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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 exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space). If tangent vectors are defined as equivalence classes of the curves γ {\displaystyle \gamma } for which γ ( 0 ) = x , {\displaystyle \gamma (0)=x,} then the differential is given by
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
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, O X) → (Y, O Y) is a morphism of ringed spaces, we obtain a direct image functor f ∗: Sh(X,O X) → Sh(Y,O Y) from the category of sheaves of O X-modules to the category of sheaves of O Y-modules.
0-9 22 The 22 m line, marking 22 metres (72 ft) from the tryline. 89 An "89" or eight-nine move is a phase following a scrum, in which the number 8 picks up the ball and transfers it to number 9 (scrum-half). 99 The "99" call was a policy of simultaneous retaliation by the 1974 British Lions tour to South Africa, (the 99 comes from the British emergency services telephone number which is 999 ...
People often use the simple mnemonic spring forward, fall back to remember to set clocks forward one hour (e.g., from 2 a.m. to 3 a.m.) in the spring and backward one hour (e.g., from 2 a.m. to 1 ...
A three-year congressional effort to make daylight saving time permanent in the United States appears to have halted, with President Donald Trump saying on Thursday that Americans are evenly ...
We build the pushforward homomorphism as follows (for singular or simplicial homology): . First, the map : induces a homomorphism between the singular or simplicial chain complex and () defined by composing each singular n-simplex: with to obtain a singular n-simplex of , # =:, and extending this linearly via # = # ().