Sentence examples for proof noting from inspiring English sources

Proof Noting the continuity of f, this follows in a standard step-by-step process and so is omitted.

Proof Noting that I − h is a ( 1 + L ) -Lipschitzian and ( 1 − α ) -strongly monotone mapping, so the variational inequality problem (3.4) has a unique solution, which is denoted by x ∗.

Proof Noting the compatible condition and that F ( t, z ) is an analytic function with respect to z, using the Pompeiu formula [12], it is not difficult to verify by direct calculation that u ( t, z ) expressed by (31) is the general solution of the system (30).

Proof Noting that 〈 λ T x + C x + μ J φ x, x 〉 ≥ μ 〈 J φ x, x 〉 = μ ∥ x ∥ γ + 1 ≥ 0. for every x ∈ L and μ > 0, it is clear that hypothesis (t2) in Theorem 3.1 is satisfied.

Proof Noting that for GED ( 1 ), i.e., the Laplace distribution with pdf f 1 ( x ) = 2 − 1 / 2 exp ( − 2 1 / 2 | x | ), we have lim n → ∞ F 1 n ( a n x + b n ) = Λ ( x ). with normalizing constants a n = 2 − 1 / 2 and b n = 2 − 1 / 2 ( log n − log 2 ).

Proof Noting B is full column rank, a p × p constant nonsingular matrix Z can be found such that Z B = [ I m 0 ] ′.

For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([1 6]) where further references are given.

For valid proofs of NE, BF, and CBF can still be generated in SQML from the proofs noted by simply replacing free occurrences of the variable 'x' with occurrences of a constant 'c'c

Proof Note that E ↪ L 1.

Proof Note that γ ≤ α, β.

Proof Note that γ satisfies the relation 0 < γ < 1.