Sentence examples for provided there exist from inspiring English sources

provided there exist non-negative functions p +, p − ∈ L with norms ∥ p + ∥ L = T +, ∥ p − ∥ L = T − (12).

This will be so provided there exist two circumstances, in connection with problems considered here, which ensure achieving the provisos mentioned.

We give a new construction for (n×n)/k semi-Latin squares where n is a prime power, k is any value in the range n⩽k⩽(α+1)(n−1) provided there exist α integers ri such that 1<r1<r2<⋯<rα<n−1 where ri and n−1 are relatively prime.

Let us also recall that the space X is an absolute neighborhood retract ((X in operatorname {ANR})), provided there exist an open set U in a normed space E and two maps: r colon U to X quadmbox{and}quad s colon X to U such that (r circ s = mathrm {id}_{X}); if U is an arbitrary convex set, then X is called an absolute retract ((Xin operatorname {AR})).

A multi-valued mapping φ: X ⊸ Y is called admissible provided there exists a selected pair (p, q) of φ.

is said to be (delta) differentiable at provided there exists a constant such that for any, there is a neighborhood of (i.e., for some ) with (23).

If (n=3), then there are extremals for (mu _{s,gamma, lambda }(Omega )) provided there exists (x_0) in (Omega {setminus }{0}) such that (R_{gamma,-lambda }(x_0)>0).

(2) If (n=3), then there are extremals for (mu _{s,gamma, lambda }(Omega )) provided there exists (x_0) in (Omega {setminus }{0}) such that (R_{gamma,-lambda }(x_0)>0).  .

We recall that a Banach -algebra is said to be an -algebra provided there exists on a second norm, not necessarily complete, which is a -norm.

This spectral representation is valid provided there exists M > 0 such that ∥∑i = 1NPi ∥ ⩽ M, N = 1, 2, …, and generalizes results that apply to self-adjoint, normal, and spectral operators.

We say that a module M is present in C [ → Y 〉 provided there exists a right minimal map f : M → Y which is right C -determined; similarly, we say that M is present in C [ → Y 〉 t provided there exists a right minimal map f : M → Y which is right C -determined and | f | C = t.