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This linear map is known as the pullback (by ), and is frequently denoted by . More generally, any covariant tensor field – in particular any differential form – on N {\displaystyle N} may be pulled back to M {\displaystyle M} using ϕ {\displaystyle \phi } .
In general, any measurable function can be pushed forward. The push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator.In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" operations it defines
Any vector field Y on N defines a pullback section φ ∗ Y of φ ∗ TN with (φ ∗ Y) x = Y φ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φ ∗ X = φ ∗ Y as vector fields along φ. In other words, for all x in M, dφ x (X) = Y φ(x).
Stanley Druckenmiller says "animal spirits" have been unleashed as Trump takes office. Business leaders have shown optimism for Trump's pro-business policies.
When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the transpose or composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator.
The Democratic Party plans to push forward with a virtual roll call in which delegates to its convention can choose a presidential nominee before they gather next month in Chicago — despite Vice ...
Image source: The Motley Fool. Synchrony Financial (NYSE: SYF) Q4 2024 Earnings Call Jan 28, 2025, 8:00 a.m. ET. Contents: Prepared Remarks. Questions and Answers. Call Participants