Sentence examples for functional is defined on from inspiring English sources

The energy functional is defined on Sobolev spaces of order −1, so the subdifferential mathematically formulates the energyʼs gradient which formally involves 4th order spacial derivatives of the surfaceʼs height.

Let the nonnegative continuous convex functionals, be defined on the cone as Theorem 3.1 and the nonnegative continuous concave functional be defined on the cone by (3.15).

Let the nonnegative continuous convex functionals, and the nonnegative continuous concave functional be defined on the cone by (3.3).

Let the nonnegative continuous concave functional, the nonnegative continuous convex functionals,, and nonnegative continuous functional be defined on cone by (4.1).

Let the nonnegative continuous concave functional, the nonnegative continuous convex functionals and, and the nonnegative continuous functionals be defined on the cone by (41).

Additionally, the objective functional is defined also on the design boundaries.

The functional is defined based on the hypothesis that the activity of the units reflects only two simple constraints: The minimization of the variability of the maps across space, that is, a preference for smooth maps.

Associated with, the functional is defined as.

Let D 0 1, p be the completion of C 0 ∞ with respect to the norm ( ∫ Ω ∣ ∇ u ∣ p d x ) 1 ∕ p. The energy functional of (1.1) is defined on D 0 1, p by J λ ( u ) = 1 p ∫ Ω ∣ ∇ u ∣ p - μ ∣ u ∣ p ∣ x ∣ p d x - λ q ∫ Ω f ∣ u ∣ q d x - 1 p * ∫ Ω g ∣ u ∣ p * d x.

First the energy functional associated to (1.1) is defined on W 1, p ( R N ), which is not a Hilbert space for p ≠ 2. Another difficulty is the lack of a powerful regularity theory.

Let the nonnegative continuous concave functional φ, the nonnegative continuous convex functionals γ, θ and the nonnegative continuous functional ψ be defined on the cone by γ ( u ) = max 0 ≤ t ≤ 1 | D 0 + β u ( t ) |, θ ( u ) = ψ ( u ) = max 0 ≤ t ≤ 1 | u ( t ) |, φ ( u ) = min 1 2 ≤ t ≤ 1 | u ( t ) |.