Sentence examples for assumption that the function from inspiring English sources

The following theorem provides the existence of a homoclinic solution under the assumption that the function in (1.1) has a linear behaviour near.

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.

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.

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.

Remark 3.1 In [1], the Grünwald-Letnikov fractional derivative, under the assumption that the function x ( t ) must be m + 1 times continuously differentiable, can be obtained from (4) under the same assumption by performing repeatedly integration by parts and differentiation.

Despite the stochastic disturbances the awareness share is bounded between 0 and 1 with the assumption that the function ( sigma :(0,1) to {mathbb{R}} ) is continuous and Lipschitz on every subinterval of (0,1), and ( aleft( t right), a_{M} left( t right) ge 0 ) and ( sigma left( 0 right) = sigma left( 1 right) = 0 ).

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.

Previous studies typically predict the functions of proteins under the assumption that the functions of labeled proteins are complete; that is, there are no missing labels.