Sentence examples for trivial direction from inspiring English sources

The less trivial direction is to assume the conditions R1 R5 hold for (langle X,rangle ;) and it turns out that the naïve choice of taking (mathcal{R }) to be ({[a,b]: a,bin X}) is a good one.

This can be explained as a kind of denoising operation like principal component analysis [27]: since data in trivial component directions usually result from noise or inaccuracy, discarding these parts should improve data purity.

The first direction is trivial.

The reverse direction is trivial since (C_{varphi}) and (T_{psi}) are invertible.

In Section 2, we show a global bifurcation phenomena from the trivial branch with rightward direction.

In this section, we state some preliminary results and show a global bifurcation phenomenon from the trivial branch with rightward direction.

Let { α n } be a sequence of nonnegative numbers converging to α ∈ [ 0, ∞ ) such that α n ≠ α for all n ∈ N. Then the following representation of the set of all common fixed points of ℱ holds F ( F ) = ⋂ n = 1 ∞ F ( T α n ). (3.9). Proof We only need to prove that ⋂ n = 1 ∞ F ( T α n ) ⊂ F ( F ) as the other direction is trivial.

The other direction is trivial.

One direction is trivial: if two trees have the same generalized Laplacian immanantal polynomial for all characters, then their generalized Laplacians have the same characteristic polynomial and hence the same spectrum.

The patterns associated to best performances are individuated and it is shown that they correspond to non-trivial alignment to wind direction.

Covering pregnancy, birth, love, and death, the stories are compact yet often novelistic in scope, deftly alighting on the seemingly trivial moments that determine the direction of relationships and lives.