Sentence examples for 1 nullity from inspiring English sources

Both consider the set of active fluxes, Z ¯ v = i : v i ≠ 0, 1 ≤ i ≤ R. For example, the rank test states that the flux vector v :  Sv = 0,  v ≥ 0 is elementary, if (1) nullity S *, Z ¯ v = Z ¯ v − rank S *, Z ¯ v = 1 Alternatively, we can consider the set of inactive fluxes Z v) = { i :  v i  = 0, 1 ≤  i ≤  R}.

where the relevance function ρ should satisfy the following properties: – Non-negativity: i.e., ρ ( X, C, x ij ) ≥ 0. – Nullity: the function ρ ( X, C, x ij ) is null if the feature x ij has not relevance at all.

– Non-redundancy: if x ij ′ = α x ij + ς, where the real valued α≠0 and ς is some noise with mean zero and unit variance, then | ρ ( X, C, x ij ) − ρ ( X, C, x ij ′ ) | → 0. – Non-negativity: i.e., ρ ( X, C, x ij ) ≥ 0. – Nullity: the function ρ ( X, C, x ij ) is null if the feature x ij has not relevance at all.

If x 0 ∈ K and Φ ″ ( x 0 ) is a Fredholm operator and the Morse index μ 0 and nullity v 0 of x 0 are finite, then we have.

Let 0 be an isolated critical point of Φ ∈ C 2 ( E, R ) with a finite Morse index m 0 and nullity n 0. Assume that Φ has a local linking at 0 with respect to a direct sum decomposition E = E − ⊕ E +, κ = dim E −, i.e., there exists r > 0 small such that Φ ( z ) > 0 for z ∈ E +, 0 < ∥ z ∥ ⩽ r, Φ ( z ) ⩽ 0 for z ∈ E −, ∥ z ∥ ⩽ r.

Let 0 be an isolated critical point of J ∈ C 2 ( E, R ) with finite Morse index μ ( 0 ) and nullity ν ( 0 ).

Suppose that u 0 is an isolated critical point of J ∈ C 2 ( E, R ) with finite Morse index μ ( u 0 ) and nullity ν ( u 0 ).

It follows from ( f 0 ) that u = 0 is a degenerate critical point of J with Morse index μ ( 0 ) = ω m − 1 and nullity ν ( 0 ).

Let 0 be an isolated critical point of J ∈ C 2 ( H, R ) with the Morse index μ 0 and the nullity ν 0. Assume that J has a local linking at 0 with respect to H = H − ⊕ H +, l = dim H − < ∞, i.e., there exists ρ > 0 such that J ( u ) ≤ 0, for  u ∈ H −, ∥ u ∥ ≤ ρ ; J ( u ) ≥ 0, for  u ∈ H +, ∥ u ∥ ≤ ρ.

Note that is the unique critical point of with Morse index (see Remark 3.1) and nullity.

Figure 4 displays the expectations of those outputs at the seventh iteration, while Figure 5 shows the expectations of tracking errors, which convey that expectations of tracking errors with respect to the proposed NILC scheme (4) with (2) and (3) converge to nullity very well.