Sentence examples for for any functions from inspiring English sources

for any functions f and φ = 0.

Conversely, for any functions defined in which satisfy (i)–(iii) in above and is compact, there exists a unique such that,.

Lemma 3.7 Let S i, j be the multiplier operators defined in (2.24) for any i, j ∈ Z. Then (i) for each fixed 1 < p < 2 and for any functions { g i, j ; s, t ; κ, ℓ } (3.14).

Since the results stated above show that it is important for any functions of three variables to find the appropriate superpositions which can approximate most efficiently to the original function.

It is worthwhile to mention that if F and G are solutions to (8), then F + F 1 ( k, l ) and G + G 1 ( k, l ) are also solutions for any functions F 1 and G 1. One can see from (8) that F and G take different arguments.

Then, with a non-vanishing q ∈ Q, for any functions f ∈ W l − 2 m, p ( R + n ; E ) and g j ∈ B p, p l − m j − 1 p ( R n − 1 ; E ) there is a unique solution u ∈ W l, p ( R + n ; E ) of problem (28 - 29).

These formulas are true for any function.

Another matter of concern was the abuse of the library for any function the council saw fit.

Because by then I had learned, rubbing polish into an altar, that active citizenship is essential for any functioning democracy.

So if anyone fancies a loud covers band for any function have them give me a call".

The Leibniz rule is true for any function, continuous or not.