The Euclidean space itself carries a natural structure of Riemannian manifold ( the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space ).
42.
When the scalar product is expressed in terms of the right ascension \ alpha and declination \ delta of the Sun ( index \ odot ) and of the extrasolar object this becomes:
43.
For two einselected elements of the entangled system's state to interfere, both the original system and the measuring in both elements device must significantly overlap, in the scalar product sense.
44.
The least-squares estimator is simply an estimation of an orthogonal projection, using the Euclidean scalar product in \ mathbb { R } ^ n instead of the one defined for random variables.
45.
Denotes the microcanonical equilibrium distribution . " Fast " variables, by definition, are orthogonal to all functions G ( A ( \ Gamma ) of A ( \ Gamma ) under this scalar product.
46.
Now I would like to add axioms concerning the relation of perpendicularity between lines in order to reach Euclidean geometry ( with a scalar product that is well-defined up to a constant multiple ).
47.
The sum and scalar action are defined on say the direct sum ( equivalently the direct product ) of vector spaces ( and even then only over a common base field for the scalar product ).
48.
And by projecting the momentum equation on the flow direction, i . e . along a " streamline ", the cross product disappears due to a vector calculus identity of the triple scalar product:
49.
Here \ hat { P } is an ordering operator, the dot stands for a scalar product and \ mathbf { q } and \ mathbf { q _ 0 } are two points on \ Gamma.
50.
However, I do use probability theory to prove that the regression problem is equivalent to an orthogonal projection given the scalar product \ langle X, Y \ rangle : = \ mathbb { E } [ XY ].
