Maslov index¶
Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
Core Idea¶
Maslov index is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. It is associated most classically with a loop or path of Lagrangian subspaces in a symplectic vector space, and measures how that path meets a distinguished singular hypersurface known as the Maslov cycle.
Scope of Application¶
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Fourier integral operators. If an oscillatory integral is written using two different phase functions that.
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Microlocal analysis. In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions.
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Infinite-dimensional generalizations. intersections in real Hilbert spaces, and a widely cited functional-analytic definition for continuous curves in the.
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Fourier integral operators. parametrize the same Lagrangian submanifold, the method of stationary phase.
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Documented setting. The Maslov index was introduced in the context of asymptotic methods in mathematical physics and later reformulated in topological and geometric terms.
Clarity¶
A clear use of Maslov index names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. The strongest recognition evidence in the frozen account is: Then the edge AkA{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid.
Manages Complexity¶
Maslov index compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—when an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2 .—and the practical consequence—a path of Lagrangian lines is therefore a path of unoriented lines through the origin. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
- Check operation and conditions. At a transverse crossing time t for which \gamma(t)\cap L0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t)\cap L0.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Maslov index transfers literally when a new case preserves the same carrier type, relation, and recognition test. If an oscillatory integral is written using two different phase functions that. In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions. Beyond the home domain. No canonical parent is asserted for Maslov index. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.
Neighborhood in Abstraction Space¶
Maslov index sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Continuum Mechanics & Field Models (42 abstractions)
Nearest neighbors
- Coons patch — 0.86
- Filling radius — 0.86
- Glauber dynamics — 0.85
- Homoclinic bifurcation — 0.85
- Incidence (geometry) — 0.85
Computed from structural-signature embeddings · 2026-10-08