Sentence examples for by these two theorems from inspiring English sources

Motivated by these two theorems we have the following for stationary points.

By combining these two theorems, we can derive the following result dealing with bi-inductive properties of the set of fixed points.

These two theorems can be proved by Lemma 1.1 and Lemma 1.2 following the same pattern as in [1].

To understand Skolem's Paradox, we need to start by recalling two theorems from classical logic.[1] The first comes from the late 19th century.

This is demonstrated by two theorems which provide constructions for BCAs from suitable distributive lattices.

This result is justified by the next two theorems.

Our approach is supported by two theorems showing how to decide whether a new component should be added to the ensemble or not, based on the assumption that such an action should increase the accuracy of the ensemble not only for the current portion of observations but also for the whole (infinite) data stream.

The correspondence expressed by the previous two theorems between intuitionistic validity and typability is known as the Curry-Howard-de Bruijn correspondence, after three logicians who noticed it independently.

The main results are given by two theorems in Section 3: one is the existence of eigenvalues and eigenfunctions for an arbitrarily dimensional case; the other is a more concrete conclusion for a one-dimensional case.

The main results of the paper are given by the following two theorems: partially convex function theorem (PCF-Theorem) and weighted partially convex function theorem (WPCF-Theorem).

This theorem does not give any information if, This case is covered by two theorems: if an entire holomorphic function has order then (Wiman [1]) and if is an entire holomorphic function of order and is a number satisfying the conditions then there exists a sequence of circular arcs along which tends to uniformly with respect to (Arima [2]).