Sentence examples for maps over mathbb from inspiring English sources

that is a set of contraction maps over (mathbb {H}) indexed by the symbol s.

An (mathrm {ad} -invariant bilinear form on a Lie superalgebra (mathfrak {ad} -invariant{g}_0 oplus mathfrak {g}_1) over (mathbilinearis a bilinear mapping (B :, mathform {g} times mathfrak {g} rightarron mathbb {K}) with the properties 1. (mathfrak {g}_0) and (mathfrak {g}_1) are B-orthogonaLieo each other;   2.

A Lie superalgebra over (mathbb {K}) is a (mathbb {Z}_2 -graded (mathbb {K})-vector space (mathfrak {g} = mathfrak {g}_0 oplus mathfrak {g}_1) equipped with a bilinear map (:, mathfrak {g} times mathfrak {g} rightarrow mathfrak {g}) satisfying 1. ([mathfrak {g}_s, mathfrak {g}_{s^prime } ] subset mathfrak {g}_{s + s^prime },), i.e., (|[X,Y]|= |X| + |Y|) (mod 2) for homogeneous elements X, Y.   2.

In particular, if (mathfrak {t}) is a Cartan subalgebra of (mathfrak {sp}) which is defined over (mathbb {R},), then (pi vert _{exp (mathrm {i} mathfrak {t}_mathbb {R})}) is just the squaring map (t mapsto t^2).

It can be seen that k contains linear maps taking (mathbb {L}^{2M+1}) to (mathbb {L}^{2K}).

Take (R = M_2 (mathbb {Z})), the (2 times 2) matrix ring over (mathbb {Z}).

by subspaces with rank 1 over (mathbb {Z}/Nmathbb {Z}).

All algebras will be considered over (mathbb {k}).

A (mathbb {Z}_2) -grading of a vector space V over (mathbb {K} = mathbb {R}) or (mathbb {C}) is a decomposition (V = V_0oplus V_1) of V into the direct sum of two (mathbb {K} -vector spaces (V_0) and (V_1,).

Namely, (mathbb {Z}/Nmathbb {Z}) in OSS signature scheme is replaced by a quaternion algebra over (mathbb {Z}/Nmathbb {Z}).

It follows that P maps (operatorname{KAA}^(mathbb {R},mathbb {R})) into itself.