Sentence examples for proofs using from inspiring English sources

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.

This book teaches students how to read and write proofs using Athena, a freely downloadable computer language.

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

In particular we detail the fact that they preserve proofs in equational logic, as well as proofs in other logical systems like propositional logic with modus ponens or proofs using resolution rule.

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

The proofs use associative techniques and the Gelfand Naimark representation.

The proofs use an interesting combination of combinatorial, algebraic and analytic tools.

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

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

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