The phrase "assumption that the functions" is correct and usable in written English.
It can be used when discussing a belief or premise regarding the nature or behavior of certain functions, often in mathematical or theoretical contexts.
Example: "The assumption that the functions are continuous is crucial for the validity of the theorem."
Alternatives: "belief that the functions" or "premise that the functions".
For functions V→R2, where V is a bounded domain in 3D space, recently a classification of singular fibers and their topological changes has been obtained in [15], under the assumption that the functions are stable.
Most physical settings lead to the default assumption that the functions defined for are nonnegative and continuous, and that the initial data, are nonnegative, which are mathematically convenient and currently followed throughout this paper.
Note that, with the assumption that the functions from (mathcal{F}_{1}), (mathcal{F}_{2}), (mathcal{F}_{3}) are differentiable, the results from Theorem 14, Corollary 1, and Corollary 2 still hold when two of the points (z_{0},z_{1},ldots,z_{2l}in [ alpha,beta ]) coincide.
The only work we can find about the uniqueness of solutions to VEPs is [3], in which Khanh and Tung established sufficient conditions for the local uniqueness of solutions to VEPs by using approximations as generalized derivatives under the assumption that the functions have first and second Fréchet derivatives.
Previously, prediction of protein functions was typically operated with the assumption that the functions of labeled proteins are complete; that is, there are no missing labels [ 7, 8].
Automated protein function prediction is typically implemented under the assumption that the functions of labeled proteins are complete; that is, there are no missing labels.
Most of known results concerning existence of iterative roots for a continuous function were given under the assumption that the function has finitely many non-monotonic points.
The following theorem provides the existence of a homoclinic solution under the assumption that the function in (1.1) has a linear behaviour near.
In fact, only under the assumption that the function (theta_{2}(cdot)) is strongly convex, Xu [17] proved that the new method has (mathcal{O}(1/t)) convergence rate with constant parameters and enjoys (mathcal {O}(1/t^{2})) convergence rate with adaptive parameters.
In [4], applying Galerkin's method, Antontsev and Shmarev obtained the existence and uniqueness of weak solutions with the assumption that the function a ( u ) in div ( a ( u ) | ∇ u | p ( x ) − 2 ∇ u ) was bounded.
(2) The authors [26, 28] obtained fixed point theorems for multivalued maps T defined on cone metric spaces (X, d) under assumption that the function I x) = infx ∈ Tx||d x,y)|| is lower semicontinuous, and the author [27] obtained a fixed point theorem for multivalued maps T under assumptions that the function I x), x ∈ X is lower semicontinuous and a dynamic process is given.
