The

endomorphism [[pi].sub.[??]] is a unique z [member of] B such that zy = yz = z for each y [member of] B.

By an

endomorphism f: X [right arrow] X, we mean a morphism from a projective variety X to itself.

It is not difficult to check that the

endomorphism [End.sub.k](H, [alpha]) in [??]([M.sub.k]) forms a monoidal Hom-algebra with Hom-multiplication convolution *, twisting map id and unit [eta][epsilon].

Conversely, define an

endomorphism D of A by D(x) = x, D(y) = -y, [for all]x [member of] B, y [member of] C.

We remark that if A is the usual tangent bundle of N then (1.24) - (1.26) reduce to the formulae (1.7) - (1.9) since [rho] is the Kronecker

endomorphism (1.17).

In our approach, we do not consider any possible mixing

endomorphism. Contrary to existing methods we fix a surjective transformation 9.

Recall that any

endomorphism of f of H has a matrix representation A = ([a.sub.ij]) [member of] [Z.sup.nxn], where [mathematical expression not reproducible] for i > j.

If G = G1 = G2 then the homomorphism is called an

endomorphism and the isomorphism is called an automorphism.

One can show that [[kappa].sup.-1.sub.h] is the norm of the H-orthogonal projection onto [V.sub.h] viewed as an

endomorphism on V, and therefore [[kappa].sub.h] is bounded form below for some commonly used finite element spaces [3, Lemma 6.2].

where [mathematical expression not reproducible], B(Y) = A(QY) and Q is the symmetric

endomorphism of the tangent space at a point corresponding to the Ricci tensor S

An almost Hermitian manifold ([bar.M], g, J) is a manifold endowed with an almost complex structure J, that is, compatible with the metric g, that is, an

endomorphism J: T[bar.M] [right arrow] T[bar.M] such that [J.sup.2.sub.P] = -Id for every p [epsilon] [bar.M] and g(JX, JY) = g(X,Y).