Sentence examples for i assumptions from inspiring English sources

Ahmadet al.[15] formulated a unified higher-order dual of (P) and established appropriateduality theorems under higher-order ( F, α, ρ, d ) -type I assumptions.

In this paper, we formulate a higher-order dual for a non-differentiable minimaxfractional programming problem and establish weak, strong and strict converseduality theorems under generalized higher-order ( F, α, ρ, d ) -type I assumptions.

Ahmad, Husain and Sharma [16] formulated a unified higher-order dual of (P2) and established weak, strong and strict converse duality theorems under higher-order ( F, α, ρ, d ) -Type I assumptions.

Part I: Assumptions on the setting of C, F, A, and T: (A1) C is a nonempty closed convex subset of a real Hilbert space H; (A2) F : C × C → R is a bifunction satisfying the conditions (C1 - C4); (A3) A : C → H is an α-inverse-strongly monotone mapping; (A4) T : C → C is a nonexpansive mapping.

(F – I ) Assumptions and replacement rules of the computational model.

In the next sections we will present an alternative computational method which uses all available Chip-seq data and estimates the specificity of Chip-seq assay without (i) assumptions regarding physical distribution non-specific Chip-seq DNA clusters (noise BEs) and (ii) the need for a validation of the results in ChIP-q-PCR assay.

Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.

(i) Assumption 3.1 and the definition of (x_{n}) ((ninmathbb {N})) ensure the boundedness of ((x_{n})_{ninmathbb{N}}).

The proposed solution focuses on two types of assumption: (i) application assumptions; and (ii) environmental assumptions.

Assumption II follows from the standard coloring argument, we only need to prove Assumption I and Assumption III.

end{cases} Simply, assumption (i) and assumption (ii) in Theorem 2.5 are satisfied thanks to Lemma 3.2 and Lemma 3.3.