Sentence examples for example we prove from inspiring English sources

For example, we prove (12).

As an example, we prove the following.

As an example, we prove that the standard Podleś sphere is equivariantly Poincaré dual to itself.

As an example, we prove a closed high-order form of the classical Jacobiʼs formula.

For example, we prove the equivalence between the chromatic number of S- and G-IDNC graphs.  .

As an example, we prove this for the following result from [2].

As examples, we prove the reduced group C⁎-algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.

[14] In the following examples, we prove that label-free MST is a valuable tool to study the binding of numerous types of ligands to different protein classes.

Additionally, in specific examples, we prove whether the non-informative margins condition is satisfied or not, and in a simulation study we demonstrate the consequences when the conditional models are compatible but do not satisfy the non-informative margins condition.

Let E = [0, 1/ 4) ∪ [1/ 2, 3/ 4) and let J ( x, y ) = d ( x, y ) if { x, y } ∩ E = { x, y }, 2 if { x, y } ∩ E ≠ { x, y }. By Example 12.1, J is a J-generalized pseudodistance. Using analogous considerations as in Example 12.3, we prove that T and J satisfy conditions (C1' - C4' - C4e above implies that Theorem 12.3 holds.

As an application (see Example 5.5), we prove the asymptotic stability of the nonlinear Matukuma equation from astrophysics [8, 9].