of Grassmann variables we may equivalently write [etc.]

2016 E. Gozzi et al. Path Integrals for Pedestrians App. D. 119 Gaussian integrals over complex Grassmann variables appear frequently in path integrals.

This entry has been updated (OED Third Edition, March 2019; most recently modified version published online March 2022).

单词	grassmann
释义	Grassmannn. Brit. /ˈɡrasmən/, /ˈɡrɑːsmən/, U.S. /ˈɡræsmən/, /ˈɡrɑsmən/ Origin: From a proper name. Etymon: proper name Grassmann. Etymology: < the name of Hermann Grassmann (1809–77), German mathematician and linguist. 1. Grassmann's Law n. (also with lower-case initial in second element) [compare German Grassmanns Gesetz (1880 or earlier)] Linguistics a phonetic law formulated by Hermann Grassmann which states that where two aspirated plosives occurred in neighbouring syllables in Indo-European, dissimilation took place and the first of the two lost its aspiration in Greek and Sanskrit.E.g. the case cited in quot. 1886, where an earlier χ (/kh/) became unaspirated κ ( /k/) owing to the influence of the following θ ( /th/).This law was formulated to explain apparent exceptions to Grimm's law (see Grimm's law n. at law n.¹ 17c(c)) and published in 1863 ( Zeitschrift für vergleichende Sprachforschung 12 81–138). ΘΚΠ the mind > language > linguistics > study of speech sound > speech sound > sound changes > [noun] > rule or law > specific Grimm's law1838 Verner's law1878 Grassmann's Law1886 Lachmann's law1889 the law of Hobson-Jobson1898 Sievers1934 1886 Academy 20 Mar. 206/3 F. Holthausen also gives some interesting etymologies—e.g.,..κανθύλη, which stands (by Grassmann's law) for χανθύλη. 1891 F. M. Müller Lect. Sci. Lang. II. v. 270 The following are the principal words in which, according to Grassmann's Law, double media in Gothic can be accounted for: Sk. [= Sanskrit] Gardh [etc.]. 1939 L. H. Gray Found. Lang. 70 A noteworthy form of incontiguous dissimilation is seen in the operation of Grassmann's Law. 1960 S. Potter Lang. in Mod. World x. 152 Grassmann's law cannot fail to warm the heart of every true-born philologist. 2002 Glotta 78 25 At first sight the theory that in Greek stops are aspirated before /s/ appears to cause difficulties for the understanding of Grassmann's Law. 2. Mathematics and Physics. attributive and in the genitive. Designating mathematical concepts and structures introduced by or associated with Hermann Grassmann. a. Grassmann product n. = exterior product n. at exterior adj. and n. Additions.Sometimes used to refer solely to exterior products that are defined on vector spaces. ΚΠ 1902 Proc. Royal Irish Acad. 1900–02 6 17 We have the Grassmann ‘product’ when we divide by Ω. 1959 Nuovo Cimento 14 Suppl. No. 1 180 The Grassmann product of two equal elements is zero. 2011 E. Zeidler Quantum Field Theory III. ii. 143 If a, b ∈ E₃, then the Grassmann product..is called a bivector. b. Grassmann manifold n. (also Grassmann's manifold) a Grassmannian (Grassmannian n.) which has been equipped with the structure of a differentiable manifold. ΚΠ 1930 Jrnl. London Math. Soc. 5 34 On representations of the S_k's of S_n and of the Grassmann manifolds G(k, n). 1981 Jrnl. Geom. 17 51 The space (S, R) is isomorphic with the Grassmann's Manifold representing the lines of a Galois Space. 2017 G. Aubrun & S. J. Szarek Alice & Bob meet Banach App. B. 316 Each Grassmann manifold carries a natural probability measure which can be constructed in two different but equivalent ways. c. Grassmann algebra n. (also Grassmann's algebra) = exterior algebra n. at exterior adj. and n. Additions.Sometimes used solely to refer to exterior algebras defined on vector spaces. ΚΠ 1891 Nature 28 May 80/1 The ‘Ausdehnungslehre’ affords a system of geometrical algebra..which for convenience I shall call Grassmann's algebra of points.] 1941 Amer. Jrnl. Math. 63 788 We construct over ℒ a Grassmann algebra, in which we denote the operation of multiplication by the sign □. 1960 Ann. Math. 72 460 For more details the reader may consult standard books..concerning Grassmann algebra. 2011 M. Combescure & D. Robert Coherent States & Applic. Math. Physics xi. 310 After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. 2014 E. C. Banks Realistic Empiricism of Mach, James, & Russell vi. 169 This tradition..reaches its intellectual apex in Grassmann algebra. d. Grassmann variety n. a Grassmannian regarded as an algebraic variety (algebraic variety n. at algebraic adj. Compounds). ΚΠ 1949 Ann. Math. 50 77 It might also aid in extending our present results to general Grassmann varieties and similar spaces. 2007 C. Procesi Lie Groups xiii. 533 This theorem is used to prove the projective normality of the Grassmann variety. e. Grassmann variable n. any of the elements of an exterior algebra (exterior algebra n. (a) at exterior adj. and n. Additions).Grassmann variables have the property that xy = −yx and x² = 0 (where multiplication is defined using the exterior product). ΚΠ 1974 Physical Rev. D. 494 In terms of Grassmann variables we may equivalently write [etc.] 2016 E. Gozzi et al. Path Integrals for Pedestrians App. D. 119 Gaussian integrals over complex Grassmann variables appear frequently in path integrals. This entry has been updated (OED Third Edition, March 2019; most recently modified version published online March 2022). < n.1886
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