Sentence examples for proofs use from inspiring English sources

The proofs use associative techniques and the Gelfand Naimark representation.

Our proofs use the Leray-Schauder nonlinear alternative and Krasnoselkii's fixed-point theorem in cones.

Our proofs use Brouwer's fixed point theorem and properties of minimum points of convex functions.

Both proofs use a reduction of the partition problem, an NP-complete problem [21].

Our proofs use partially some decomposition techniques already used in [30] where the constant exponent case was studied.

Not surprisingly, in addition to estimates of Temple Kato type, the proofs use a variant of quadratic form methods.

In classical implicational logic, each ϕn have normal proofs using at most one assumption occurrence.

Here we give new and complete proofs using the methods of double operator integrals.

Hilbert there also presented his approach to giving consistency proofs using his so-called ε-substitution method.

Stability proofs using Lyapunov theory and passivity theory as well as simulations demonstrate asymptotic stability, eliminating oscillatory or divergent behaviors.

It is to be noted that almost all proofs used the idea imaginated by Hyers.