Change of Even and Odd Variables
Let a coordinate transformation be given by, where are even and are odd polynomials of depending on even variables The Jacobian matrix of this transformation has the block form:
where each even derivative commutes with all elements of the algebra ; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks and are even and the entries of the offdiagonal blocks are odd functions, where mean right derivatives. The Berezinian (or the superdeterminant) of the matrix is the even function
defined when the function is invertible in Suppose that the real functions define a smooth invertible map of open sets in and the linear part of the map is invertible for each The general transformation law for the Berezin integral reads
where is the sign of the orientation of the map The superposition is defined in the obvious way, if the functions do not depend on In the general case, we write where are even nilpotent elements of and set
where the Taylor series is finite.
Read more about this topic: Berezin Integral
Famous quotes containing the words change of, change, odd and/or variables:
“The change of Philomel, by the barbarous king
So rudely forced; yet there the nightingale
Filled all the desert with inviolable voice
And still she cried, and still the world pursues,
Jug Jug to dirty ears.”
—T.S. (Thomas Stearns)
“Its pretty hard to change once youve burned away your youth and your illusions.”
—John Farrow. Captain (John Wayne)
“Young people love what is interesting and odd, no matter how true or false it is. More mature minds love what is interesting and odd about truth. Fully mature intellects, finally, love truth, even when it appears plain and simple, boring to the ordinary person; for they have noticed that truth tends to reveal its highest wisdom in the guise of simplicity.”
—Friedrich Nietzsche (18441900)
“The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
—George Steiner (b. 1929)