When applying a rule to a number of indices ( differentiation, symmetrization etc ., shown next ), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.
12.
The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X _ i ( sometimes referred to as a " ghost sample " ) and replacing the inner expectation of the LHS by these copies.
13.
A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to \ mathbb { P } _ n ^ 0 and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties.
14.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
15.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
16.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
17.
The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
18.
The next deeper, " third stratum " is one where different classes are identified ( thus containing a fair amount of asymmetrical thinking ) but in which . . . parts of a class are always taken as the whole class symmetrization ( plus a degree of timelessness ).
19.
Finally, the deepest, " fifth stratum " is that in which processes of symmetrization tend towards the mathematical limit of " indivisibility " thinking, which requires asymmetrical relations, is greatly impaired and becomes the realm of psychotic functioning : without asymmetrical logic, play breaks down into delusion.
20.
K, K are constants, square brackets in indices U _ { [ \ alpha, \ mu ] } represent anti-symmetrization, \ lambda is a Lagrange multiplier ( calculated elsewhere ), and is a Lagrangian translated from flat spacetime onto the metric \ tilde g ^ { \ alpha \ beta }.
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