Sentence examples for a simpler proof for from inspiring English sources

In 1972, Hedberg [9] gave a simpler proof for all α.

A simpler proof for the special case of 2-players and 2-channels is given in [32].

Here, we present a simple proof for this fact.

Next, we give a simple proof for Theorem 1.3.

We also formulate a simple proof for the averaging formula for the Lefschetz coincidence number.

Here we give a simple proof for the reader's convenience.

In Theorem 3.1, we give a simple proof for this case.

As an application, we give a simple proof for the denseness of the Schwartz class in such spaces.

Fixed-point theory has attracted many mathematicians since it provides a simple proof for the existence and uniqueness of the solutions to various mathematical models (integral and partial differential equations, variational inequalities etc).

In case of (r=2), (gamma_{m}(langle xrangle )=sum_{k=1}^{m-1} frac{1}{k m-k }G_{k}( langle x rangle )G_{m-k}( langle x rangle )), and hence our problem here is a natural extension of the previous work, which leads to a simple proof for the important Faber-Pandharipande-Zagier and Miki identities (see [16, 22]).

See also [7] for a simpler proof of this result.