If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally.
If, additionally, the signal space consists only of finite-length signals, all samples long, i.e., or, then every linear filter may be called a linear transformation, which is representable by a constant matrix.
If additionally A∗A dμ⩽I and B∗B dμ⩽I, then I−A∗A dμ XI−B∗B dμ∈|||·|||(H) andI−A∗A dμI−B∗B dμ⩽AB .Applications include Young's and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions.
If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms of the derivative of F. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals.
If additionally we assume that.
If additionally a consequence relation has the substitution property (7), then it is called structural.
If additionally (f3) holds, then the problem (1.1) possesses infinitely many negative nontrivial solutions.
If additionally H ∈ ( 3 4, 1 ), then the process z 1 has an L ∞ ( 0, T ; V 1 ) version.
If, additionally, (p_{3}(1)ne 0), that is, (a+cne1), then none of them is equal to 1.
