monodromyn.

Brit.

/məˈnɒdrəmi/, U.S.

/məˈnɑdrəmi/

Forms: 1800s– monodromy, 1900s– monodromie (rare).

Origin: Formed within English, by derivation. Etymons: monodrome adj., -y suffix³.

Etymology: < monodrome adj. + -y suffix³.

Mathematics.

1. A property of certain functions f(z) of a complex variable. Usually attributive.Monodromy is the property of a function f(z) defined in some domain S, that if it is analytically continued along any paths in a larger domain D (containing S) that begin and end at a point p in S, then its values f(z) return to the value f(p).

ΚΠ

1891 Amer. Jrnl. Math. 13 117 If the point t describes all possible closed curves, the letters x₁, x₂,..x_n undergo a series of different substitutions, which are easily seen to form a group, the monodromy-group of the equation.

1942 Ann. Math. 43 234 Corollary 6.1 cannot be extended to give the general monodromy property, ‘continuation of E_a along any two descending paths from a to b gives the same result’.

1970 Ann. Math. 92 342 Condition D, Theorem 4, and a standard monodromy argument show that Ψ is one-to-one in Δ⁰.

1994 Communic. Math. Physics 160 393 The matrix Jost solutions (and hence the monodromy matrix) lack simple analyticity properties in the spectral parameter.

2009 H. Kroto in J. Laane Frontiers Molecular Spectroscopy i. 6 The top of the barrier to linearity..forms the critical monodromy point at which there is a discontinuity in the mathematical description of the eigenstates that lie above and below this point.

2. The property that a curve described by rotation of a point through four right angles about an axis is closed. rare. disused.

ΘΚΠ

the world > relative properties > number > algebra > [noun] > expression > function > value or set of values of > property of

monodromy1897

monotony1902

monotoneity1926

monotonicity1934

minorant1938

1897 B. Russell Ess. Found. Geom. i. 24 As regards independence of rotation in rigid bodies (Monodromy). If (n−1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed.

1903 Nature 19 Feb. 382/2 It is pointed out that in the non-Pythagorean geometries devised by Hilbert, Helmholtz's axiom of monodromy is not verified, inasmuch as it is possible by rotation through four right angles, to bring the points of a line into positions which they do not occupy before the rotation.

Compounds

monodromy group n. a group each of whose elements is a transformation taking one branch of a multivalued function f(z) to another when the variable z is taken round a closed path surrounding a branch point of f; spec. one in which the function concerned is a solution of a differential equation.

ΚΠ

1891Monodromy group [see sense 1].

1929 Amer. Jrnl. Math. 51 310 It is impossible to construct upon an irreducible branch curve an algebraic function possessing an abelian non-cyclic monodromie group.

2010 J. Belk & S. Koch in M. Bonk et al. In Trad. of Ahlfors-Bers, V: Triennial Ahlfors-Bers Colloquium 1 We compute the iterated monodromy group for a postcritically finite endomorphism F : P² → P².

monodromy theorem n. the theorem that if a function of a complex variable is analytic at a point inside a simply connected domain, and can be uniquely extended to an analytic function along all paths in the domain leading from the point, then it can be uniquely extended to a single-valued analytic function on the whole domain.

ΚΠ

1934 Trans. Amer. Math. Soc. 36 225 By the monodromy theorem..the function ϕ[f(z)] is analytic and single-valued in the circle |z| < 1.

1992 Amer. Jrnl. Math. 114 537 Since O is simply-connected, the monodromy theorem implies that these continuations of h do not depend on the choice of path.

2011 J. L. Taylor Complex Variables vii. 227 The concept of analytic continuation along a curve and the Monodromy Theorem serve to bring order to this process.

This entry has been updated (OED Third Edition, December 2002; most recently modified version published online March 2022).

单词	monodromy
释义	monodromyn. Brit. /məˈnɒdrəmi/, U.S. /məˈnɑdrəmi/ Forms: 1800s– monodromy, 1900s– monodromie (rare). Origin: Formed within English, by derivation. Etymons: monodrome adj., -y suffix³. Etymology: < monodrome adj. + -y suffix³. Mathematics. 1. A property of certain functions f(z) of a complex variable. Usually attributive.Monodromy is the property of a function f(z) defined in some domain S, that if it is analytically continued along any paths in a larger domain D (containing S) that begin and end at a point p in S, then its values f(z) return to the value f(p). ΚΠ 1891 Amer. Jrnl. Math. 13 117 If the point t describes all possible closed curves, the letters x₁, x₂,..x_n undergo a series of different substitutions, which are easily seen to form a group, the monodromy-group of the equation. 1942 Ann. Math. 43 234 Corollary 6.1 cannot be extended to give the general monodromy property, ‘continuation of E_a along any two descending paths from a to b gives the same result’. 1970 Ann. Math. 92 342 Condition D, Theorem 4, and a standard monodromy argument show that Ψ is one-to-one in Δ⁰. 1994 Communic. Math. Physics 160 393 The matrix Jost solutions (and hence the monodromy matrix) lack simple analyticity properties in the spectral parameter. 2009 H. Kroto in J. Laane Frontiers Molecular Spectroscopy i. 6 The top of the barrier to linearity..forms the critical monodromy point at which there is a discontinuity in the mathematical description of the eigenstates that lie above and below this point. 2. The property that a curve described by rotation of a point through four right angles about an axis is closed. rare. disused. ΘΚΠ the world > relative properties > number > algebra > [noun] > expression > function > value or set of values of > property of monodromy1897 monotony1902 monotoneity1926 monotonicity1934 minorant1938 1897 B. Russell Ess. Found. Geom. i. 24 As regards independence of rotation in rigid bodies (Monodromy). If (n−1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed. 1903 Nature 19 Feb. 382/2 It is pointed out that in the non-Pythagorean geometries devised by Hilbert, Helmholtz's axiom of monodromy is not verified, inasmuch as it is possible by rotation through four right angles, to bring the points of a line into positions which they do not occupy before the rotation. Compounds monodromy group n. a group each of whose elements is a transformation taking one branch of a multivalued function f(z) to another when the variable z is taken round a closed path surrounding a branch point of f; spec. one in which the function concerned is a solution of a differential equation. ΚΠ 1891Monodromy group [see sense 1]. 1929 Amer. Jrnl. Math. 51 310 It is impossible to construct upon an irreducible branch curve an algebraic function possessing an abelian non-cyclic monodromie group. 2010 J. Belk & S. Koch in M. Bonk et al. In Trad. of Ahlfors-Bers, V: Triennial Ahlfors-Bers Colloquium 1 We compute the iterated monodromy group for a postcritically finite endomorphism F : P² → P². monodromy theorem n. the theorem that if a function of a complex variable is analytic at a point inside a simply connected domain, and can be uniquely extended to an analytic function along all paths in the domain leading from the point, then it can be uniquely extended to a single-valued analytic function on the whole domain. ΚΠ 1934 Trans. Amer. Math. Soc. 36 225 By the monodromy theorem..the function ϕ[f(z)] is analytic and single-valued in the circle \|z\| < 1. 1992 Amer. Jrnl. Math. 114 537 Since O is simply-connected, the monodromy theorem implies that these continuations of h do not depend on the choice of path. 2011 J. L. Taylor Complex Variables vii. 227 The concept of analytic continuation along a curve and the Monodromy Theorem serve to bring order to this process. This entry has been updated (OED Third Edition, December 2002; most recently modified version published online March 2022). < n.1891
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