Sentence examples for which follow from from inspiring English sources

We need the following inequalities which follow from (2): (17).

Proof To prove the inequalities 0 ≤ NH X ≤ 2, it suffices to show that 0 ≤ NH X (x) ≤ 2 hold for each x ∈ S X, which follow from the following observation: for each number α > 0, ∥ x + α y ∥ - ∥ x - α y ∥ α ≤ ∥ ∣ x + α y ∥ - ∥ x - α y ∥ ∣ α ≤ min { 2, 2 α } α ≤ 2. By Theorem 3, it is clear that if NH X = 0 then X is a Hilbert space.

The algorithm has the following steps (see Fig. 4), which follow from Lemma 2.1.

Our thinking - and Numenta's software - is based on forecasts, which follow from memories.

Fig. 2a illustrates our hypothesis tests, which follow from the retinotopic coding in mIPS and SPOC.

These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.

We begin with the following lemma, which follows from Lemma 2.10 immediately.

The following result which follows from slight modification of Chowdhury and Tan [6, Lemma 3]: Lemma 2.3.

In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem 2.4, Lemma 2.5, and (2.2).

The following identity, which follows from (2.5) when we replace q by q 2 and z by q α, α > 0, is essential in our investigations.

We apply the following existence principle which follows from [11 13] to prove the solvability of problem (1.12), (1.2).