For 1/(1 - v) < p [less than or equal to] [infinity], the operator [mathematical expression not reproducible] is compact, and since by the assumption (A3), the nullspace
of I - T is trivial in C[0,1] implying that the null space of I - [??] is trivial in [??](R), the bounded inverse [mathematical expression not reproducible] exists.
The kernel is simply the collection of k-simplices which form the nullspace
of the matrix which correspond to cycles (note that this agrees with the notion of graph-theoretic cycles).
Where [[omega].sub.k] presents the precoding unit-norm beamforming vector for user k is chosen in the direction of the projection of [h.sub.k] on the nullspace
of [h.sub.j], j[not equal to]k.
We write N(T) and R(T) for the nullspace
and range of an operator T [member of] B(H).
Thus for each eigenvalue pair ([gamma], 1/[gamma]) of [C.sup.T] and -K*, respectively, the coordinates of the eigenvectors of [C.sup.T] associated to [gamma] determine which linear combinations of the coordinate functions of y must satisfy a Fredholm-like orthogonality condition with elements of the nullspace
of the resolvent 1/[gamma] + K*.
By [sigma](A), R(A), D(A), and N we denote the spectrum, range, domain, and nullspace
of A, respectively.
In general, jamming signals are deliberately designed in nullspace
of the legitimate channel.
Wang, "A nullspace
based L1 minimizing Kalman filter approach to sparse CS reconstruction," in Proceedings of the 11th European Conference on Synthetic Aperture and Radar, June 2016.
(ii) from Corollary 12, [D.sup.-1]p is in the nullspace
of the Laplacian Lap and [D.sup.-1] d = j;
Observe that the columns of the matrix U associated with null eigenvalues form a basis of the nullspace
of A and that means that each individual (each column of A), if expressed in the base formed by the columns of U, will have at most [r.sub.A] nonnull coefficients; that is, we will not find more than [r.sub.A] independent elements in the Pareto front.
Because H and E have the same continuous condition on the tangential components, H will not be compatible with the nullspace
of the curl operator if H and E also have the same order.
The first Q columns and the last N - Q columns of V span the row space and nullspace
of H, respectively.