Sentence examples for completeness conditions from inspiring English sources

This process also clearly establishes consistency and completeness conditions.

In [15] and [16], Li and the first author have shown that the affine Kähler-Ricci flat graph hypersurface has a rigidity property under different completeness conditions.

Using this notion, they established best proximity point theorems under some mild conditions; indeed, their hypotheses were a combination of compactness and completeness conditions.

We proved also that our proximal orbital completeness condition is weaker than the compactness condition and the proximal condition of second kind.

To satisfy the completeness condition, that all Cauchy sequences of functions converge (in the mean) to some function in that space, he specified that integration must be defined in the manner of Lebesgue.

The completeness condition requires that there are no further disturbing factors i.e., it requires an exclusive cp-clause (see section 3) Canfield and Lehrer also present an argument for why these cp-clauses cannot be defined away (any such attempt leads to an infinite regress of further cp-clauses).

We note ({|alpha,irangle }) the eigenbasis constructed from the eigenstates of (H_{mathrm{mat}}^{alpha}) that satisfied (H_{mathrm{mat}}|alpha,irangle = E^{alpha}_{i} |irangle ) and the completeness condition (sum_{i} |alpha, irangle langle alpha,i| = I_{d}^{alpha}), where (I_{d}^{alpha}) is the identity matrix acting on the subspace of the (alpha^{th}) emitter.

Applying twice the completeness condition on the Hamiltonian (H_{mathrm{mat}}^{alpha}), one finds (H_{mathrm{mat}}^{alpha} = frac {E_+E_{2}I_{d}^{alpha} + frac{hbaromega^{alpha}}{2} sigma ^{alpha}_{z} ) where (hbaromega^{alpha} = E_^{alpha} - E_^{alpha}) and (sigma_{z}^{alpha} = |alpha,+rangle langle alpha,+| - |alpha,-rangle langle alpha,-|).

(21)–(22) is expanded over shape functions ϕ i(r) satisfying the completeness condition Σ iϕ i = 1 inside a cell.

FV scheme can be obtained from DG scheme by replacing u i and ρ i in Eq. (25) with their arithmetic averages and using the completeness condition for shape functions.

In the following theorem, the chain-completeness condition for the underlying space is replaced by the chain-completeness property of the range of the considered mapping, where the range of a set-valued mapping (F: P rightarrow 2^{P}backslash{emptyset}) is defined as F(P) = bigcup bigl{ F x): x in Pbigr}.