A closer inspection of the convolution of two basis elements as shown in the equation above reveals that the multiplication in L ^ 1 ( G ) corresponds to that in \ C [ G ].
32.
Terms appear in the expansion of the transition probability above which involve i \ ne j; these can be thought of as representing " interference " between the different basis elements or quantum alternatives.
33.
The relation to complex numbers becomes clearer, too : in 2D, with two vector directions ? 1 and ? 2, there is only one bivector basis element ? 1 ? 2, so only one imaginary.
34.
It is easy to check that f is a solution to Cauchy's functional equation given a definition of f on the basis elements, f : \ mathcal { B } \ rightarrow \ mathbb { R }.
35.
This is the defining condition for a generating set of a Clifford algebra . ? } } } of a ( ? ?) representation of .-- > Further basis elements of the Clifford algebra are given by
36.
We have a base B at \ theta for a translation invariant topology, so the set of all basis elements for the whole set are of the form x + B . In this specific part, we are trying to show
37.
If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements ( known as the syzygies ) is finitely generated over k [ V ].
38.
The problem can also be studied from the abstract point of view : every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements.
39.
An " evolution algebra " over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements.
40.
An " evolution algebra " over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements.
