| 释义 |
▪ I. norm, n.|nɔːm|
[Anglicized form of norma: cf. also norme.]
1. a. A standard, model, pattern, type. (Common since c 1855.)
1821Coleridge in Blackw. Mag. X. 257 Each after its own norm or model. 1828Pusey Hist. Enq. i. 21 Every expression of his upon controverted points became a norm for the party. 1857P. Freeman Princ. Div. Serv. II. 143 The norm and measure of all our eucharistic thoughts, and words, and actions. 1877E. Caird Philos. Kant iv. 66 The mind must find in itself the norm or principle of unity upon which it works. 1911R. Brooke Coll. Poems (1918) 154 All of the accents upon all the norms!—And ah! the stress on the penultimate! We never knew blank verse could have such feet. 1941J. P. Marquand H.M. Pulham, Esquire iv. 44 Beatrice considered that I was utterly characteristic, completely true to type; once she called me a norm. 1961S. R. Herman in J. A. Fishman Readings Sociol. of Lang. (1968) 505 Frequently the newcomer arrives with the belief that the prevailing norm about using Hebrew to the exclusion of other languages is more rigidly observed than is actually the case... Soon, however, the newcomer becomes aware of the wide range of deviations from the norm. 1964M. Argyle Psychol. & Social Probl. iii. 38 One particular respect in which a group equilibrium develops is in the formation of norms—shared patterns of behaving, feeling and thinking. All social groups develop norms, particularly about matters connected with the group's main purposes and activities... When group members deviate from the norms, various kinds of persuasion, pressure and sanctions are exerted in order to make them conform. 1965Economist 13 Feb. 645/2 The other vital point will be the ‘norms’ or ‘guiding lights’ [for incomes] recommended for each year.
b. Algebra. (See quots.) Also defined analogously for other quantities. [Introduced as L. norma by Gauss 1832, in Commentationes Recentiores Soc. R. Scient. Gottingensis VII. Class. math. 98.]
1856W. R. Hamilton Notebook in Halberstam & Ingram Math. Papers Sir W. R. Hamilton (1967) III. 657, a + ib is said to be a complex number, when a and b are integers, and i = √ - 1; its norm is a2 + b2; and therefore the norm of a product is equal to the product of the norms of its factors. 1866Brande & Cox Dict. Sci., etc. II. 228/2 The product a2 + b2 of a complex number a + b√- 1 , and its conjugate a - b√- 1 is called its norm. 1932Turnbull & Aitken Introd. Theory Canonical Matrices iv. 38 This fundamental Hermitian inner product of x and x is often called the norm of the complex vector x... The square root of the norm, taken with positive sign, (xx)½ , is sometimes denoted by {vb}x{vb}. 1949A. Albert Solid Analytic Geom. i. 3 The norm of a vector P is defined to be the inner product P·P = x12 +{ddd}+ xn2. 1952C. Møller Theory of Relativity iv. 99 It then follows..that the ‘square of the magnitude of the four-vector’, or the norm of the vector, {Summ}iai2 = {Summ}ia′i2 , is an invariant. 1967MacLane & Birkhoff Algebra v. 187 Each quadratic field Q(√d), with the elements σ = r + s√d, has as automorphisms the identity and σ {vb}→ σ = r - s√d . The product σσ = r2 + s2d is called the norm N(σ) of σ.
(ii) The positive square root of the quantity defined above; more generally, a quantity defined on a vector space over the real or complex field which represents a generalization of the concept of length or magnitude and has the properties that ‖ u‖ > 0 if u ≠ 0 (‖ u‖ being the norm of the vector u), ‖ u‖ = 0 if u = 0, ‖ au‖ = {vb}a{vb} ‖ u‖ (a being a real number), and ‖ u + v‖ {slle}‖ u‖ + ‖ v‖ (v being another vector).
1921Proc. Nat. Acad. Sci. VII. 84 The notion of norm or numerical value of a complex quantity, c = a + b √ - 1, namely, {vb}c{vb} = √(a2 + b2), as it arises in algebra, has a more or less immediate generalization to more extensive matric systems. 1951P. R. Halmos Introd. Hilbert Space i. 13 The norm of a vector α in the inner product space..coincides with the absolute value of the complex number α. 1955L. F. Boron tr. I. P. Natanson's Theory of Functions of Real Variable I. vii. 199 Let f(x) {elem} Lp. The number ‖ f‖ = p√(∫ b / a {vb}f(x){vb}pdx) is called the norm of the function f(x) (considered as an element of Lp). 1962Kacinskas & Counts tr. L. S. Pontryagin's Ordinary Differential Equations iv. 152 The maximum modulus of this function..will be called its norm. 1965Patterson & Rutherford Elem. Abstract Algebra v. 184 A norm can be defined in a vector space having an inner product by writing ‖ x‖ = √(x·x)... There exist norms which cannot be expressed in this way in terms of an inner product. In the case where an inner product exists, it is clear that the length of a vector satisfies the requirements for a norm. Ibid. 185 Let M2, 2(C) be the vector space of 2 × 2 matrices over the complex field. Then the mapping ⎥ a b / c d ⎥ → max ({vb}a{vb}, {vb}b{vb}, {vb}c{vb}, {vb}d{vb}) defines a norm in M2, 2(C). The mapping ⎥ a b / c d ⎥ → √({vb}a{vb}2 + {vb}b{vb}2 + {vb}c{vb}2 + {vb}d{vb}2) defines another norm. 1970F. A. Matsen Vector Spaces & Algebras i. 11 The norm Nc of c is defined by Nc = (cc*)1/2 = √(a2 + b2).
c. In Communist countries, a standard unit of work prescribed.
1935S. & B. Webb Soviet Communism II. ix. 706 They [sc. piece-work rates] are, in some cases, even progressive, the rate rising by stages for output beyond the norm. 1952Manch. Guardian 6 June, Stakhanovite women miners in the Donetz basin are performing four, nine, and eleven norms each. 1959Times 12 Mar. 13/6 This moulding process may vary from the crudest regimentation and subordination..to factory-like specification and ‘norms’ to mere pilotage..towards the Marxist haven.
2. Petrol. A hypothetical mineral composition of a rock calculated by assigning the compounds present to certain relatively simple minerals in accordance with prescribed rules.
1902[see mode n. 5 b]. 1932A. Johannsen Descr. Petrogr. Ign. Rocks II. 272 (heading) Table 135. Norms of rhyolites. 1973Jrnl. Petrol. XIV. 35 The norms used are not based on stoichiometric formulae..but approach the compositions of the constituents as actually found in natural rocks. Ibid. 250 The C.I.P.W. norms which accompany the analyses were calculated using an Fe2O3/FeO ratio estimated to be appropriate for the quartz-fayalite-magnetite-buffered charges.
3. attrib. and Comb.
1934Archit. Rev. LXXVI. 42/1 It represents a return to the machine, regarded realistically and as mechanism to produce mechanical norm-types. 1958M. Argyle Relig. Behaviour v. 57 All these findings can be regarded as instances of social learning, mediated by the usual processes of persuasion, imitation and norm-formation. 1961J. N. Findlay Values & Intentions ix. 399 The religious object..must tend more and more towards the pattern of a detached, suprapersonal, norm-setting mind. 1964I. L. Horowitz New Sociology 32 Why do people choose rapid industrialization with its attendant psychological turmoil over social stability and norm adherence? 1966Mathematical Rev. XXXI. 8/2 Since norm-sentences are neither true nor false, logical connectives undergo re⁓interpretation. 1969J. F. Szwed in Halpert & Story Christmas Mumming in Newfoundland 116 During socialization, each person acquires an awareness of social sanctions and at the same time becomes ‘norm-oriented’.
▪ II. norm, v. Math.|nɔːm|
[f. the n.]
1. trans. = normalize v. 3 a. ? Obs.
1931P. Dienes Taylor Series viii. 274 We ‘norm’ the mapping with respect to u = a by requiring that a1k = 1. 1941R. V. Churchill Fourier Series iii. 38 The functions of the set are normed by dividing each function gn(x) by [N(gn)]1/2.
2. trans. [back-formation from normed a.] To define a norm on (a space).
1959L. F. Boron tr. Naimark's Normed Rings i. 73 The space X/𝔐, normed by formula (1), will be called a normed factor-space. 1964D. E. Brown tr. Kantorovich & Akilov's Functional Analysis in Normed Spaces ii. 51 The metric spaces..are also linear sets and can be normed. 1972A. G. Howson Handbk. Terms used in Algebra & Analysis xxii. 111 The vector space {scrL}(E, F) can be normed in the following way.
Hence ˈnorming vbl. n.
1967L. Rédei Algebra I. iv. 307 In different rings norming is carried out in different ways. |