Sentence examples for commuting diagram from inspiring English sources

Consider the following commuting diagram: (4.4).

We have the commuting diagram property div ∘ Π h = P h ∘ div : V → W h and div ( I − Π h ) V ⊥ W h, (2.30).

The commuting diagram means that the homeomorphism h takes each point x ∈ A F with address ω : = τ F ( x ) to the point y ∈ A G with the same address ω = τ G ( y ).

We have the commuting diagram property operatorname{div}circPi_{h}=R_{h}circ operatorname{div}:mathbf{V} rightarrow W_{h} quad text{and}quad operatorname{div} I-Pi _{h})mathbf{V}perp W_{h}, (2.56) where I denoperatorname{div} I-Piator.

Furthermore, we present a commuting diagram of intermediate analysis steps, which relates different strategies and enables the reuse of soundness proofs between them.

(ii Conversely, for any homomorphism of Lie algebras, there exists a unique homomorphism of Lie groups, making the above diagram commuting.

If you have an early printing of the book where the next-to-last commutative diagram on this page is a small diagram consisting of two short exact sequences joined by vertical maps alpha, beta, and gamma, then add the hypothesis that these maps are chain maps, commuting with boundary homomorphisms.

That is, the following diagram is commuting: (3.31).

Then the automorphism induces a surjection on the nilmanifold and the following diagram is commuting: (3.5).

Moreover, the automorphism on induces a surjection so that the following diagram is commuting: (4.3).

The homomorphism associated with induces a homomorphism and in turn induces a homomorphism so that the following diagram is commuting: (4.2).