We say that { S i } i = 1 m satisfies the open set condition (OSC) if there exists a nonempty bounded open set U ⊆ R d such that S i ( U ) ⊆ U and S i ( U ) ∩ S j ( U ) = ∅ for i ≠ j.
When the matrices ((boldsymbol{varPhi}^{i}_{k})), (kinmathbb{I}^{i}), (i = 1,ldots, M), satisfy the rank restricted isometry property [24], the trace norm relaxation converges to the true solution of (32), provided that the number of measurements is of order (mathcal{O} (operatorname{dim}(mathcal{H}_{1} log (operatorname{dim }( mathcal{H}_{1} ) ) )) [24].
Definition 3.4 We say that the condition E 2 is satisfied if there exists a constant C E such that for every z > 0 the system { D i, j ( z ) } i, j = 1 m satisfies the ellipticity condition.
Definition 3.9 We will say that the condition E q ′ is satisfied if there exists a constant C E > 0 and for every z > 0 such that the system { D i, l j ( z ) } i, l = 1 m satisfies the ellipticity condition for j = 1, 2, …, m.
Definition 3.6 We will say that the condition E p is satisfied if there exists a constant C E > 0 such that for every z > 0 the system { D i, l j ( z ) } i, l = 1 m satisfies the ellipticity condition for j = 1, 2, …, m.
Corollary 5.3 Let f and I n be as in the Lemma 5.2, then ∫ a b f ( x ) d x = A M ( f, I n ) + R M ( f, I n ), where A M denotes the mid point quadrature rule i.e., A M ( f, I n ) = ∑ i = 0 n − 1 f ( x i + x i + 1 2 ) h i. and the remainder R M satisfies the estimation | R M ( f, I n ) | ≤ 1 2 ∥ f ′ ∥ p ( q + 1 ) 1 q ( ∑ i = 0 n − 1 h i q + 1 ) 1 q.
It is clear that the sequence of operators {I − K m } m satisfies lim m → ∞ ( I - K m ) f = 0 for each f ∈ H (U n ), and the space H(U n ) endowed with the compact open topology τ is a Fréchet space.
Step 4. If x i n + 1 and u i j n + 1 for i, j = 1, 2, …, m satisfy (3.2) to sufficient accuracy, stop.
The model M satisfies ψ if S 0 ⊆ S ψ, otherwise, the model checker will output a counterexample (i.e., a sequence of transitions which starts from a state in S 0 ) that falsifies the formula ψ .
