Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry.
12.
By Euler's second theorem for homogeneous functions, \ bar { Z _ i } is a homogeneous function of degree 0 which means that for any \ lambda:
13.
By Euler's second theorem for homogeneous functions, \ bar { Z _ i } is a homogeneous function of degree 0 which means that for any \ lambda:
14.
Given a homogeneous polynomial of degree " k ", it is possible to get a homogeneous function of degree 1 by raising to the power 1 / " k ".
15.
(This is equivalent to saying that extensive composite properties are homogeneous functions of degree 1 with respect to \ { A _ j \ } . ) It follows from Euler's homogeneous function theorem that
16.
(This is equivalent to saying that extensive composite properties are homogeneous functions of degree 1 with respect to \ { A _ j \ } . ) It follows from Euler's homogeneous function theorem that
17.
A function ( defined on some open set ) on \ mathbb P ( V ) gives rise by pull-back to a 0-homogeneous function on " V " ( again partially defined ).
18.
The aforementioned equivalence of metric functions remains valid if } } is replaced with, where is any convex positive homogeneous function of degree 1, i . e . a vector norm ( see Minkowski distance for useful examples ).
19.
This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescaling, or " " independent of the radial coordinate " ".
20.
In the case the constraint on the particle is time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy has no time-dependence and is a homogeneous function of degree 2 in the generalized velocities;
