Sentence examples for imposed on the functions from inspiring English sources

The restrictions imposed on the functions γ, Q, and the right-hand side of Eq. (1) guarantees that, by virtue of (11) and (15), the kernel k 1 ( x, t ) is a kernel with weak singularity.

Under the assumptions imposed on the functions r ( x, s ) and ξ ( x, y ) in Section 1.2, from Lemma B.1 (in Appendix 2) we obtain the following statement.

Suppose the conditions imposed on the function (psi(I)) are satisfied.

In these fixed point theorems, very simple conditions are imposed on the function ϕ.

Under certain assumptions imposed on the function q, we obtain necessary conditions for the existence of nontrivial solutions to (1.1).

The following assumptions are imposed on the function f: (i) f ′ ( N t ) < 0 for N t ∈ [ 0, ∞ ) ; i.e., as the density increases, f decreases continuously.

In this paper, inspired by Sedghi et al. and Hu's work mentioned above, we prove some common fixed point theorems for ϕ-contractive mappings in fuzzy metric spaces, in which a very simple condition is imposed on the function ϕ.

M. Rassias [4] extended the problem to ∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ≤ ε ( ∥ x ∥ p + ∥ y ∥ p ) x, y ∈ E 1, for some ε ≥ 0 and some 0 ≤ p < 1. Subsequently, in 1994, P. Gavruta [5] generalized the problem to ∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ≤ ϕ ( x, y ) with certain conditions imposed on the function ϕ.

The mode of convergence of the series depends on the topology imposed on the function space.

In general, we can prove that the two scenarios above are the only possible outcomes, given the biological requirements imposed on the function G.

The lower bound of ε was imposed on the function to reduce round-off errors in floating point arithmetic and to allow for robustness against contaminating peaks when used in a Bayesian system.