Sentence examples for a assume that from inspiring English sources

(b) ⇒ (a) Assume that G is S-orbitally lower semicontinuous at (x_{0}) and H is T-orbitally lower semicontinuous at (y_{0}).

(a) Assume that add C = add C ′. Then f is right C -determined if and only if f is right C ′ -determined.

(a) Assume that X ( t ) are DRHR for all t and X ( 0 ) = 0. Further, assume that a 1 and a 2 are increasing and convex functions, with a 1 ( 0 ) = a 2 ( 0 ) = 0 and Y is IFR.

In order to make a bifurcation with a nontrivial, one parameter family of solutions, we need (iii) in the following assumptions: (A) Assume that: (i) There exist C and α such that begin{aligned} bigl|V_{i}(x bigr|leq Clangle xrangle ^{-alpha},quad i=1,2, forall xinmathbb{R}^{2}.

Theorem A Assume that Φ is even and ( Φ 0 )-( Φ 2 ) are satisfied.

Proof (a) Assume that the conclusion of (a) is not true.

Let (A D(A)subset Xrightarrow X) be an unbounded self-adjoint and invertible operator satisfying (sigma(A)=sigma_{d}(A)). Assume that Y is a Banach space with the norm (Vert cdot Vert _{Y}) satisfying (D(A)subset Ysubset X), the inclusion map from (D(A)) to Y is compact and the inclusion from Y to X is continuous.

For two positive real numbers a and b, assume that there exists a constant number 0 ≤ η < 1 such that 0 < b ≤ η a. Assume that z ( t ) is a nonnegative continuous function on [ t 0 − τ, t 0 ] and satisfies the following inequality: D + z ( t ) ≤ − a z ( t ) + b ∥ z t ∥, for t > 0, (6).

Then there is a ∈ L such that d ( x ) = x ∧ a. Assume that b ∈ L and x ∈ d − 1 ( b ).

Let C be nonempty closed and convex subset of E. Let A : C → E ∗ a single valued, monotone and hemicontinuous operator with VI ( C, A ). Assume that C has the normal structure.

We have a right to assume that Prohibition will stay..