Synonyms for pushforward or Related words with pushforward
Examples of "pushforward"
measure is also denoted as formula_2.
The Grothendieck–Riemann–Roch theorem relates the
where "X" is a vector on "M", and "d" denotes the
measure formula_8---called a distribution of "X" in this context---is a probability measure
We build the
homomorphism as follows (for singular or simplicial homology):
where formula_6 is the
along π of the vector "v" ∈ TUT"M".
The law of the process "X" is then defined to be the
A measurable function formula_76 induces a
measure, – the probability measure formula_77 on formula_78 defined by
In the case of a fiber bundle, one can also define a
where formula_61 denotes the
measure of formula_62 induced by the canonical projection map formula_63.
along a vector function f with respect to vector v in R is given by
Conversely, if "f" is proper, for "Y" a subvariety of "X" the
is defined to be
If "μ" is finite, "ν" can be taken to be the
"π""μ", and then the "λ" are probabilities.
Further unwinding the definitions, the
of a vector field under a map between manifolds is defined by
An -valued random vector is called "comonotonic", if its multivariate distribution (the
measure) is comonotonic, this means
The pullback is naturally defined as the dual (or transpose) of the
. Unraveling the definition, this means the following:
from the tangent space of "M" at "x" to the tangent space of "N" at "φ"("x"). The application of d"φ" to a tangent vector "X" is sometimes called the
of "x" by "φ". The exact definition of this
depends on the definition one uses for tangent vectors (for the various definitions see tangent space).
The measure that associates to the set "A" the number γ("A"−"h") is the
measure, denoted ("T")(γ). Here "T" : R → R refers to the translation map: "T"("x") = "x" + "h".. The above calculation shows that the Radon–Nikodym derivative of the
measure with respect to the original Gaussian measure is given by
Nevertheless, it follows from this that if Φ is invertible, pullback can be defined using
by the inverse function Φ. Combining these two constructions yields a
operation, along an invertible linear map, for tensors of any rank ("r","s").
at each point "v" ∈ T"M"; here π : TT"M" → T"M" denotes the
(differential) along the projection π : T"M" → "M" associated to the tangent bundle.
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