Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional.
In particular, we assume throughout the paper that and are real locally convex Hausdorff topological vector spaces, and let denote the dual space, endowed with the weak -topology By we will denote the value of the functional at, that is,.
Both (J_{operatorname {diag}}) and (J_{gamma}) take values (in[0, 4]), hence the normalization factor (frac{1}{8}) in Eq. (11) to bring the value of the functional closer to that of (J_{T}^{mathrm {sm}}).
By (langle x^{ast},xrangle), we shall denote the value of the functional (x^{ast}in X^{ast}) at (xin X); i.e., (langle x^{ast},xrangle=x^{ast}(x)).
quad (nin mathbb{N}), (1.5) where, in a similar way to above, (langlealpha_{n},v rangle) denotes the value of the functional (alpha_{n}) at (vin C[0,t_{0}]).
Let X be a Banach space, the set-valued mapping (F_{X}:Xrightarrow 2^{X^{ast}}), defined by F_{X} ( x ) = bigl{ x^{ast}in X^{ast}: bigllangle x^{ast },x bigrrangle = bigl| x^{ast} bigr| ^{2}= | x| ^{2} bigr} for (xin X), is called the dual mapping of X, where (langle x^{ast }, xrangle) denotes the value of the functional (x^{ast}in X^{ast }) on (xin X) (see [9]).
In this case, the values of the functional characteristics depend on the gap situations, and are not explicitly formulated as a function of part deviations.
In this case, the values of the functional characteristics depend on the gap situations and are not explicitly formulated with respect to part deviations.
In this process it was assumed that the values of the functional status (θ k ) formed a Normal distribution, resulting in marginal maximum likelihood estimates.
Thus the value of the optimization functional in MI registration for MR-CT substantially increases after our distortion correction.
We also note that for simplicity we denote again by T 1 the value of the linear functional T 1 in (4.4).
