Sentence examples for as proved in section from inspiring English sources

However [19] is missing the feedback scheme needed, as proved in Section 4, against parameter changes; in [9] the efficiency levels are lower than those achieved by our scheme.

As proved in Section "Problem statement", energy-aware bandwidth allocation and routing is an NP-hard problem with a high complexity due to the exponential number of nodes and routes.

Concerning the positioning of sink nodes, a lot of research papers already exist as proven in Section 2. As depicted in Figure9, we assume that the sinks nodes are placed at the middle of each side of the grid structure.

As proven in Sections 6.2 and 6.3, this contributes to the higher accuracy of identification for each individual's handwriting.

More precisely, the main result of this paper can be stated as follows, which will be proved in Section 3 later.

The above theorem will be proved in Section 3. As is well known that degenerate and singular equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

Since these sources have a maximal PLF with each other and the mixture components do not (as motivated in Section 2.4 above and proved in Section 3.3 below), we can unmix them by searching for projections that maximize the resulting PLFs.

This paper is organized as follows: Theorem 1.1 and Theorem 1.2 are proved in Section 2.

In this section, we study the existence of unique solution of nonlinear integral equations, as an application of the fixed point theorem proved in Section 3.

The results are proved in Section 3.

The stability of endemic equilibrium is proved in Section 4.