Sentence examples for proofs we use from inspiring English sources

To establish the proofs, we use several computer-aided searches.

For the proofs, we use the upper and lower solution method.

In particular, for achievability, we use stochastic encoding [2] in conjunction with superposition coding [24]; and for the converse proofs, we use outer bounding techniques in [1, 2], more specifically, the Csiszar-Korner identity, [2, Lemma 7].

In the proof, we use Krein's theory of the spectral shift function.

For the proof we use new tools of stochastic analysis, in particular fractional Sobolev spaces in Malliavin calculus and martingale methods.

In our NP-hardness proof we use a very general instance which makes the proof applicable to a large set of special cases of the RDAP, including several important scheduling problems whose complexity was unresolved heretofore.

In the proof we use oscillatory integrals, the Cotlar Stein almost orthogonality theorem, a sort of Littlewood Paley decomposition for a certain operator, some basic facts about Fourier integral operators and pseudodifferential operators.

Proof We use mathematical induction.

Proof We use the method of [18].

Proof We use the idea from [35].

Proof We use mathematical induction to prove inequality (2.1).