Sentence examples for mapping is quadratic from inspiring English sources

Thus the mapping is quadratic.

Therefore, the mapping is quadratic.

By Lemma 2.1, the mapping is quadratic.

Since the mapping is quadratic (see [54, Lemma ]), we get that the mapping is quadratic.

By [45, Lemma 2.1], the mapping,  :  is quadratic.

By Proposition 2.1 (by letting ), the mapping is quadratic.

It is proven in [8]; that for the vector spaces and and the fixed positive integer, the map is quadratic if and only if the following equality holds: (2.1).

Conversely, if a mapping f is quadratic, then it is obvious that f satisfies equation (2).

Conversely, if a mapping f is quadratic, then it is obvious that f satisfies (4).

δ is a quadratic mapping, δ is quadratic homogeneous, that is, δ ( λ a ) = λ 2 δ ( a ) for all a ∈ A and all, δ ( a b ) = δ ( a ) b 2 + a 2 δ ( b ) for all a, b ∈ A, δ ( a ∗ ) = δ ( a ) ∗ for all a ∈ A. Example 2.2 Let A be a commutative ∗-normed algebra.

The mapping Q is quadratic because it satisfies equation (1.2) as follows: ∥ D ˜ 1 Q ( x, y ) ∥ β = lim n → ∞ 1 | k | 2 n β ∥ D ˜ 1 f ( k n x, k n y ) ∥ β ≤ lim n → ∞ 1 | k | 2 n β φ ( k n x, k n y ) = 0. for all x, y ∈ X ; therefore, by Lemma 2.2, it is quadratic.